\not\hskip3.0pt\mid
\not\hskip3.0pt\mid
We find a different method to compute the generating series in Shimura's conjecture for $Sp_3$, proved by Andrianov in 1967. Formulas %by A. N. Andrianov for the Satake spherical map for $\Sp_3$ are used.
spinor-v8.tex
%% Created by Maple 9.50 (IBM INTEL NT)
%% Source Worksheet: spinor8.mws
%% Generated: Sat Apr 15 18:50:46 2006
%\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[a4paper, 14pt, noxcolor
,draft]
{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Output}
\DefineCharStyle{2D Math}
\DefineCharStyle{2D Output}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\documentclass[fancybox]{seminar}

%\documentstyle[fancybox,article]{seminar} 
%%%%%%%%%%%%%%%%%%%%%%%%
%\documentclass[fancybox,article]{seminar} 
\usepackage[a4paper]{hyperref}

%\def\printlandscape{\special{landscape}}    % Works with dvips.

%\articlemag{1}
%\twoup   % Try me.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\slideframe{none}
%\slideframewidth=0.5pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\slideframe{none}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\slideframe{plain}
\newcommand{\BS}{}
\newcommand{\ES}{}
\newcommand{\EB}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\pagestyle{empty}
%\textheight=200pt
%\textwidth=280pt


\addtolength{\textwidth}{2cm}
%\addtolength{\textwidth}{4cm}
\addtolength{\oddsidemargin}{-1.9cm}
\addtolength{\evensidemargin}{-1.9cm}
%\addtolength{\textheight}{1cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\addtolength{\topmargin}{-1cm}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\renewcommand{\slideleftmargin }{.2in}
%\renewcommand{\sliderightmargin}{.2in}
%\renewcommand{\slidetopmargin}{.2in}  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\addtolength{\textwidth}{3cm}
%\addtolength{\oddsidemargin}{-1cm}
%\addtolength{\evensidemargin}{-1.9cm}
%\usepackage[a4paper]{hyperref}
%\def\printlandscape{\special{landscape}}    % Works with dvips.
%\articlemag{1}
%\textheight=200pt
%\textwidth=300pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\usepackage[francais]{babel}
%\usepackage[french]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{fancybox}
\usepackage{xy}
\usepackage{epsfig}
%%%%%%%%%%%%%%%%
\usepackage[T1]{fontenc} 
\usepackage[latin1]{inputenc}
\usepackage{amsfonts,amssymb,amsmath}
\usepackage{graphics} 
%\usepackage{pstricks} 
\usepackage{multido, xcolor} 
\newcommand{\red}{}
\newcommand{\black}{}
\newcommand{\magenta}{}
\newcommand{\blue}{}
\newcommand{\purple}{}
\newcommand{\orange}{}
\newcommand{\violet}{}
\newcommand{\brown}{}
%\usepackage{showkeys} 
%\input{fabi}
%\newfam\persofam
%\font\tenperso=msam10
%\font\sevenperso=msam7
%\font\fiveperso=msam5
%\textfont\persofam=\tenperso
%\scriptfont\persofam=\sevenperso
%\scriptscriptfont\persofam=\fiveperso
%\mathchardef\geqslant="393E
%\mathchardef\leqslant="3936
%\usepackage{showkeys} 
%\usepackage{multido, color} 
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\documentclass[draft, definitions, 12pt]{article}
%\textheight=200pt
%\textwidth=300pt

%\input eight.tex
%\usepackage{amssymb}
%\usepackage{amsfonts}
%\usepackage{isolatin1}
%
%\usepackage{amsmath}
%\usepackage{hyperref}
%\usepackage[latin1]{inputenc}
%\usepackage[T1]{fontenc}

%\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{tpc-e.tex}

%\documentclass[12pt,definitions,draft]{article}
%\usepackage[latin1]{inputenc}
%\usepackage[T1]{fontenc}
%\usepackage{amssymb,amsfonts}




%\renewcommand{\theequation}{\thesubsection.\arabic{equation}}
%\renewcommand{\thetheoc}{\thesubsection.\arabic{theoc}}


%\documentclass[a4paper,12pt]{article}

%\hsize=15cm
%\vsize=10cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newtheorem{theo}{\sc Theorem}[section]



%\newcommand{\theo}
\renewcommand{\thesection}{\arabic{section}}
%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thesubsection}{\thesection.\arabic{subsection}}
\renewcommand{\theequation}{\arabic{equation}}
\renewcommand{\thetheo}{\thesection.\arabic{theo}}
\newcommand{\Equation}{\setcounter{equation} {0}\theequation}
%\newcommand{\soussection}{\setcounter{theo} {0}\subsection}
\newcommand{\Section}{\setcounter{equation}{0} \section}
%\renewcommand{\thetheo}{\thesection\arabic{theo}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\soussection}{\setcounter{theoc}{0} \subsection}
\renewcommand{\thesubsubsection}{\thesubsection.\arabic{subsubsection}}







%\def\rem{{\bf Remarque : }}
%\newcounter{ncours}
%\newcommand{\ncours}
% {\bigskip Cours \Numero \arabic{ncours})\addtocounter{ncours}{1} Lundi \ }




%\newtheorem{theo}{\sc Theorem}[section]

\newtheorem{coro}[theo]{\sc Corollary}
\newtheorem{defi}[theo]{\sc Definition}
\newtheorem{lemm}[theo]{\sc Lemma}
\newtheorem{prop}[theo]{\sc Proposition}
\newtheorem{rema}[theo]{\sc Remark}
\newtheorem{rems}[theo]{\sc Remarks}
\newtheorem{main}[theo]{\sc Main Theorem}
\newtheorem{conj}[theo]{\sc Conjecture}
\newtheorem{exam}[theo]{\sc Example}


%\bibliographystyle{amsplain}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%\def\defi#1{ {\it #1 } \index{#1} }

\def\qed{ \ \hfil$\square$}
\def\pgcd{{\hskip0.3mm\rm  pgcd\hskip0.5mm}}
\def\ppcm{{\hskip0.3mm\rm  ppcm\hskip0.5mm}}
\def\Ker{{\hskip0.3mm\rm  Ker\hskip0.5mm}}
\def\Im{{\hskip0.3mm\rm  Im\hskip0.5mm}}
\def\ostar{\otimes}
\makeindex

\def \F {{\mathbb F}}
\def\Si{\Sigma}
\def\St{{\widetilde\Sigma}}
\def\Nt{{\widetilde N}}

\def\ph{\varphi}
\def\W{A}
\def\End{{\rm End}}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\is{\buildrel \sim   \over \rightarrow}

\newcommand{\Ft}{\widetilde{F}}
\newcommand{\Gat}{\widetilde{\Gamma}}
\newcommand{\Mt}{\widetilde{\mathcal M}}
\newcommand{\mt}{\tilde{m}}


\newcommand{\bi}[2]{{#1 \choose #2}}
\newcommand{\di}[1]{\displaystyle{#1}}
\newcommand{\dkr}{\delta_k^{(r)}}
\newcommand{\Dpi}{\frac{1}{2\pi i}}
\newcommand{\Kpi}{\left(\frac{-1}{4\pi}\right)}
\newcommand{\kpiy}{\frac{k}{4\pi y}}
\newcommand{\ny}{\infty}
\newcommand{\pa}{\partial}
\newcommand{\pt}{\cdot}
\newcommand{\tpa}{\tilde{\partial}}
\newcommand{\tr}{{\rm tr}}


\newcommand{\dd}{{\rm d}}
\newcommand{\cont}{{\rm cont}}
\newcommand{\Gal}{{\rm Gal}}
\newcommand{\Gl}{{\rm Gl}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\GSp}{{\rm GSp}}
\newcommand{\Hol}{{\mathcal Hol}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\M}{{{\rm M}}}
%\newcommand{\mod}{{\rm ~mod~ }}
\newcommand{\oo}{{\rm o}}
\newcommand{\Og}{{\rm O}}
\newcommand{\ord}{{\rm ord}}
\newcommand{\SL}{{\rm SL}}
\newcommand{\SO}{{\rm SO}}
\newcommand{\SP}{{\rm Sp}}
\newcommand{\Stab}{{\rm Stab}}
\newcommand{\tors}{{\rm tors}}
\newcommand{\Tr}{{\rm Tr}}
%\newcommand{\U}{{\rm U}}
\newcommand{\lcm}{{\rm LCM~\!\!}}
%\newcommand{\gcd}{{\rm GCD~\!\!}}





\newcommand{\A}{{\mathbb A}}
%\newcommand{\C}{{\mathbb C}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
%\newcommand{\Hb}{{\mathbb H}}
\newcommand{\Hb}{{\frak H}}


%\newcommand{\Hm}{{\mathbb H}_m}
\newcommand{\Hm}{{\mathbb H}_m}

\newcommand{\Fb}{\hat{F}}
\newcommand{\Mb}{\bar{M}}
\newcommand{\Qb}{\overline{\mathbb Q}}
\newcommand{\Sb}{\bar{S}}
\newcommand{\chib}{\bar{\chi}}
\newcommand{\psib}{\bar{\psi}}
\newcommand{\omb}{\bar{\omega}}

\newcommand{\Ar}{{\mathcal A}}
\newcommand{\Ac}{{\mathcal A}}
\newcommand{\Br}{{\mathcal B}}
\newcommand{\Cr}{{\mathcal C}}
\newcommand{\Cc}{{\mathcal C}}
\newcommand{\Ci}{{\mathcal C}^{\infty}}
\newcommand{\Dr}{{\mathcal D}}
\newcommand{\Hr}{{\mathcal H}}
\newcommand{\Mc}{{\mathcal M}}
\newcommand{\Mr}{{\mathcal M}}
\newcommand{\Lr}{{\mathcal L}}
\newcommand{\Or}{{\mathcal O}}
\newcommand{\Sr}{{\mathcal S}}
\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Vr}{{\mathcal V}}
\newcommand{\Xr}{{\mathcal X}}
\newcommand{\X}{{\mathcal X}}





%\newcommand{\bi}[2]{{#1 \choose #2}}
%\newcommand{\di}[1]{\displaystyle{#1}}
%\newcommand{\dkr}{\delta_k^{(r)}}
%\newcommand{\Dpi}{\frac{1}{2\pi i}}
%\newcommand{\Kpi}{\left(\frac{-1}{4\pi}\right)}
%\newcommand{\kpiy}{\frac{k}{4\pi y}}
%\newcommand{\ny}{\infty}
%\newcommand{\pa}{\partial}
%\newcommand{\pt}{\cdot}
%\newcommand{\tpa}{\tilde{\partial}}
%\newcommand{\tr}[1]{{\rm tr}\left(#1\right)}


\newcommand{\al}{\alpha}
\newcommand{\De}{\Delta}
\newcommand{\de}{\delta}
\newcommand{\e}{\varepsilon}
\newcommand{\ep}{\varepsilon}
\newcommand{\Ga}{\Gamma}
\newcommand{\ga}{\gamma}
\newcommand{\GaN}{\Gamma_0^m(N)}
\newcommand{\ka}{\kappa}
\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\vph}{\varphi}


%\newcommand{\bm}{\underline{m}}
\newcommand{\derpa}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\dv}{\:|\:}
\newcommand{\fraga}[2]{\frac{\Gamma(#1)}{\Gamma(#2)}}
\newcommand{\fragab}[2]{\frac{\Gamma(\beta+#1)}{\Gamma(\beta+#2)}}
\newcommand{\fragaf}[2]{\frac{\Gamma(\alpha+#1)}{\Gamma(\alpha+#2)}}
\newcommand{\fragafb}[2]{\frac{\Gamma(\alpha+\beta+#1)}
                        {\Gamma(\alpha+\beta+#2)}}
\newcommand{\fragam}[2]{\frac{\Gamma_m(#1)}{\Gamma_m(#2)}}
\newcommand{\luv}{\lambda(u,v)}
\newcommand{\mabcd}{\left(\begin{array}{cc}a&b\\c&d\end{array}\right)}
\newcommand{\mn}{\setminus}
\newcommand{\ndv}{\not{\:|\:}}
\newcommand{\nuv}{\eta(u,v)}
\newcommand{\rabcd}{\left(\frac{az+b}{cz+d}\right)}

\font\titlefont=cmss10 scaled \magstep2
\font\mathfont=cmmi10 scaled \magstep2
\font\authorfont=cmss10 scaled \magstephalf
\font\sectionfont=cmss10 scaled \magstep1
\font\Itemfont=cmss10
\font\ninerm=cmr9

%Script (in place of caligraphic) letters:
\font\teneusm=eusm10 \font\seveneusm=eusm7 
\font\fiveeusm=eusm5 
\newfam\eusmfam 
\textfont\eusmfam=\teneusm 
\scriptfont\eusmfam=\seveneusm 
\scriptscriptfont\eusmfam=\fiveeusm 





%%%%%%%%%%%%%Notation of M.Courtieu
\def\nd{\not\hskip3.0pt\mid}
\def\nds{\not\hskip1.5pt\mid}
\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\bra#1,{{\left\lbrace {#1}\right\rbrace}}
\def\ph{\varphi}
\def\Ph{\Phi}
\def\si{\sigma}
\def\z{\zeta}
\def \sign {{\rm sign}}
\def \sgn {{\rm sgn}}
\def \inv {{\rm inv}}
\def \diag {{\rm diag}}
%\def \rem {{\rm rem}}
\def \Inf {{\rm Inf}}
%\def \Im {{\rm Im}}
%\def \Ker {{\rm Ker}}
\def \Proj {{\rm Proj}}
\def \Mor {{\rm Mor}}
\def \Div {{\rm Div}}
\def \div {{\rm div}}
\def \Pic {{\rm Pic}}
\def \Res {{\rm Res}}
\def \vol {{\rm vol}}
\def \Ver {{\rm Ver}}
%\def \di {{\rm div}}
\def \Spin {{\rm Spin}}
\def \Step {{\rm Step}}
\def \Spl {{\rm Spl}}
\def \M{{\rm M}}
\def \Gal{{\rm Gal}}
\def \suc{{\rm suc}}
\def \Gab{G^{\rm ab}}
\def \rk{{\rm rk}}
\def \Supp{{\rm Supp}}
\def \Id{{\rm Id}}
\def \Sp{{\rm Sp}}
\def\tm{\tilde m}
\def\tGa{\tilde{\Ga}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\bs{\backslash}
\def\rr{\rangle}
\def\l1{\langle}

\newcommand{\B}{\left(\begin{array}{cc}}
\newcommand{\E}{\end{array}\right)}
%\newcommand{\e}{{\varepsilon\over N}}
\newcommand{\p}{\tilde}

%\newcommand{\f}{{\epsilon\over N}}


%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\def \fns{{${}^{*)}$}}
%%%%%%%%%%%%%%%%%%%
%\newcommand{\marginalfootnote}[1]{%
%   \fns \footnotetext{*{#1}}
%   \marginpar{\hskip0.5cm{\sf *\thefootnote}}%
%}
%\newcommand{\edit}[1]{\marginalfootnote{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT
%\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\comm}[1]
%{\fns  
%\marginpar{
%$\boxed{{\scriptsize {\sf \red
%\begin{tabular} {l} 
% #1
%\end{tabular}}}
%}$
%}
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\marginalfootnote}[1]{%
   \fns \footnotetext{{#1}}
   \marginpar{\hskip0.5cm{\sf \fns}}
%\thefootnote}}%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\edit}[1]{\marginalfootnote{#1}}
%%%%%%%%%%%%%%%%%%%
\newcommand{\comm}[1]
{\fns\marginpar{$\boxed
{\hskip-6pt
%\begin{center}
%\parbox{3
{\small {\sf 
\begin{tabular} {l}
 #1
\end{tabular}
%\end{center}
}
}
}
$
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \?  {\comm{check ?}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\def \dem  {\noindent {\it Proof}}
%%%%%%%%%%%%%%%%%%%%%
\newcommand{\abst}[1]
{\scriptsize \begin{description}
\item{\par\noindent} 
 #1
 \end{description}
\normalsize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{euscript}
\let\scr=\EuScript
\def\scrA{{\scr A}}   \def\scrB{{\scr B}}
\def\scrC{{\scr C}}   \def\scrD{{\scr D}}
\def\scrE{{\scr E}}   \def\scrF{{\scr F}}
\def\scrG{{\scr G}}   \def\scrH{{\scr H}}
\def\scrI{{\scr I}}   \def\scrJ{{\scr J}}
\def\scrK{{\scr K}}   \def\scrL{{\scr L}}
\def\scrM{{\scr M}}   \def\scrN{{\scr N}}
\def\scrO{{\scr O}}   \def\scrP{{\scr P}}
\def\scrQ{{\scr Q}}   \def\scrR{{\scr R}}
\def\scrS{{\scr S}}   \def\scrT{{\scr T}}
\def\scrU{{\scr U}}   \def\scrV{{\scr V}}
\def\scrW{{\scr W}}   \def\scrX{{\scr X}}
\def\scrY{{\scr Y}}   \def\scrZ{{\scr Z}}
\def\Mr{{\scr M}}
\def\Hol{{\scr Hol}}
\let\mathcal=\scr           \def\Or{{\scr O}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ang#1,{{\left\langle {#1}\right\rangle}} 
\newcommand{\ds}{\displaystyle}
\def\G{{\dbG}}
\def\ca {c'est \`a dire}
\def \po {polyn\^ome }
\def \ssi {si et seulement si }
\def \dtfq { de telle fa\c con que }
\def \Mq { Montrer que }
\def \Mqe { Montrer qu'il existe }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \preuve  {\noindent {\sc Preuve}}
\def \proof  {\noindent {\it Proof}}
\def \dem  {\noindent {\it  Demonstration}}
\def \exe  {\medskip\noindent {\sc Example}}
\def \exo  {\medskip\noindent {\sc Exercise}}
\def \rem  {\medskip\noindent {\sc Remark}}
\def \nota  {\medskip\noindent {\sc Notations.}\  }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\thmref}[1]{theorem{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{lemma~\ref{#1}}
\newcommand{\prref}[1]{proposition~\ref{#1}}
%\footnotetext [1]{
%Soutenu par le programme de coop�ration franco-allemand de
%J.-P.Demailly}




%\noindent Institut Fourier, B.P.74, 38402 St.--Martin d'H\`eres, FRANCE

%\noindent 
%e-mail : panchish$@$mozart.ujf-grenoble.fr, 
%FAX:  33 (0) 4 76 51 44 78

%\special {red}
%\abstract
%\abstract{ R�sum�}


%\input{2pt-0.tex}
%\input{2pt-1.tex}
%\input{2pt-2.tex}
%\input{2pt-3.tex}
%\input{2pt-4.tex}
%\input{2pt-5.tex}
%\input{2pt-6.tex}
%\input{2pt-7.tex}
%\input{2pt-8.tex}
%\input{2pt-9.tex}%diop23 vector24 triple26 gauss21maass28 
% + edits made on 27.08.03 
%\input{2pt-aa.tex}%  the latest version see in twist c 
%\input{2pt-ab.tex}
%\input{2pt-ac.tex}
%\input{2pt-b.tex} 


%\NoBlackBoxes

%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{tpc-ed.tex}
\def\qed{ \ \hfil$\square$}
\def\pgcd{{\hskip0.3mm\rm  pgcd\hskip0.5mm}}
\def\ppcm{{\hskip0.3mm\rm  ppcm\hskip0.5mm}}
\def\Ker{{\hskip0.3mm\rm  Ker\hskip0.5mm}}
\def\Im{{\hskip0.3mm\rm  Im\hskip0.5mm}}
\def\ostar{\otimes}
%\makeindex

\def \F {{\mathbb F}}
\def\Si{\Sigma}
\def\St{{\widetilde\Sigma}}
\def\Nt{{\widetilde N}}

\def\ph{\varphi}
\def\W{A}
\def\End{{\rm End}}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\Sym{{\rm Sym}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%














\font\titlefont=cmss10 scaled \magstep2
\font\mathfont=cmmi10 scaled \magstep2
\font\authorfont=cmss10 scaled \magstephalf
\font\sectionfont=cmss10 scaled \magstep1
\font\Itemfont=cmss10
\font\ninerm=cmr9

%Script (in place of caligraphic) letters:
\font\teneusm=eusm10 \font\seveneusm=eusm7 
\font\fiveeusm=eusm5 
\newfam\eusmfam 
\textfont\eusmfam=\teneusm 
\scriptfont\eusmfam=\seveneusm 
\scriptscriptfont\eusmfam=\fiveeusm 


\font\tengothic=eufm10
\font\sevengothic=eufm7
\font\fivegothic=eufm5
\newfam\Gothic
\textfont\Gothic=\tengothic
\scriptfont\Gothic=\sevengothic
\scriptscriptfont\Gothic=\fivegothic
\def\go{\fam\Gothic\tengothic}



%%%%%%%%%%%%%Notation of M.Courtieu
\def\nd{\not\hskip3.0pt\mid}
\def\nds{\not\hskip1.5pt\mid}
\def\mat #1,#2,#3,#4,{\left({#1\atop #3}{#2\atop #4}\right)}
%\def\frac #1,#2,{{{#1}\over {#2}}}
\def\bra#1,{{\left\lbrace {#1}\right\rbrace}}
\def\ph{\varphi}
\def\Ph{\Phi}
\def\si{\sigma}
\def\z{\zeta}
\def\w{\omega}
\def\eps{\varepsilon}
\def\W{\Omega}
\def \sign {{\rm sign}}
\def \sgn {{\rm sgn}}
\def \inv {{\rm inv}}
\def \diag {{\rm diag}}
%\def \rem {{\rm rem}}
\def \Inf {{\rm Inf}}
%\def \Im {{\rm Im}}
%\def \Ker {{\rm Ker}}
\def \Proj {{\rm Proj}}
\def \Mor {{\rm Mor}}
\def \Div {{\rm Div}}
\def \div {{\rm div}}
\def \Pic {{\rm Pic}}
\def \Res {{\rm Res}}
\def \Re {{\rm Re}}
\def \vol {{\rm vol}}
\def \Ver {{\rm Ver}}
%\def \di {{\rm div}}
\def \Spin {{\rm Spin}}
\def \Card {{\rm Card}\,}
\def \Step {{\rm Step}}
\def \Spl {{\rm Spl}}
\def \M{{\rm M}}
\def \Gal{{\rm Gal}}
\def \suc{{\rm suc}}
\def \Gab{G^{\rm ab}}
\def \rk{{\rm rk}}
\def \Supp{{\rm Supp}}
\def \Id{{\rm Id}}
\def \Sp{{\rm Sp}}
\def\m{{\go m}}
\def\tGa{\tilde{\Ga}}
\def\is{\buildrel \sim   \over \rightarrow}
\def\bs{\backslash}
\def\rr{\rangle}
\def\l1{\langle}
\def\Eq{\Longleftrightarrow}
%\newcommand{\B}{\left(\begin{array}{cc}}
%\newcommand{\E}{\end{array}\right)}
%\newcommand{\e}{{\epsilon\over N}}
%\newcommand{\p}{\tilde}
\def\Numero{${\rm N\sp\circ}$}




\let\scr=\EuScript
\def\scrA{{\scr A}}   \def\scrB{{\scr B}}
\def\scrC{{\scr C}}   \def\scrD{{\scr D}}
\def\scrE{{\scr E}}   \def\scrF{{\scr F}}
\def\scrG{{\scr G}}   \def\scrH{{\scr H}}
\def\scrI{{\scr I}}   \def\scrJ{{\scr J}}
\def\scrK{{\scr K}}   \def\scrL{{\scr L}}
\def\scrM{{\scr M}}   \def\scrN{{\scr N}}
\def\scrO{{\scr O}}   \def\scrP{{\scr P}}
\def\scrQ{{\scr Q}}   \def\scrR{{\scr R}}
\def\scrS{{\scr S}}   \def\scrT{{\scr T}}
\def\scrU{{\scr U}}   \def\scrV{{\scr V}}
\def\scrW{{\scr W}}   \def\scrX{{\scr X}}
\def\scrY{{\scr Y}}   \def\scrZ{{\scr Z}}
\def\Mr{{\scr M}}
\def\Hol{{\scr Hol}}
\let\mathcal=\scr           \def\Or{{\scr O}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\db=\mathbb
\def\dbA{{\db A}}   \def\dbB{{\db B}}
\def\dbC{{\db C}}   \def\dbD{{\db D}}
\def\dbE{{\db E}}   \def\dbF{{\db F}}
\def\dbG{{\db G}}   \def\dbH{{\db H}}
\def\dbI{{\db I}}   \def\dbJ{{\db J}}
\def\dbK{{\db K}}   \def\dbL{{\db L}}
\def\dbM{{\db M}}   \def\dbN{{\db N}}
\def\dbO{{\db O}}   \def\dbP{{\db P}}
\def\dbQ{{\db Q}}   \def\dbR{{\db R}}
\def\dbS{{\db S}}   \def\dbT{{\db T}}
\def\dbU{{\db U}}   \def\dbV{{\db V}}
\def\dbW{{\db W}}   \def\dbX{{\db X}}
\def\dbY{{\db Y}}   \def\dbZ{{\db Z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\bold=\boldsymbol
\def\bal{{\bold \alpha}}  \def\bbeta{{\bold \beta}} 
\def\bga{{\bold \gamma}}  \def\bde{{\bold \delta}} 
\def\bepsilon{{\bold \epsilon}}  \def\bep{{\bold \varepsilon}} 
\def\bzeta{{\bold \zeta}}  \def\boeta{{\bold \eta}} 
\def\btheta{{\bold \theta}}  \def\bth{{\bold \vartheta}} 
\def\biota{{\bold \iota}}  \def\bkappa{{\bold \kappa}} 
\def\bvarkappa{{\bold \varkappa}}  \def\blambda{{\bold \lambda}} 
\def\bmu{{\bold \mu}}  \def\bnu{{\bold \nu}} 
\def\bomicron{{\bold \omicron}}  \def\bpi{{\bold \pi}} 
\def\brho{{\bold \rho}}  \def\bsigma{{\bold \sigma}} 
\def\btau{{\bold \tau}}  \def\bipsilon{{\bold \ipsilon}} 
\def\bphi{{\bold \phi}}  \def\bchi{{\bold \chi}} 
\def\bpsi{{\bold \psi}}  \def\bxi{{\bold \xi}} 
\def\bvarphi{{\bold \varphi}}  \def\bomega{{\bold \omega}} 
\def\Bgamma{{\bold \Gamma}}  \def\Bdelta{{\bold \delta}} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\G{{\dbG}}
\def\ca {c'est \`a dire}
\def \po {polyn\^ome }
\def \ssi {si et seulement si }
\def \dtfq { de telle fa\c con que }
\def \Mq { Montrer que }
\def \Mqe { Montrer qu'il existe }
\def\im{\Im}
\def \Pf{{\rm Pf}}
%%%%%%%%%%%%%%%%%%%%%%%%%
%\def\vin{{ \rotate[90]\in  }}
\def\vin{{ {\tiny \mid }  
%\kern-4.75pt 
\kern-7.29pt 
\bigcup }}
\def \Ann{{\rm Ann}}
\def \id{{\rm id}}
\def \Ad{{\rm Ad}}
\def \Sel{{\rm Sel}}
\def \Ext{{\rm Ext}}
\def \Coker{{\rm Coker}}
\def \bs{{\backslash}}
\def \PGL{{\rm PGL}}
\def\ang#1,{{\left\langle {#1}\right\rangle}} 
\def \sec{{'{\!}'}}
\def \cds{{\cdot{\dots}\cdot}}





%\newcommand{\f}{{\epsilon\over N}}
%\newenvironment{slide}{\rule{12cm}{.1cm}} 
%\def \preuve  {\noindent {\sc Preuve}}
%\def \exe  {\medskip\noindent {\sc Exemple.} \ }
%\def \exo  {\medskip\noindent {\sc Exercice.}\  }
%\def \rem  {\medskip\noindent {\sc Remarque.}\  }
%\def \nota  {\medskip\noindent {\sc Notations.}\  }
%\def \rep  {\medskip\noindent {\it R�ponse~:}\  }
%\def \sol {\medskip\noindent {\it Solution~:}\  \small}
%\def \pgcd {{\rm pgcd}}
%\def \ppcm {{\rm ppcm}}
\def \Aut {{\rm Aut}}
\def\m{{\frak m}}


%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\kr}[2]{\left(\frac{#1},{#2},\right)}
\newcommand{\con}{\equiv}
\newcommand{\union}{\cup}
\newcommand{\Tl}{T_{\ell}}
\newcommand{\into}{\rightarrow}
\newcommand{\onto}{\rightarrow\!\!\!\!\rightarrow}
\newcommand{\intersect}{\cap}
\newcommand{\cross}{\times}
%\newcommand{\ds}{\displaystyle}
%%%%%%%%%%%%%%%%%%%%%%%%%
\newenvironment{exercises}
{\setcounter{ex}{0}\vspace{1em}\par\noindent{\sc Exercices}             
  \renewcommand{\labelenumi}{\thesection\theenumi}\small
  \begin{enumerate}
}{\end{enumerate}}\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \prodp {\mathop {\prod{}^{\displaystyle  {}^{\prime}}}}
\def \sump {\mathop {\sum{}^{\displaystyle  {}^{\prime}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\AAA}{\mathbb A}
\newcommand{\CC}{\mathbb C}
\newcommand{\FF}{\mathbb F}
\newcommand{\HH}{\mathbb H}
\newcommand{\NN}{\mathbb N}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\ZZ}{\mathbb Z}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\cb{\C}
\def\ab{\A}
\def\nb{\N}
\def\pb{\Pb}
\def\qb{\Q}
\def\rb{\R}
\def\zb{\Z}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\wh{\widehat}
\def\la{\displaystyle\mathop{\longrightarrow}}
\def\oc{\Or}
\def\cc{\Cr}
\def\dc{\Dr}
\def\hc{\Hr}
\def\lc{\Lr}
\def\mc{\Mr}
%\def\lc{\Lr}
\def\pc{\Pr}
\def\scc{\Sr}
\def\vc{\Vr}
\def\xc{\Xr}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{ncours}{\setcounter{ncours} {1}}
\newcommand{\ncours}{\bigskip Cours \Numero \arabic{ncours}.\addtocounter{ncours}{1} Jeudi\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\kp}{\kappa}
\newcommand{\vkp}{\varkappa}
\newcommand{\chiba}{\bar{\chi}}
\newcommand{\chibar}{\bar{\chi}}
\newcommand{\Mba}{\bar{M}}
\newcommand{\mba}{\underline{m}}
\newcommand{\omba}{\bar{\omega}}
\newcommand{\psiba}{\bar{\psi}}
\newcommand{\Sba}{\bar{S}}
\newcommand{\SE}{{\rm SE}}
\newcommand{\lb}{\lambda}
\newcommand{\vr}{}
\def\dbQbar{\overline {\dbQ}}
\def\Lb{\Lambda}
\def\exptr{{\rm exptr}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Alphabet rond``\cal"     %Alphabet double``\ Bbb"      %Alphabet gras``\bf"  
\def\ac{{\cal A}}           \def\ab{{\Bbb A}}            \def\aaf{{\bf A}}
\def\bc{{\cal B}}           \def\bbb{{\Bbb B}}           \def\bbf{{\bf B}}
\def\cc{{\cal C}}           \def\cb{{\Bbb C}}            \def\ccf{{\bf C}}
\def\dc{{\cal D}}           
                                   %\def\db{{\Bbb D}}            
                                                                          \def\ddf{{\bf D}}
\def\ec{{\cal E}}           \def\eb{{\Bbb E}}            \def\eef{{\bf E}}
\def\fc{{\cal F}}           \def\fb{{\Bbb F}}            \def\fff{{\bf F}}
\def\gc{{\cal G}}           \def\gb{{\Bbb G}}            \def\ggf{{\bf G}}
\def\hc{{\cal H}}           \def\hb{{\Bbb H}}            \def\hhf{{\bf H}}
\def\ic{{\cal I}}           \def\ib{{\Bbb I}}            \def\iif{{\bf I}}
\def\jc{{\cal J}}           \def\jb{{\Bbb J}}            \def\jjf{{\bf J}}
\def\kc{{\cal K}}           \def\kb{{\Bbb K}}            \def\kkf{{\bf K}}
\def\lc{{\cal L}}           \def\lb{{\Bbb L}}            \def\llf{{\bf L}}
\def\mc{{\cal M}}           \def\mb{{\Bbb M}}            \def\mmf{{\bf M}}
\def\nc{{\cal N}}           \def\nb{{\Bbb N}}            \def\nnf{{\bf N}}
\def\oc{{\cal O}}           \def\obb{{\Bbb O}}           \def\oof{{\bf O}}
\def\pcc{{\cal P}}          \def\pb{{\Bbb P}}            \def\ppf{{\bf P}}
\def\qc{{\cal Q}}           \def\qb{{\Bbb Q}}            \def\qqf{{\bf Q}}
\def\rcc{{\cal R}}           \def\rb{{\Bbb R}}            \def\rrf{{\bf R}}
\def\scc{{\cal S}}          \def\sbb{{\Bbb S}}           \def\ssf{{\bf S}}
\def\tcc{{\cal T}}           \def\tb{{\Bbb T}}            \def\ttf{{\bf T}}
\def\uc{{\cal U}}           \def\ub{{\Bbb U}}            \def\uuf{{\bf U}}
\def\vc{{\cal V}}           \def\vb{{\Bbb V}}            \def\vvf{{\bf V}}
\def\wc{{\cal W}}           \def\wb{{\Bbb W}}            \def\wwf{{\bf W}}
\def\xc{{\cal X}}           \def\xb{{\Bbb X}}            \def\xxf{{\bf X}}
\def\yc{{\cal Y}}           \def\yb{{\Bbb Y}}            \def\yyf{{\bf Y}}
\def\zc{{\cal Z}}           \def\zb{{\Bbb Z}}            \def\zzf{{\bf Z}}



\def\diag {\mathop{\rm diag}\nolimits}
\def\SL {\mathop{\rm SL}\nolimits}
\def\Sp {\mathop{\rm Sp}\nolimits}
\def\npn{Np^n}
\def\ol{\overline}
\def\summ{\sum}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\slide}{
%\red \rule{12cm}{.1cm}\par\black  
%\newpage  
%\rule{12cm}{.1cm\green}
%\par 
%} 
%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\def\frac #1,#2,{{{#1}\over {#2}}}

%%%%%%%%%%%%%%%%%%%%%%%
\def\qed{\quad\hbox{\hskip 1pt\vrule width 4pt height 6pt
          depth 1.5pt\hskip 1pt}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\Pc}{{\mathcal P}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{padic-m.tex}


%\magnification=1200

\input amssym.def
\input amssym.tex

%\input eight.tex


\font\titlefont=cmss10 scaled \magstep2
\font\mathfont=cmmi10 scaled \magstep2
\font\authorfont=cmss10 scaled \magstephalf
\font\sectionfont=cmss10 scaled \magstep1
\font\Itemfont=cmss10
\font\ninerm=cmr9

%Script (in place of caligraphic) letters:
\font\teneusm=eusm10 \font\seveneusm=eusm7 
\font\fiveeusm=eusm5 
\newfam\eusmfam 
\textfont\eusmfam=\teneusm 
\scriptfont\eusmfam=\seveneusm 
\scriptscriptfont\eusmfam=\fiveeusm 
\def\scr#1{{\fam\eusmfam\relax#1}}

\def\scrA{{\scr A}}   \def\scrB{{\scr B}}
\def\scrC{{\scr C}}   \def\scrD{{\scr D}}
\def\scrE{{\scr E}}   \def\scrF{{\scr F}}
\def\scrG{{\scr G}}   \def\scrH{{\scr H}}
\def\scrI{{\scr I}}   \def\scrJ{{\scr J}}
\def\scrK{{\scr K}}   \def\scrL{{\scr L}}
\def\scrM{{\scr M}}   \def\scrN{{\scr N}}
\def\scrO{{\scr O}}   \def\scrP{{\scr P}}
\def\scrQ{{\scr Q}}   \def\scrR{{\scr R}}
\def\scrS{{\scr S}}   \def\scrT{{\scr T}}
\def\scrU{{\scr U}}   \def\scrV{{\scr V}}
\def\scrW{{\scr W}}   \def\scrX{{\scr X}}
\def\scrY{{\scr Y}}   \def\scrZ{{\scr Z}}

\def\gr#1{{\fam\eufmfam\relax#1}}

\def\grm{{\gr m}}

%\font\symbols=cmsy10 scaled \magstep2
%\def\SEarrow{\hbox{\symbols \char'046}}
%\def\SWarrow{\hbox{\symbols \char'056}}
%\def\NEarrow{\hbox{\symbols \char'045}}
%\def\NWarrow{\hbox{\symbols \char'055}}


\def\dspace{\lineskip=2pt
  \baselineskip=18pt\lineskiplimit=0pt}
%\def\leaderfill{\leaders\hbox to 
%   1em{\hss.\hss}\hfill}
\def\Proclaim#1{\medbreak
  \noindent{\Itemfont #1\enspace}\it\ignorespaces}
\def\finishproclaim{\par\rm
    \ifdim\lastskip<\medskipamount\removelastskip
     \penalty55\medskip\fi}
\def\proof{\par\noindent{\it Proof}}
\def\proofof#1:{\par\medskip
   \noindent{\it Proof of {\rm #1}}}
\def\condition#1{\hangindent%
  \parindent{\rm#1}\enspace\ignorespaces}
\def\Firstcondition#1{\hangindent%
  \parindent{\rm #1}\enspace\ignorespaces}
%\def\mapright#1{\smash{
%  \mathop{\,\,\longrightarrow\,\,}\limits^{#1}}}
%\def\maprightsub#1{
%  \mathop{\,\,\longrightarrow\,\,}\limits_{#1}}
%\def\lmapdown#1{\Big\downarrow\llap{%
%    $\vcenter{\hbox{$\scriptstyle#1\enspace$}}$}}
%\def\rmapdown#1{\Big\downarrow\rlap{%
% $\kern-1.0pt\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\vrulesub#1{\hbox{$\,$\vrule height7pt 
   depth5pt$\,$}_{#1}}
%\def\longmapright#1#2{\smash{\mathop
%  {\hbox to #1pt{\rightarrowfill}}\limits^{#2}}}
%\def\longmaprightsub#1#2{%
%  \mathop{\hbox to #1pt{\rightarrowfill}}%
%  \limits_{#2}}
\def\nspace{\lineskip=1pt%
  \baselineskip=12pt\lineskiplimit=0pt}
\def\references#1{\bigskip\par
  \centerline{\sectionfont References}\bigskip
     \parindent=#1pt\nspace}
\def\Ref[#1]{\par\smallskip\hang\indent%
  \llap{\hbox to\parindent{[#1]\hfil\enspace}}%
     \ignorespaces}
%\def\ref#1{\par\smallskip
%  \hang\indent\llap{\hbox to\parindent
%     {#1\hfil\enspace}}\ignorespaces}
\def\Item#1{\par\smallskip
  \hang\indent\llap{\hbox to\parindent
     {#1\hfill\enspace}}\ignorespaces}
\def\ItemItem#1{\par\indent\hangindent2\parindent
\hbox to\parindent{#1\hfill\enspace}\ignorespaces}
%\def\lmapup#1{\Big\uparrow\llap{%
%  $\vcenter{\hbox{$\scriptstyle#1\enspace$}}$}}
%\def\rightcapdownarrow{\raise9pt\hbox{%
%   $\scriptstyle\cap$}\kern-7.75pt\Big\downarrow}
%\def\longmaprightsubsup#1#2#3{%
%\mathop{\hbox to #1pt{\rightarrowfill}}%
%\limits_{\raise3pt\hbox{$\scriptstyle #2$}}^{#3}}
%\def\BIGdownarrow{\raise15pt\hbox{%
%  ${\Bigg\vert}\kern-5.0pt%
%     \lower23pt\hbox{$\Bigg\downarrow$}$}}
%\def\BIGdownarrow{\raise15pt\hbox{%
%  ${\Bigg\vert}\kern-5.0pt%
%    \lower23pt\hbox{$\Bigg\downarrow$}$}}
%\def\Rmapdown#1{\Bigg\downarrow\rlap{%
% $\kern-1.0pt\vcenter{\hbox{$\scriptstyle#1$}}$}}
%\def\uphat{{\hat{\phantom a}}}
%\def\Piop{\mathop{\Pi}}
%\font\apostrophe=cmtt10 
%\def\ap{\hbox{\apostrophe\char'015}}
\font\dotless=cmr10 %for the roman i or j to be
                    %used with accents on top.
                    %(\dotless\char'020=i)
                    %(\dotless\char'021=j)
%\font\biggermath=cmmi10 at 12pt
%\def\hati{{\^{\dotless\char'020}}} %roman 
%\def\Matrix#1{\null\,\vcenter{\normalbaselines
% \ialign{\hfil$##$\hfil &&$\,$\hfil$##$\hfil\crcr
%      \mathstrut\crcr
%\noalign{\kern-\baselineskip}
%  #1\crcr\mathstrut\crcr
%  \noalign{\kern-\baselineskip}}}\,}
%\def\Pmatrix#1{\left(\Matrix{#1}\right)}
%\def\Limright#1{\mathop{\vtop{\ialign{##\crcr
%   \hfil\rm lim\hfil\crcr
%   \noalign{\vskip2.0 true pt}
%   \noalign{\nointerlineskip}\rightarrowfill\crcr
%    \noalign{\nointerlineskip}
%    \hfil$\scriptstyle #1$\hfil\crcr
%    \noalign{\nointerlineskip}\crcr}}}}
\def\Limleft#1{\mathop{\vtop{\ialign{##\crcr
    \hfil\rm lim\hfil\crcr
    \noalign{\vskip2.0 true pt}
    \noalign{\nointerlineskip}\leftarrowfill\crcr
    \noalign{\nointerlineskip}
    \hfil$\scriptstyle #1$\hfil\crcr
    \noalign{\nointerlineskip}\crcr}}}}
%\def\restprod{\mathop{%
%  \hbox{$\prod\kern-11.0pt\coprod$}}}
%\def\Coprod{\mathop{\textstyle \coprod}}
%\def\Prod{\mathop{\textstyle \prod}}
%\def\ra{\mathop{\rightarrow}}
%\def\lmapdown#1{\Big\downarrow\llap{%
%  $\vcenter{\hbox{$\scriptstyle#1\enspace$}}$}}
%\def\rmapdown#1{\Big\downarrow\kern-1.0pt
%  \vcenter{\hbox{$\scriptstyle#1$}}}
%\def\longmapright#1#2{\smash{\mathop
%   {\hbox to #1pt{\rightarrowfill}}\limits^{#2}}}
%\def\longmaprightsub#1#2{\mathop{%
%  \hbox to #1pt{\rightarrowfill}}\limits_{#2}}
%\def\longRightarrow#1{\raise1.57pt\hbox to #1pt{%
% \hrulefill}\kern-4pt\lower.40pt\hbox{%
%  $\longrightarrow$}}
\def\otimesop{\mathop{\otimes}\limits}
\def\arrowsim{\smash{\mathop{\longrightarrow}
 \limits^{\lower1.5pt \hbox{$\scriptstyle\sim$}}}}
\def\Le{{\mathchoice{\,{\scriptstyle\le}\,}
  {\,{\scriptstyle\le}\,}%
  {\,{\scriptscriptstyle\le}\,}%
  {\,{\scriptscriptstyle\le}\,}}}
\def\Ge{{\mathchoice{\,{\scriptstyle\ge}\,}
  {\,{\scriptstyle\ge}\,}%
  {\,{\scriptscriptstyle\ge}\,}%
  {\,{\scriptscriptstyle\ge}\,}}}

\def\Hom{{\rm Hom}} \def\ord{{\rm ord}}
\def\End{{\rm End}} \def\Tr{{\rm Tr}}
\def\cont{{\rm cont}} \def\diag{{\rm diag}}
\def\Ch{{\rm Ch}}
%\def\Image{\hbox{ Image }}
%\def\Ad{{\rm Ad}} \def\reg{\hbox{\rm reg}}
\def\Quot{{\rm Quot\,}}
\def\Gal{{\rm Gal}} \def\im{{\rm Im}} 
\def\tors{{\rm tors}} \def\re{{\rm Re}}
\def\Distr{{\rm Distr}} \def\SL{{\rm SL}}
\def\SE{{\rm SE}}
\def\mod{{\rm mod}}
%\def\Ind{{\rm Ind}} \def\ad{{\rm ad}} 
%\def\adg{{\rm adg}}
%\def\SC{{\rm sc}} \def\st{{\rm st}}
%\def\SL{{\rm SL}} 
\def\GL{{\rm GL}}
%\def\im{{\rm Im}} \def\mes{{\rm mes}}
%\def\reele{{\rm r\'eele}} \def\Ker{{\rm Ker}}
%\def\Ext{{\rm Ext}} \def\SU{{\rm SU}}

\def\dbA{{\Bbb A}}
\def\dbC{{\Bbb C}}
\def\dbN{{\Bbb N}}
\def\dbQ{{\Bbb Q}}
\def\dbR{{\Bbb R}}
\def\dbZ{{\Bbb Z}}

\def\dbQbar{\overline{\dbQ}}
\def\scrLbar{\overline{\scrL}}
\def\chibar{\overline{\chi}}

\def\hatdbQbar{\widehat{\overline{\dbQ}}}
\def\qbrack{[\![q]\!]}
\def\Ybrack{[\![Y]\!]}

\def\underLam{\underline{\Lambda}}
\def\underScrL{\underline{\scrL}}


%\null\vskip2.0 true cm
%\footline={\hfill}

%\centerline{\titlefont On 
%{\mathfont\char'160}-adic
%   integration in spaces of modular forms}
%\centerline{\titlefont and its applications}

%\bigskip
%\centerline{{\authorfont A. A.  Panchishkin$^*$}%
%\footnote{}{\null\kern-20pt\baselineskip=9pt
%\eightpoint $^*$ Supported by The von Neumann Fund to the
%Institute for Advanced Study during a
%membership of the author at IAS in
%1999-2000.}}


%%% Local Variables: 
%%% mode: plain-tex
%%% TeX-master: "PADIC"
%%% End: 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}


\title{On the numerator of the symplectic Hecke series of degree three}

\author{ Alexei  Panchishkin, Kirill Vankov\\
{\tt http://www-fourier.ujf-grenoble.fr/\~{}panchish}
\\
\small e-mail : panchish$@$mozart.ujf-grenoble.fr, 
FAX:  33 (0) 4 76 51 44 78}



\date{}




\maketitle


%{\it To dear Anatoli Nikolayevich Andrianov for his seventieth birthday}
\begin{abstract}
We find a different  method to compute
the generating series in 
Shimura's conjecture for $Sp_3$, proved by Andrianov in 1967.
Formulas 
%by A. N. Andrianov 
for the Satake spherical map for $\Sp_3$ are used.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tableofcontents
\thispagestyle{empty}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A formula for the  numerator of the generating series of Hecke operators of $Sp_3$}
We obtain the following  formula for the polynomial $P(v)$ 
of degree 6 in the numerator of the Hecke-Shimura generating series 
(see \cite{An87}, (3.3.72), p. 163; this series corresponds to $D_p(s)$ in  \cite{Sh}, p.825):
%%%%%%%%%%%%%%%%%%%%%%%%

\begin{equation}
\begin{split}
P(x_0,\, x_1&,\, x_2,\, x_3,\, v)=\\
 =1&-\left({\displaystyle\frac{sym_{2,1,1}}{p}}+{\displaystyle\frac{(p^2+p+1)\, sym_{1,1,1}}{p^2}}+{\displaystyle\frac{sym_{1,1,0}}{p  }}\right) \, x_0^2\, v^2\\
   &+{\displaystyle\frac{p+1}{p^2}}\left(sym_{2,2,2}+sym_{2,2,1}+sym_{2,1,1}+sym_{1,1,1}\right) \, x_0^3\, v^3\\ 
   &-\left({\displaystyle\frac{sym_{3,2,2}}{p^2}}+{\displaystyle\frac{(p^2+p+1)\, sym_{2,2,2}}{p^3}}+{\displaystyle\frac{sym_{2,2,1}}{p^2}}\right) \, x_0^4\, v^4\\
   &+{\displaystyle\frac{sym_{3,3,3}}{p^3}}\, x_0^6\,v^6 \, .
\end{split}
\end{equation}

\smallskip
In this formula the notation $sym_{i_1,i_2,i_3}$ represents the symmetric polynomial of three variables 
$x_1$, $x_2$ and $x_3$ constructed in the following way:
$$
sym_{i_1,i_2,i_3} = \sum_{\sigma\in S_n / \,{\rm Stab}(x_1^{i_1}x_2^{i_2}x_3^{i_3})} \sigma (x_1^{i_1}x_2^{i_2}x_3^{i_3}),
$$
where the summation of permuted monomials $x_1^{i_1}x_2^{i_2}x_3^{i_3}$ is normalized using the stabilizer
${\rm Stab}(x_1^{i_1}x_2^{i_2}x_3^{i_3})$ so that all coefficients are equal to 1 and $i_1\ge i_2\ge i_3\ge 0$.  
The total degree of the polynomial is $i_1+i_2+i_3$.  Here $S_n = S_3$ is the symmetric group that acts naturally on polynomials in $n$ variables, where $n=3$ in our case. For example
\begin{equation}\nonumber
\begin{split}
sym_{0,0,0}&=1\\
sym_{1,0,0}&=x_1+x_2+x_3\\
sym_{1,1,0}&=x_1x_2+x_1x_3+x_2x_3\\
sym_{1,1,1}&=x_1x_2x_3\\
sym_{4,3,2}&=x_1^4x_2^3x_3^2+x_1^4x_2^2x_3^3+x_1^3x_2^4x_3^2+x_1^3x_2^2x_3^4+x_1^2x_2^4x_3^3+x_1^2x_2^3x_3^4 \, .
\end{split}
\end{equation}
Many computations presented in this article were performed using Maple 9.50 (IBM INTEL NT).  
Symmetric polynomials $sym_{i_1i_2i_3}$ (up to total degree 18) were computed using 
the coefficient of $t$ of the generating function 
\begin{equation}\nonumber
\begin{split}
&\prod_{\sigma\in S_3}(1+tx_{\sigma(1)}^{i_1}x_{\sigma(2)}^{i_2}x_{\sigma(3)}^{i_3})=(1+tx_1^{i_1}x_2^{i_2}x_3^{i_3})(1+tx_1^{i_1}x_2^{i_3}x_3^{i_2}) \\
&\qquad \cross (1+tx_1^{i_2}x_2^{i_1}x_3^{i_3})(1+tx_1^{i_2}x_2^{i_3}x_3^{i_1})(1+tx_1^{i_3}x_2^{i_1}x_3^{i_2})(1+tx_1^{i_3}x_2^{i_2}x_3^{i_1}) \, ,
\end{split}
\end{equation}
where $i_1=0,\dots ,6$, $i_2=0,\dots ,i_1$ and $i_3=0,\dots ,i_2$. Then it is normalized by dividing out 
its leading coefficient.
%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us state our result directly in terms of the Hecke operators for the symplectic group $\Sp_n$, 
defined at p. 142 of \cite{An87}.
Consider the group of positive symplectic similitudes
\begin{align}&
\mathrm{S}=\mathrm{S}^n=\GSp_n^+(\Q)=\{M\in \mathrm{M}_{2n}(\Q)\ |\ 
{}^tMJ_nM=\mu(M)J_n, \mu(M)>0\}. 
\end{align}
For the  Siegel  modular group 
$\Gamma=\Sp_n(\Z)$ consider the double cosets
\begin{align}&
(M)=\Gamma M\Gamma \subset 
\mathrm{S}, 
\end{align}
and the Hecke operators
\begin{align}&
\mathbf{T}(a)= \sum_{M\in \mathrm{SD}_n(a)}(M), 
\end{align}
where $M$ runs through the following integral matrices 
\begin{align}&
\mathrm{SD}_n(a)=\{\diag(d_1, \cdots, d_n;e_1, \cdots, e_n)\ |\ d_i| d_{i+1}, d_n| e_{n}, e_{i+1}|e_i, d_ie_i=a\}. 
\end{align}
Let us use the notation
\begin{align}&
\mathbf{T}(d_1, \cdots, d_n;e_1, \cdots, e_n)=(\diag(d_1, \cdots, d_n;e_1, \cdots, e_n)).
%d_i| d_{i+1}, d_n| e_{n}, e_{i+1}|e_i, d_ie_i=a\}.
\end{align}
In particular we have the operators
(see p. 149 of \cite{An87}):
\begin{align}&
\mathbf{T}(p)= \mathbf{T}(\underbrace{1, \cdots, 1}_{n},\underbrace{p, \cdots, p}_{n}),\\ &
\mathbf{T}_i(p^2)= \mathbf{T}(\underbrace{1, \cdots, 1}_{n-i},\underbrace{p, \cdots, p}_{i},\underbrace{p^2, \cdots, p^2}_{n-i},\underbrace{p, \cdots, p}_{i}),
\mbox{ for } i=1, 2, \cdots, n.
\end{align}
Then their images by the spherical map $\Omega$ are given at p.159 of  \cite{An87}:
\begin{align}&
\Omega(\mathbf{T}(p))= x_0\prod_{i=0}^n(1+x_i)=\sum_{j=0}^nx_0s_j(x_1, x_2, \cdots, x_n),\\ &
\Omega(\mathbf{T}_i(p^2))= \sum_{a+b\le n, a\ge i}p^{b(a+b+1)}\mathrm{sm}_p(a-i, a)x_0^2\omega(\pi_{a, b}).
\end{align}
Here 
$\pi_{a, b}=\left(\begin{pmatrix}E_{n-a-b} & &\\ & pE_{a}&\\ & &p^2E_{b}\end{pmatrix}\right)$ is a Hecke operator for $\GL_n$, and
the coefficient $\mathrm{sm}_p(r, a)$ denotes the number of symmetric matrices of rank $r$ and order $a$ over the field $\F_p$.
This coefficient is evaluated at p.205 of  \cite{An87}:
\begin{align}\label{phi}&
\mathrm{sm}_p(r, a)=\mathrm{sm}_p(r, r)\frac{\phi_a(p)}{\phi_{r}(p)\phi_{a-r}(p)}, 
\\ & \nonumber
\mbox{ with  } \phi_{r}(x)=(x-1)(x^2-1)\cds(x^r-1) \mbox{ for   } r\ge 1, 
\mbox{ and  } \phi_{0}(x)=1.
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In particular, we have in  the case $n=3$ that 
\begin{align}\label{Tp}&  \nonumber
\Omega(\mathbf{T}(p)) = \mathit{x}_0\,({\mathit{sym}_{1, \,1, \,0}} + 
{\mathit{sym}_{1, \,0, \,0}} + {\mathit{sym}_{1, \,1, \,1}} + 1),
\\ & \nonumber
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Omega(\mathbf{T}_1(p^2))= {\displaystyle \frac {\mathit{x}_0^{2}\,(p^{
2} - 1)\,{\mathit{sym}_{2, \,1, \,1}}}{p^{3}}}  + {\displaystyle 
\frac {\mathit{x}_0^{2}\,{\mathit{sym}_{2, \,2, \,1}}}{p}}  + 
{\displaystyle \frac {\mathit{x}_0^{2}\,{\mathit{sym}_{2, \,1, \,0
}}}{p}}  \\ &  \nonumber
\mbox{} + 
{\displaystyle \frac {\mathit{x}_0^{2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%(p-1)(2p^2+4 p+1)
(p-1)(3p^2+2 p+1)
%%%%%%%%%%%%%%%%%%%%%%%%
{\mathit{sym}_{1, \,1, \,1}}}{p^{4}}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\,(2\,p^{4} - p^{3
%} + 5\,p^{2} - 4\,p - 2)\,{\mathit{sym}_{1, \,1, \,1}}}{p^{5}}} 
 + {\displaystyle \frac {\mathit{x}_0^{2}\,(p^{2} - 1)\,{\mathit{
sym}_{1, \,1, \,0}}}{p^{3}}}  + {\displaystyle \frac {\mathit{x}_0
^{2}\,{\mathit{sym}_{1, \,0, \,0}}}{p}},  \nonumber
\\ & 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Omega(\mathbf{T}_2(p^2))= p^{0}\mathrm{sm}_p(0, 2)x_0^2\omega(\pi_{2, 0})
\\ &  \nonumber
+p^{4}\mathrm{sm}_p(0, 2)x_0^2\omega(\pi_{2, 1})+p^{0}\mathrm{sm}_p(1, 3)x_0^2\omega(\pi_{3, 0})
\\ & \nonumber
=x_0^2\omega (t(1,p, p))+p^{4}x_0^2\omega (t(p, p, p^2))+
\mathrm{sm}_p(1, 3)x_0^2\omega (t(p, p, p))
\\ & \nonumber
=p^{-3}x_0^2 sym_{1,1,0}+p^{-3}x_0^2 sym_{2,1,1}+
p^{-6}(p-1)(p^2+p+1)x_0^2 sym_{1,1,1},  
\\ &  \nonumber
\Omega(\mathbf{T}_3(p^2))=p^{0}\mathrm{sm}_p(0, 3)x_0^2\omega(\pi_{3, 0})= p^{-6}x_0^2x_1x_2x_3=p^{-6}x_0^2 sym_{1,1,1}, 
\end{align}
because of the equality (\ref{phi}) with $a=3, r=1$ implying $\mathrm{sm}_p(1, 3)= (p-1)(p^2+p+1)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consider the polynomial $Q_3(v)$ defining the spinor zeta function
$Z(s)$ of genus three:
\begin{align}&
Q_3(v)=\\ &\nonumber (1-x_0v)(1-x_0x_1v)(1-x_0x_2v)(1-x_0x_3v)\\ &\nonumber
(1-x_0x_1x_2v)(1-x_0x_1x_3v)
(1-x_0x_2x_3v)(1-x_0x_1x_2x_3v) .
\end{align}
Following the proof at  p.159 of  \cite{An87},
  threre exist   Hecke operators  $\mathbf{t}_j\in {\mathcal L}_\Z$  with ${\mathcal L}_\Z=\Z[ \mathbf{T}(p), \mathbf{T}_1(p^2), \cdots, \mathbf{T}_{n}(p^2)]$ such that 
\begin{align}&
\sum_{j=0}^{2^n}\Omega(\mathbf{t}_j)v^j = Q_n(v)=
(1-x_0v)(1-x_0x_1v)(1-x_0x_2v)\cds (1-x_0x_1x_2\cds x_nv) .
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider the series 
$\mathbf{R}_n(v)  = \sum_{\delta=0}^{\infty} \mathbf{T}(p^\delta )v^\delta  \in {\mathcal L}_\Z[\![v]\!]$ and 
the polynomial $\mathbf{Q}_n(v)=\sum_{j=0}^{2^n}\mathbf{t}_jv^j$ over the Hecke ring ${\mathcal L}_\Z$.

It was established by A.N.Andrianov, 
that there exist polynomials  $\mathbf{P}_n(v)$, $\mathbf{Q}_n(v)$ such that 
\begin{align}&
\mathbf{R}_n(v)  = \sum_{\delta=0}^{\infty} \mathbf{T}(p^\delta )v^\delta  =
\frac{\mathbf{P}_n(v)}{\mathbf{Q}_n(v)},
\end{align}
with the above polynomial $\mathbf{Q}_n(v)=\sum_{j=0}^{2^n}\mathbf{t}_jv^j$ of degree $2^n$, and  such that
 $\ds \mathbf{P}_{n}(v)=\sum_{j=0}^{2^n-2}\mathbf{u}_jv^j$ is a polynomial
 of degree $2^n-2$ 
with the leading term $(-1)^{n-1}p^{{n(n+1)}2^{n-2}-n^2}[\mathbf{p}]^{2^{n-1}-1} v^{2^n-2}$
(as stated in  Theorem 6 at p. 451 of \cite{An70} and at p.61 of \S 1.3, \cite{An74}).
In the following theorem we denote by $[\mathbf{p}]_n=(pE_{2n})=\mathbf{T}_n(p^2)$  the  element (3.4.48) in \cite{An70}), so that 
$\Omega([\mathbf{p}]_n)= p^{-n(n+1)/2}x_0^2x_1\cds x_n$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theo}[see also  \cite{An67}]\label{ThP} If $n=3$, there is the following explicit polynomial presentation: 
	\begin{align}&
\mathbf{R}_3(v)  = \sum_{\delta=0}^{\infty} \mathbf{T}(p^\delta )v^\delta 
=\frac{\mathbf{P}_3(v)}{\mathbf{Q}_3(v)},
\end{align}
where
\begin{align}\label{P3}&
\mathbf{P}_3(v) 
 =1-
p^2\left( \mathbf{T}_2(p^2)+
(p^2-p+1)(p^2+p+1)
%(p^4+p^2+1){p^2}
[\mathbf{p}]_3\right)v^2
+(p+1)p^4\mathbf{T}(p)[\mathbf{p}]_3v^3
\\ \nonumber 
      &     \qquad %-p^5\left(p^2\mathbf{T}_2(p^2)[\mathbf{p}]_3+
-p^7[\mathbf{p}]_3\left( \mathbf{T}_2(p^2)+
(p^2-p+1)(p^2+p+1)
%(p^4+p^2+1){p^2}
[\mathbf{p}]_3\right) v^4 
   +p^{15}[\mathbf{p}]_3^3\,v^6 \, \in {\mathcal L}_\Z[v].
\end{align}

\end{theo}
\begin{rema} It was pointed out to the first author by S.Boecherer, that
difficulties in the problem of analytic continuation of the spinor $L$-function of genus 3
%are closely related to the explicit form of 
could come from
the polynomial $\mathbf{P}_3(v)$.
Indeed, this is clearly indicated  by Kurokawa's paper \cite{Ku88}, Theorem 2
in the case of the Siegel-Eisenstein series of genus 3.
A similar polynomial is discussed in the paper of Maass \cite{Maa76}.
\end{rema}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Explicit form of Shimura's conjecture for $Sp_3$}
Our result gives a complement to the solution  of Shimura's conjecture by A.N.Andrianov in \cite{An67} (see  also \cite{An68} and \cite{An69}).
This conjecture was stated in  \cite{Sh}, at p.825 as follows:
\begin{quote}
``In general, it is plausible that $D_p(X)=E(X)/F(X)$ with polynomials $E(X)$  
and $F(X)$ in $X$ with integral coefficients of degree $2^n-2$ and $2^n$, respectively'' 
\end{quote}
({\it i.e. with coefficients in } ${\mathcal L}_\Z=\Z[ \mathbf{T}(p), \mathbf{T}_1(p^2), \cdots, \mathbf{T}_{n}(p^2)]$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theo}[see also  \cite{An67}] If $n=3$, one has the following explicit polynomial presentation: 
\begin{align}&
\mathbf{Q}_3(v) 
 = 1 - \mathbf{T}(p)\,v 
\\ & \nonumber
+p\big(\,
\mathbf{T}_1(p^2) 
+ (p^{2} + 1)\,\mathbf{T}_2(p^2) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
(1+p^2)^2
%(p^{6} -2p^{5} +2\,p^{4} + 2\,p^{3} +  p)\,
%(p^{6} + p^{5} - 6\,p^{4} + 5\,p^{3} + 2\,p^{2} + p)\,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- p^{3}\mathbf{T}(p)\,\big( \mathbf{T}_2(p^2)
+ [\mathbf{p}]_3\big)
v^{3} 
\\ & \nonumber
+ p^6\big(
 - 2\,p\,\mathbf{T}_1(p^2)\,[\mathbf{p}]_3 + \,\mathbf{T}_2(p^2)
\mbox{}  - 2( p -1)\,\mathbf{T}_2(p^2)\,[\mathbf{p}]_3
 \\ &\nonumber
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-(p^2+2p-1)(p^2-p+1)(p^2+p+1))
%(- 3\,p^{12}+4\,p^{11}-4\,p^{10}- 2\,p^{9}- 2\,p^{7}+p^{6})\,
%( - 8\,p^{9} - 3\,p^{12} - 2\,p^{11} - 4\,p^{8} + 12\,p^{10} + p^{6} - 2\,p^{7})\,
[\mathbf{p}]_3^{2} +
[\mathbf{p}]_3\,
\mathbf{T}(p)^{2}\big)v^{4} \\ &\nonumber
\mbox{} 
- \,p^{9}\,[\mathbf{p}]_3^{}\,\mathbf{T}(p)\,\big ( 
\mathbf{T}_2(p^2) +[\mathbf{p}]_3\big )\,
v^{5}   \\ &\nonumber
+p^{13}\,[\mathbf{p}]_3^{2}\,\big(\,\mathbf{T}_1(p^2) + (
p^{2} + 1)\,\mathbf{T}_2(p^2) + 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(p^2+1)^2
%(p^{6} -2p^{5} +2\,p^{4} + 2\,p^{3} +  p)\,
%(p^{6} + p^{5} - 6\,p^{4} + 5\,p^{3} + 2\,p^{2} + p)\,
[\mathbf{p}]_3\big)\,v^{6} \\ &\nonumber
\mbox{} - \,p^{18}\,[\mathbf{p}]_3^{3}\,\mathbf{T}(p)
\,v^{7} +\,p^{24}\,[\mathbf{p}]_3^{4}\,v^{8}
\in {\mathcal L}_\Z[v].
\end{align}

\end{theo}
Proof follows the same lines as that of Theorem \ref{ThP}.
We compute an expression for $\Omega(\mathbf{Q}_3(v))$ in terms of
$sym_{i_1,i_2,i_3}$:
\begin{align}& \nonumber
%\mathit{Q81}
\Omega(\mathbf{Q}_3(v))
=1 - \mathit{x}_0^{}\,({\mathit{sym}_{1, \,1, \,1}} + {\mathit{sym}_{1, \,1, \,0}} + {
\mathit{sym}_{1, \,0, \,0}} +  1)\,
v \\ &  \nonumber
\mbox{} + \mathit{x}_0^{2}\,(4\,{\mathit{sym}_{1, \,1, \,1}} + {
\mathit{sym}_{1, \,0, \,0}} + 2\,{\mathit{sym}_{2, \,1, \,1}} + 2
\,{\mathit{sym}_{1, \,1, \,0}} + {\mathit{sym}_{2, \,1, \,0}} + {
\mathit{sym}_{2, \,2, \,1}})\,v^{2} - \\ &  \nonumber
-\mathit{x}_0^{3}({\mathit{sym}_{3, \,1, \,1}} + {\mathit{sym}_{1, 
\,1, \,0}} + 4\,{\mathit{sym}_{2, \,2, \,1}} + 4\,{\mathit{sym}_{
1, \,1, \,1}} + {\mathit{sym}_{2, \,1, \,0}} + {\mathit{sym}_{2, 
\,2, \,0}} + 4\,{\mathit{sym}_{2, \,2, \,2}} \\ &  \nonumber
\mbox{} + {\mathit{sym}_{3, \,2, \,2}} + {\mathit{sym}_{3, \,2, 
\,1}} + 4\,{\mathit{sym}_{2, \,1, \,1}})v^{3}\mbox{} + 
\mathit{x}_0^{4}({\mathit{sym}_{3, \,1, \,1}} + {\mathit{sym}_{1, \,1, \,1}}
 + {\mathit{sym}_{3, \,3, \,1}} \\ &  
\mbox{} + {\mathit{sym}_{4, \,2, \,2}} + 2\,{\mathit{sym}_{2, \,1
, \,1}} + 4\,{\mathit{sym}_{3, \,2, \,2}} + 2\,{\mathit{sym}_{3, 
\,2, \,1}} + {\mathit{sym}_{2, \,2, \,0}} + 8\,{\mathit{sym}_{2, 
\,2, \,2}} \\ &  \nonumber
\mbox{} + 2\,{\mathit{sym}_{3, \,3, \,2}} + {\mathit{sym}_{3, \,3
, \,3}} + 4\,{\mathit{sym}_{2, \,2, \,1}})v^{4}\mbox{} - 
\mathit{x}_0^{5}({\mathit{sym}_{4, \,3, \,3}} + {\mathit{sym}_{4, \,3, \,2
}} + {\mathit{sym}_{2, \,2, \,1}} \\ &  \nonumber
\mbox{} + 4\,{\mathit{sym}_{2, \,2, \,2}} + 4\,{\mathit{sym}_{3, 
\,3, \,2}} + {\mathit{sym}_{3, \,3, \,1}} + 4\,{\mathit{sym}_{3, 
\,2, \,2}} + 4\,{\mathit{sym}_{3, \,3, \,3}} + {\mathit{sym}_{4, 
\,2, \,2}} + {\mathit{sym}_{3, \,2, \,1}} %\\ &  \nonumber 
)v^{5}
 \\ &  \nonumber
 + \mathit{x}_0^{6}\,(2\,{\mathit{sym}_{3, \,3, \,2}}
 + {\mathit{sym}_{3, \,2, \,2}} + 2\,{\mathit{sym}_{4, \,3, \,3}}
 + 4\,{\mathit{sym}_{3, \,3, \,3}} + {\mathit{sym}_{4, \,3, \,2}}
 + {\mathit{sym}_{4, \,4, \,3}})\,v^{6} \\ &  \nonumber
\mbox{} - \mathit{x}_0^{7}\,({\mathit{sym}_{4, \,3, \,3}} + {
\mathit{sym}_{3, \,3, \,3}} + {\mathit{sym}_{4, \,4, \,4}} + {
\mathit{sym}_{4, \,4, \,3}})\,v^{7} + \mathit{x}_0^{8}\,{\mathit{
sym}_{4, \,4, \,4}}\,v^{8}.
 \end{align}
Then we use the polynomial expressions (\ref{Tp}) 
for the generators of the Hecke ring.
Using these generators,  we may look for a solution 
in the following form:
\begin{align}&
%\mathit{Q87} :
\Omega(\mathbf{Q}_3(v))
= 1 - \Omega(\mathbf{T}(p))v
 + 
\big(\mathit{K}_{T1p2}\Omega(\mathbf{T}_1(p^2)) + \mathit{K}_{T2p2}\Omega(\mathbf{T}_2(p^2))
 \\ & \nonumber
 + \mathit{K}_{T3p2}\Omega([\mathbf{p}]_3) + \mathit{K}_{TpTp}\Omega(\mathbf{T}(p))^{2}\big)\,v^{2} 
  \\ & \nonumber
+\big(\mathit{K}_{TpT1p2}\Omega(\mathbf{T}(p)\mathbf{T}_1(p^2)) 
+ 
\mathit{K}_{TpT2p2}\Omega(\mathbf{T}(p)\mathbf{T}_2(p^2)) 
 \\ & \nonumber
+ \mathit{K}_{TpT3p2}\,
\Omega(\mathbf{T}(p)[\mathbf{p}]_3) + \mathit{K}_{TpTpTp}\Omega(\mathbf{T}(p))^{3}\big)
v^{3}\mbox{}
 \\ & \nonumber
 + 
\big(\mathit{K}_{T1p2T1p2}\Omega(\mathbf{T}_1(p^2))^{2} 
+ \mathit{K}_{T1p2T2p2}\Omega(\mathbf{T}_1(p^2)\mathbf{T}_2(p^2)) 
\\ & \nonumber
+ 
\mathit{K}_{T1p2T3p2}\,\Omega(\mathbf{T}_1(p^2)[\mathbf{p}]_3)
\mbox{} + \mathit{K}_{T2p2T2p2}\Omega(\mathbf{T}_2(p^2))^{2}
\\ & \nonumber
 + 
%%%% 
\mathit{K}_{T2p2T3p2}\,\Omega(\mathbf{T}_2(p^2)[\mathbf{p}]_3) + \mathit{K}_{T3p2T3p2}\,
\Omega([\mathbf{p}]_3^{2}) \\ &\nonumber
\mbox{} + \mathit{K}_{T1p2TpTp}\,\Omega(\mathbf{T}_1(p^2)\mathbf{T}(p)^{2}) + 
\mathit{K}_{T2p2TpTp}\Omega(\mathbf{T}_2(p^2)\mathbf{T}(p)^{2}) 
\\ &\nonumber
+ \mathit{K}_{T3p2TpTp}\,\Omega([\mathbf{p}]_3\mathbf{T}(p)^{2}) 
\mbox{} + \mathit{K}_{TpTpTpTp}\,\Omega(\mathbf{T}(p))^{4}\big)v^{4}\\
&\nonumber
+
\Omega(\mathbf{t}_5)v^5+\Omega(\mathbf{t}_6)v^6
+\Omega(\mathbf{t}_7)v^7+\,p^{24}\,\Omega([\mathbf{p}]_3)^{4}v^8.  
\end{align} 

It is not too difficult to resolve the resulting equations in the indeterminate coefficients:\\
$
K_{TpT1p2} = 0, K_{TpTpTp} = 0, K_{TpT2p2} = -p^3, K_{TpT3p2} =
-p^3,
K_{T2p2} = p^3+p$, \\ $K_{T3p2} = 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
p(1+p^2)^2,
% p^{6} -2p^{5} +2\,p^{4} + 2\,p^{3} +  p,\, 
%p^6+p^5-6p^4+5p^3+2p^2+p,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
K_{T1p2} = p, K_{TpTp} = 0$,
$
K_{T2p2TpTp}=0$, \\ 
$K_{T1p2T3p2}=-2p^7,
K_{T2p2T3p2}= -2p^7+2p^6, 
K_{T1p2T1p2}= 0,
K_{T1p2T2p2}= 0$, \\ 
$K_{T2p2T2p2}= p^6$, 
$K_{T1p2TpTp}= 0, 
K_{T3p2TpTp}=p^6, 
K_{TpTpTpTp}= 0$,\\ $
K_{T3p2T3p2}=
-p^6(p^2+2p-1)(p^2-p+1)(p^2+p+1)$.
%- 3\,p^{12}+4\,p^{11}-4\,p^{10}- 2\,p^{9}- 2\,p^{7}+p^{6}$.
%p^6-2p^7-4p^8-8p^9+12p^{10}-2p^{11} -3p^{12}$.

Then we find the remaining coefficients $\mathbf{t}_5$, $\mathbf{t}_6$, $\mathbf{t}_7$ using the functional equation \cite{An87}, p.164
(3.3.79):   
$\mathbf{t}_{8-i}=(p^6[\mathbf{p}]_3)^{4-i}\mathbf{t}_i$ ($i=0, \cdots, 8$), 
compare with formulas in 
\cite{An67} and in 
\cite{Evd}. \qed


\section{An identity involving $\omega (t(1,p^{\lambda_2}, p^{\lambda_3}))$ and $Q_3(v)$}
The theory of Hecke rings for the symplectic group is developed in \cite{Sh}, 
\cite{An87} and \cite{AnZh95} (Ch. 3).  
At page 150 of \cite{AnZh95} we have the following identity for the spherical map 
\begin{equation}\nonumber
\begin{split}
R_n(v) & = \sum_{\delta=0}^{\infty} \Omega (\mathbf{T}^n(p^\delta ))v^\delta \\
 & = \sum_{\delta=0}^{\infty}\quad\sum_{0\le\delta_1\le\cdots\le\delta_n\le\delta}p^{n\delta_1+(n-1)\delta_2+\cdots+\delta_n}\omega(t(p^{\delta_1},\cdots,p^{\delta_n}))(x_0v)^\delta \, .
\end{split}
\end{equation}

This identity for formal generating series of elements of Hecke ring allows 
to reduce computations in the local Hecke rins of the symplectic group 
to computations in polynomial rings by applying the spherical map $\Omega$ 
to elements $\mathbf{T}^n(p^\delta ) \in L_p^n(q)$ of Hecke ring for the symplectic 
group or spherical map $\omega$ to elements  
$t(p^{\delta_1},\cdots,p^{\delta_n}) = \sum_j a_j(\Lambda g_j) \in L_{\Q}(\Lambda ^n,G_p^n)$ 
of Hecke ring for the general linear group. 
Detailed definition of spherical maps 
for Hecke elements as well as definitions and a structure of left cosets $\Lambda g_j$ 
and Hecke pairs $\Lambda ^n,G_p^n$ can be found in \cite{AnZh95} chapter 3 paragraphs 
2 and 3, see also \cite{An87} chapter 3.  
For generating elements $\pi_i(p) = \pi_i^n(p) = (\diag(1,\dots ,1,p,\dots ,p))$ with 1 
on the diagonal listed $(n-i)$ times and then $p$ listed $i$ times ($1\le i \le n$) 
the images elements under the map $\omega = \omega_p^n$ are given by the formulas
\begin{equation}\nonumber
\omega (\pi_i^n(p))=p^{- \langle i \rangle }s_i(x_1,\dots ,x_n) \quad (1\le i \le n),
\end{equation}
where
\begin{equation}\nonumber
s_i(x_1,\dots ,x_n) = \sum_{1\le \alpha_1 < \cdots < \alpha_i \le n} x_{\alpha_1} \cdots x_{\alpha_i}
\end{equation}
is the $i$th elementary symmetric polynomial (different then previously defined $sym_{i_1,i_2,i_3}$).
For an arbitrary element $t$ the map is defined by
\begin{equation}\nonumber
\omega(t)=\sum_j a_j\omega ((\Lambda g_j)) \, .
\end{equation}

Examples of computation for cases $n=1$ and $n=2$ are given in \cite{Sh}, at page 824, and in the book \cite{AnZh95}, at page 150:
\begin{align}&
R_1(v)=[(1-x_0v)(1-x_0x_1v)]^{-1}\, ,\\
&
R_2(v)=[(1-x_0v)(1-x_0x_1v)(1-x_0x_2v)(1-x_0x_1x_2v)]^{-1}\left( 1-{\displaystyle \frac{x_0^2x_1x_2v^2}{p}}\right) \, .
\end{align}
In this paper we consider $n=3$.  Acting analogously to the case $n=2$ let 
\begin{equation}
\begin{split}
\delta_2 &=\delta_1+\delta_1^\prime\\
\delta_3 &=\delta_1+\delta_2^\prime\\
\delta &=\delta_1+\delta^\prime\\
\delta^\prime &=\delta_2^\prime+\beta
\end{split}
\end{equation}
where $0\le\delta_1^\prime\le\delta_2^\prime\le\delta^\prime$, $\beta\ge0$.  Then
\begin{equation}\nonumber
\begin{split}
&R_3(v)=\sum_{\delta =0}^\infty \, \sum_{0\le \delta_1 \le \delta_2 \le \delta_3 \le \delta} p^{3\delta_1 +2\delta_2 +\delta_3} \omega (t(p^{\delta_1},p^{\delta_2},p^{\delta_3})) (x_0 v)^\delta\\
&=\sum_{\delta_1\ge 0}\sum_{\beta\ge 0, 0\le\delta_1^\prime\le\delta_2^\prime} (x_0v)^{\delta_1+\delta_2^\prime+\beta} \left(\frac{x_1x_2x_3}{p^6}\right)^{\delta_1} \omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime})) p^{3\delta_1+2(\delta_1+\delta_1^\prime)+(\delta_1+\delta_2^\prime)}\\
&=\sum_{\delta_1\ge 0}\sum_{\beta\ge 0, 0\le\delta_1^\prime\le\delta_2^\prime} (x_0v)^{\delta_1+\delta_2^\prime+\beta} \left(\frac{x_1x_2x_3}{p^6}\right)^{\delta_1} p^{6\delta_1+2\delta_1^\prime+\delta_2^\prime} \omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime}))\\
&=\sum_{\delta_1\ge 0}\sum_{\beta\ge 0} (x_0vx_1x_2x_3)^{\delta_1} (x_0v)^\beta \sum_{0\le\delta_1^\prime\le\delta_2^\prime} 
\omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime})) p^{2\delta_1^\prime+\delta_2^\prime} (x_0v)^{\delta_2^\prime}\\
&=[(1-x_0v)(1-x_0x_1x_2x_3v)]^{-1} \sum_{0\le\delta_1^\prime\le\delta_2^\prime} \omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime})) 
p^{2\delta_1^\prime+\delta_2^\prime} (x_0v)^{\delta_2^\prime}\\
&=[(1-x_0v)(1-x_0x_1v)(1-x_0x_2v)(1-x_0x_3v)(1-x_0x_1x_2v)(1-x_0x_1x_3v)\\
&\ \ \ \ \cross(1-x_0x_2x_3v)(1-x_0x_1x_2x_3v)]^{-1} P_3(v)
\end{split}
\end{equation}
Here $P_3(v)$ denotes a polynomial of degree 6 as stated in the Theorem 6 (page 451) of \cite{An70}.  
This rational polynomial presentation is proved in \cite{An69} for Hecke series and $\zeta$-functions
of the groups $GL_n$ and $SP_n$ over local fields.  Further theory and applications were developed 
for genus 2 in the work \cite{An74}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It follows that
\begin{align}&
P_3(v)= \sum_{0\le\delta_1^\prime\le\delta_2^\prime} \omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime})) p^{2\delta_1^\prime+\delta_2^\prime} (x_0v)^{\delta_2^\prime}\times \\
&\nonumber \times [(1-x_0x_1v)(1-x_0x_2v)(1-x_0x_3v)(1-x_0x_1x_2v)(1-x_0x_1x_3v)(1-x_0x_2x_3v)],
\end{align}
and our main result follows from the explicit computation of  the coefficients $\omega(t(1,p^{\delta_1^\prime},p^{\delta_2^\prime}))$
given in the next section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\iffalse
Let us notice that 
\begin{align}
\nonumber &{\mathit{sym}_{2,1,1}}
={\mathit{sym}_{1,1,1}}\cdot
{\mathit{sym}_{1,0,0}}=p^7\omega(\pi_3\cdot \pi_1)
\\ 
\nonumber &{\mathit{sym}_{1,1,1}}
=p^6\omega(\pi_3)\\
\nonumber &{\mathit{sym}_{1,1,0}}
=p^3\omega(\pi_2)\\
 &{\mathit{sym}_{2,2,2}}
=p^{12}\omega(\pi_3^2)\\
\nonumber &{\mathit{sym}_{2,2,1}}
={\mathit{sym}_{1,1,1}}\cdot
{\mathit{sym}_{1,1,0}}=p^9\omega(\pi_3\cdot \pi_2)
\\
\nonumber &{\mathit{sym}_{3,2,2}} =
{\mathit{sym}_{2,2,2}}\cdot
{\mathit{sym}_{1,0,0}}=p^{13}\omega(\pi_3^2\cdot \pi_1)
\\
\nonumber &{\mathit{sym}_{3,3,3}}=p^{18}\omega(\pi_3^3)
\end{align}
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A computation for the images of the Hecke operators under the spherical map}
The formula for $P_3$ is obtained using the following computation for the images
$\omega ({t}(1,p^{\lambda_2},p^{\lambda_3}))$ of the Hecke operators under the spherical 
map for the group $\Lambda=GL_3(\Z)$.  
Note that the notation $\Omega$ used in the article \cite{An70}
corresponds to  $\omega$ in our formulas by the substitution of 
$x_1$ by $x_1/p$, $x_2$ by $x_2/p$ and $x_3$ by $x_3/p$.  
We used formula (1.7) for $\Omega$ at 
page 432 of \cite{An70} and adopted it for $\omega$.  
In the notations of that article we have that
$W$ is the group $S_n = S_3$, the set $\Sigma = \{(1,-1,0),(1,0,-1),(0,1,-1)\}$, $q=p$ and 
$\lambda = (0,\delta_1^{'},\delta_2^{'})$.  
The expression for the polynomial $c(x)$ from \cite{An70}
takes the following form
$$c(x_1,x_2,x_3)=\frac{(x_2-x_1/p)(x_3-x_1/p)(x_3-x_2/p)}{(x_2-x_1)(x_3-x_1)(x_3-x_2)}\,.$$
In the case of $n=3$ we have 28 different possibilities for $\omega(t(1,p^{\lambda_2},p^{\lambda_3}))$,
which we only need in order to compute the polynomial $P_3(v)$ of degree 6:

\begin{itemize}
\item[1)]
 $\omega(t(1,1,1))
 = 1$
\item[2)]
 $\omega(t(1,1,p))
 = {\displaystyle \frac{\mathit{sym}_{1,0,0}}{p}}$
\item[3)]
 $\omega(t(1,p,p))
 = {\displaystyle \frac{\mathit{sym}_{1,1,0}}{p^3}}$
\item[4)]
 $\omega(t(1,1,p^2))
 = {\displaystyle \frac{(p-1)\,\mathit{sym}_{1,1,0}}{p^3}}
 + {\displaystyle \frac{\mathit{sym}_{2,0,0}}{p^2}}$
\item[5)]
 $\omega(t(1,p,p^2))
 = {\displaystyle \frac{(2p^2-p-1)\,\mathit{sym}_{1,1,1}}{p^6}}
 + {\displaystyle \frac{\mathit{sym}_{2,1,0}}{p^4}}$
\item[6)]
 $\omega(t(1,p^2,p^2))
 = {\displaystyle \frac{(p-1)\,{\mathit{sym}_{2,1,1}}}{p^7}}
 + {\displaystyle \frac{\mathit{sym}_{2,2,0}}{p^6}}$
\item[7)]
 $\omega ({t}(1,1,p^{3}))
 = {\displaystyle \frac {(p^{2}  - 2\,p + 1)\,{\mathit{sym}_{1,1,1}}}{p^{5}}}
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{2,1,0}}}{p^{5}}}
 + {\displaystyle \frac {{\mathit{sym}_{3,0,0}}}{p^{3}}}$
\item[8)]
 $\omega ({t}(1,p,p^{3}))
 = {\displaystyle \frac {(2\,p-2)\,{\mathit{sym}_{2,1,1}}}{p^{6}}}
 + {\displaystyle \frac {(p - 1)\,{\mathit{sym}_{2,2,0}}}{p^{6}}}
 + {\displaystyle \frac {{\mathit{sym}_{3,1,0}}}{p^{5}}}$
\item[9)]
 $\omega ({t}(1,p^{2},p^{3}))
 = {\displaystyle \frac {(2\,p-2)\,{\mathit{sym}_{2,2,1}}}{p^{8}}}
 + {\displaystyle \frac {(p - 1)\,{\mathit{sym}_{3,1,1}}}{p^{8}}} 
 + {\displaystyle \frac {{\mathit{sym}_{3,2,0}}}{p^{7}}}$ 
\item[10)]
 $\omega ({t}(1,p^{3},p^{3}))
 = {\displaystyle \frac {(p^{2} -2\,p + 1)\,{\mathit{sym}_{2,2,2}}}{p^{11}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{3,2,1}}}{p^{11}}} 
 + {\displaystyle \frac {{\mathit{sym}_{3,3,0}}}{p^{9}}}$
\item[11)]
 $\omega ({t}(1,1,p^{4}))
 = {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{2,1,1}}}{p^{6}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{2,2,0}}}{p^{6}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{3,1,0}}}{p^{6}}} \\
 + {\displaystyle \frac {{\mathit{sym}_{4,0,0}}}{p^{4}}}$
\item[12)]
 $\omega ({t}(1,p,p^{4}))
 = {\displaystyle \frac {(2\,p^{2} - 3\,p+1)\,{\mathit{sym}_{2,2,1}}}{p^{8}}} 
 + {\displaystyle \frac {(2\,p^{2} - 2\,p)\,{\mathit{sym}_{3,1,1}}}{p^{8}}} 
 + {\displaystyle  \frac {(p^{2}-p)\,{\mathit{sym}_{3,2,0}}}{p^{8}}} \\
 + {\displaystyle  \frac {{\mathit{sym}_{4, \,1, \,0}}}{p^{6}}} $
\item[13)]
 $\omega ({t}(1,p^{2},p^{4}))
 = {\displaystyle \frac {( - 4\,p^{2} + 3\,p^{3} + 2\,p-1)\,{\mathit{sym}_{2,2,2}}}{p^{11}}} 
 + {\displaystyle \frac {(2\,p^3 - 3\,p^{2} + p)\,{\mathit{sym}_{3,2,1}}}{p^{11}}} \\	
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{3,3,0}}}{p^{11}}} 
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{4,1,1}}}{p^{11}}} 
 + {\displaystyle \frac {{\mathit{sym}_{4,2,0}}}{p^{8}}}$
\item[14)]
 $\omega ({t}(1,p^{3},p^{4}))
 = {\displaystyle \frac {(2\,p^{2} - 3\,p+ 1)\,{\mathit{sym}_{3,2,2}}}{p^{12}}} 
 + {\displaystyle \frac {(2\,p^{2} - 2\,p)\,{\mathit{sym}_{3,3,1}}}{p^{12}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{4,2,1}}}{p^{12}}} \\
 + {\displaystyle \frac {{\mathit{sym}_{4,3,0}}}{p^{10}}}$
\item[15)]
 $\omega ({t}(1,p^{4},p^{4}))
 = {\displaystyle \frac {(p^{2} -2\,p + 1)\,{\mathit{sym}_{3,3,2}}}{p^{14}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{4,2,2}}}{p^{14}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{4,3,1}}}{p^{14}}} \\
 + {\displaystyle \frac {{\mathit{sym}_{4,4,0}}}{p^{12}}}$
\item[16)]
 $\omega (t(1,1,p^{5}))
 = {\displaystyle \frac { p^{2} -2\,p+ 1)\,{\mathit{sym}_{2,2,1}}}{p^{7}}} 
 + {\displaystyle \frac {(p^{2} -2\,p+ 1)\,{\mathit{sym}_{3,1,1}}}{p^{7}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{3,2,0}}}{p^{7}}} \\
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{4,1,0}}}{p^{7}}} 
 + {\displaystyle \frac {{\mathit{sym}_{5,0,0}}}{p^{5}}}$
\item[17)]
 $\omega ({t}(1,p,p^{5}))
 = {\displaystyle \frac {(2\,p^2 - 4\,p + 2)\,{\mathit{sym}_{2,2,2}}}{p^{9}}} 
 + {\displaystyle \frac {(2\,p^{2} - 3\,p+1)\,{\mathit{sym}_{3,2,1}}}{p^{9}}}  
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{3,3,0}}}{p^{9}}} \\ 
 + {\displaystyle \frac {(2\,p^{2} - 2\,p)\,{\mathit{sym}_{4,1,1}}}{p^{9}}} 
 + {\displaystyle \frac {(p^{2}-p)\,{\mathit{sym}_{4,2,0}}}{p^{9}}} 
 + {\displaystyle \frac {{\mathit{sym}_{5,1,0}}}{p^{7}}}$
\item[18)]
 $\omega ({t}(1,p^{2},p^{5}))
 = {\displaystyle \frac {(3\,p^3 - 5\,p^{2} + 3\,p - 1)\,{\mathit{sym}_{3,2,2}}}{p^{12}}} 
 + {\displaystyle \frac {(2\,p^3 - 4\,p^{2} + 2\,p)\,{\mathit{sym}_{3,3,1}}}{p^{12}}}  \\
 + {\displaystyle \frac {(2\,p^3 - 3\,p^{2} + p)\,{\mathit{sym}_{4,2,1}}}{p^{12}}} 
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{4,3,0}}}{p^{12}}} 
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{5,1,1}}}{p^{12}}}  
 + {\displaystyle \frac {{\mathit{sym}_{5,2,0}}}{p^{9}}}$
\item[19)]
 $\omega ({t}(1,p^{3},p^{5}))
 = {\displaystyle \frac {(3\,p^3 - 5\,p^{2} + 3\,p - 1)\,{\mathit{sym}_{3,3,2}}}{p^{14}}} 
 + {\displaystyle \frac {(2\,p^3 - 4\,p^{2} + 2\,p)\,{\mathit{sym}_{4,2,2}}}{p^{14}}} \\  
 + {\displaystyle \frac {(2\,p^3 - 3\,p^{2} + p)\,{\mathit{sym}_{4,3,1}}}{p^{14}}} 
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{4,4,0}}}{p^{14}}} 
 + {\displaystyle \frac {(p^{3} - p^{2})\,{\mathit{sym}_{5,2,1}}}{p^{14}}} 
 + {\displaystyle \frac {{\mathit{sym}_{5,3,0}}}{p^{11}}}$
\item[20)]
 $\omega ({t}(1,p^{4},p^{5}))
 = {\displaystyle \frac {(2\,p^2 - 4\,p + 2)\,{\mathit{sym}_{3,3,3}}}{p^{15}}} 
 + {\displaystyle \frac {(2\,p^2 - 3\,p + 1)\,{\mathit{sym}_{4,3,2}}}{p^{15}}} \\  
 + {\displaystyle \frac {(2\,p^2 - 2\,p    )\,{\mathit{sym}_{4,4,1}}}{p^{15}}} 
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{5,2,2}}}{p^{15}}} 
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{5,3,1}}}{p^{15}}} 
 + {\displaystyle \frac {                     {\mathit{sym}_{5,4,0}}}{p^{13}}}$
\item[21)]
 $\omega ({t}(1,p^{5},p^{5}))
 = {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{4,3,3}}}{p^{17}}} 
 + {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{4,4,2}}}{p^{17}}} 
 + {\displaystyle \frac {(p^2 -    p    )\,{\mathit{sym}_{5,3,2}}}{p^{17}}} \\ 
 + {\displaystyle \frac {(p^2 -    p    )\,{\mathit{sym}_{5,4,1}}}{p^{17}}} 
 + {\displaystyle \frac {                  {\mathit{sym}_{5,5,0}}}{p^{15}}}$
\item[22)]
 $\omega ({t}(1,1,p^{6}))
 = {\displaystyle \frac {(p^2-2\,p+1)\,{\mathit{sym}_{2,2,2}}}{p^{8}}} 
 + {\displaystyle \frac {(p^2-2\,p+1)\,{\mathit{sym}_{3,2,1}}}{p^{8}}}  
 + {\displaystyle \frac {(p^2-   p  )\,{\mathit{sym}_{3,3,0}}}{p^{8}}} \\ 
 + {\displaystyle \frac {(p^2-2\,p+1)\,{\mathit{sym}_{4,1,1}}}{p^{8}}} 
 + {\displaystyle \frac {(p^2-   p  )\,{\mathit{sym}_{4,2,0}}}{p^{8}}}  
 + {\displaystyle \frac {(p^2-   p  )\,{\mathit{sym}_{5,1,0}}}{p^{8}}} 
 + {\displaystyle \frac {              {\mathit{sym}_{6,0,0}}}{p^{6}}}$
\item[23)]
 $\omega ({t}(1,p,p^{6}))
 = {\displaystyle \frac {(2\,p^2 - 4\,p + 2)\,{\mathit{sym}_{3,2,2}}}{p^{10}}} 
 + {\displaystyle \frac {(2\,p^2 - 3\,p + 1)\,{\mathit{sym}_{3,3,1}}}{p^{10}}} \\ 
 + {\displaystyle \frac {(2\,p^2 - 3\,p + 1)\,{\mathit{sym}_{4,2,1}}}{p^{10}}} 
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{4,3,0}}}{p^{10}}} 
 + {\displaystyle \frac {(2\,p^2 - 2\,p    )\,{\mathit{sym}_{5,1,1}}}{p^{10}}} \\
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{5,2,0}}}{p^{10}}} 
 + {\displaystyle \frac {                     {\mathit{sym}_{6,1,0}}}{p^{8}}}$
\item[24)]
 $\omega ({t}(1,p^{2},p^{6}))
 = {\displaystyle \frac {(3\,p^3 - 6\,p^2 + 4\,p - 1)\,{\mathit{sym}_{3,3,2}}}{p^{13}}}  
 + {\displaystyle \frac {(3\,p^3 - 5\,p^2 + 3\,p - 1)\,{\mathit{sym}_{4,2,2}}}{p^{13}}} \\
 + {\displaystyle \frac {(2\,p^3 - 4\,p^2 + 2\,p    )\,{\mathit{sym}_{4,3,1}}}{p^{13}}}  
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{4,4,0}}}{p^{13}}} 
 + {\displaystyle \frac {(2\,p^3 - 3\,p^2 +    p    )\,{\mathit{sym}_{5,2,1}}}{p^{13}}} \\
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{5,3,0}}}{p^{13}}} 
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{6,1,1}}}{p^{13}}} 
 + {\displaystyle \frac {                              {\mathit{sym}_{6,2,0}}}{p^{10}}}$
\item[25)]
 $\omega ({t}(1,p^{3},p^{6}))
 = {\displaystyle \frac {(4\,p^3 - 7\,p^2 + 5\,p - 2)\,{\mathit{sym}_{3,3,3}}}{p^{15}}}  
 + {\displaystyle \frac {(3\,p^3 - 6\,p^2 + 4\,p - 1)\,{\mathit{sym}_{4,3,2}}}{p^{15}}} \\
 + {\displaystyle \frac {(2\,p^3 - 4\,p^2 + 2\,p    )\,{\mathit{sym}_{4,4,1}}}{p^{15}}}  
 + {\displaystyle \frac {(2\,p^3 - 4\,p^2 + 2\,p    )\,{\mathit{sym}_{5,2,2}}}{p^{15}}} 
 + {\displaystyle \frac {(2\,p^3 - 3\,p^2 +    p    )\,{\mathit{sym}_{5,3,1}}}{p^{15}}} \\
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{5,4,0}}}{p^{15}}}  
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{6,2,1}}}{p^{15}}} 
 + {\displaystyle \frac {                              {\mathit{sym}_{6,3,0}}}{p^{12}}}$
\item[26)]
 $\omega ({t}(1,p^{4},p^{6}))
 = {\displaystyle \frac {(3\,p^3 - 6\,p^2 + 4\,p - 1)\,{\mathit{sym}_{4,3,3}}}{p^{17}}}  
 + {\displaystyle \frac {(3\,p^3 - 5\,p^2 + 3\,p - 1)\,{\mathit{sym}_{4,4,2}}}{p^{17}}} \\
 + {\displaystyle \frac {(2\,p^3 - 4\,p^2 + 2\,p    )\,{\mathit{sym}_{5,3,2}}}{p^{17}}}  
 + {\displaystyle \frac {(2\,p^3 - 3\,p^2 +    p    )\,{\mathit{sym}_{5,4,1}}}{p^{17}}} 
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{5,5,0}}}{p^{17}}} \\
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{6,2,2}}}{p^{17}}} 
 + {\displaystyle \frac {(   p^3 -    p^2           )\,{\mathit{sym}_{6,3,1}}}{p^{17}}} 
 + {\displaystyle \frac {                              {\mathit{sym}_{6,4,0}}}{p^{14}}}$
\item[27)]
 $\omega ({t}(1,p^{5},p^{6}))
 = {\displaystyle \frac {(2\,p^2 - 4\,p + 2)\,{\mathit{sym}_{4,4,3}}}{p^{18}}} 
 + {\displaystyle \frac {(2\,p^2 - 3\,p + 1)\,{\mathit{sym}_{5,3,3}}}{p^{18}}} \\ 
 + {\displaystyle \frac {(2\,p^2 - 3\,p + 1)\,{\mathit{sym}_{5,4,2}}}{p^{18}}} 
 + {\displaystyle \frac {(2\,p^2 - 2\,p    )\,{\mathit{sym}_{5,5,1}}}{p^{18}}} 
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{6,3,2}}}{p^{18}}} \\
 + {\displaystyle \frac {(   p^2 -    p    )\,{\mathit{sym}_{6,4,1}}}{p^{18}}} 
 + {\displaystyle \frac {                     {\mathit{sym}_{6,5,0}}}{p^{16}}}$
\item[28)]
 $\omega ({t}(1,p^{6},p^{6}))
 = {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{4,4,4}}}{p^{20}}} 
 + {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{5,4,3}}}{p^{20}}} \\
 + {\displaystyle \frac {(p^2 - 2\,p + 1)\,{\mathit{sym}_{5,5,2}}}{p^{20}}} 
 + {\displaystyle \frac {(p^2 -    p    )\,{\mathit{sym}_{6,3,3}}}{p^{20}}} 
 + {\displaystyle \frac {(p^2 -    p    )\,{\mathit{sym}_{6,4,2}}}{p^{20}}} \\
 + {\displaystyle \frac {(p^2 -    p    )\,{\mathit{sym}_{6,5,1}}}{p^{20}}} 
 + {\displaystyle \frac {                  {\mathit{sym}_{6,6,0}}}{p^{18}}}$ \, . \qed
\end{itemize}

Note that Rhodes, J. A. and Shemanske, T. R. developed an alternative method of computing
$\omega(t(p^{\delta_1},\dots ,p^{\delta_n}))$ in \cite{RhSh}, based on counting of certain left
cosets in a given double coset (Theorem 4.3 of \cite{RhSh}). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse
\section{The use of the standard symmetric polynomials}
The standard symmetric polynomials were used 
with Maple 9.50 (IBM INTEL NT)

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 1, sym[0,0,0] = 1;}{%
\[
i=1, \,{\mathit{sym}_{0, \,0, \,0}}=1
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 2, sym[1,0,0] = {x}_1+{x}_2+x_3;}{%
\[
i=2, \,{\mathit{sym}_{1, \,0, \,0}}=\mathit{{x}_1} + \mathit{x}_2 + 
\mathit{x_3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 3, sym[1,1,0] = {x}_1*{x}_2+{x}_1*x_3+{x}_2*x_3;}{%
\[
i=3, \,{\mathit{sym}_{1, \,1, \,0}}=\mathit{{x}_1}\,\mathit{x}_2 + 
\mathit{{x}_1}\,\mathit{x_3} + \mathit{x}_2\,\mathit{x_3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 4, sym[1,1,1] = {x}_1*{x}_2*x_3;}{%
\[
i=4, \,{\mathit{sym}_{1, \,1, \,1}}=\mathit{{x}_1}\,\mathit{x}_2\,
\mathit{x_3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 5, sym[2,0,0] = {x}_1^2+{x}_2^2+x_3^2;}{%
\[
i=5, \,{\mathit{sym}_{2, \,0, \,0}}=\mathit{{x}_1^2} + \mathit{x}_2
^{2} + \mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 6, sym[2,1,0] =
{x}_1^2*{x}_2+{x}_1^2*x_3+{x}_1*{x}_2^2+{x}_1*x_3^2+{x}_2^2*x_3+{x}_2*x_3^2;}{%
\[
i=6, \,{\mathit{sym}_{2, \,1, \,0}}=\mathit{{x}_1^2}\,\mathit{x}_2
 + \mathit{{x}_1^2}\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^2} + 
\mathit{{x}_1}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,\mathit{x_3} + 
\mathit{x}_2\,\mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 7, sym[2,1,1] = {x}_1^2*{x}_2*x_3+{x}_1*{x}_2^2*x_3+{x}_1*{x}_2*x_3^2;}{%
\[
i=7, \,{\mathit{sym}_{2, \,1, \,1}}=\mathit{{x}_1^2}\,\mathit{x}_2
\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^2}\,\mathit{x_3} + 
\mathit{{x}_1}\,\mathit{x}_2\,\mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 8, sym[2,2,0] = {x}_1^2*{x}_2^2+{x}_1^2*x_3^2+{x}_2^2*x_3^2;}{%
\[
i=8, \,{\mathit{sym}_{2, \,2, \,0}}=\mathit{{x}_1^2}\,\mathit{x}_2
^{2} + \mathit{{x}_1^2}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,
\mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 9, sym[2,2,1] = {x}_1^2*{x}_2^2*x_3+{x}_1^2*{x}_2*x_3^2+{x}_1*{x}_2^2*x_3^2;}{%
\[
i=9, \,{\mathit{sym}_{2, \,2, \,1}}=\mathit{{x}_1^2}\,\mathit{x}_2
^{2}\,\mathit{x_3} + \mathit{{x}_1^2}\,\mathit{x}_2\,\mathit{x_3^2}
 + \mathit{{x}_1}\,\mathit{{x}_2^2}\,\mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 10, sym[2,2,2] = {x}_1^2*{x}_2^2*x_3^2;}{%
\[
i=10, \,{\mathit{sym}_{2, \,2, \,2}}=\mathit{{x}_1^2}\,\mathit{x}_2
^{2}\,\mathit{x_3^2}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 11, sym[3,0,0] = {x}_1^3+{x}_2^3+x_3^3;}{%
\[
i=11, \,{\mathit{sym}_{3, \,0, \,0}}=\mathit{{x}_1^3} + \mathit{{x}_2
^3} + \mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 12, sym[3,1,0] =
{x}_1^3*{x}_2+{x}_1^3*x_3+{x}_1*{x}_2^3+{x}_1*x_3^3+{x}_2^3*x_3+{x}_2*x_3^3;}{%
\[
i=12, \,{\mathit{sym}_{3, \,1, \,0}}=\mathit{{x}_1^3}\,\mathit{x}_2
 + \mathit{{x}_1^3}\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^3} + 
\mathit{{x}_1}\,\mathit{x_3^3} + \mathit{{x}_2^3}\,\mathit{x_3} + 
\mathit{x}_2\,\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 13, sym[3,1,1] = {x}_1^3*{x}_2*x_3+{x}_1*{x}_2^3*x_3+{x}_1*{x}_2*x_3^3;}{%
\[
i=13, \,{\mathit{sym}_{3, \,1, \,1}}=\mathit{{x}_1^3}\,\mathit{x}_2
\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^3}\,\mathit{x_3} + 
\mathit{{x}_1}\,\mathit{x}_2\,\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 14, sym[3,2,0] =
{x}_1^3*{x}_2^2+{x}_1^3*x_3^2+{x}_1^2*{x}_2^3+{x}_1^2*x_3^3+{x}_2^3*x_3^2+{x}_2^2*x_3^3;}{%
\[
i=14, \,{\mathit{sym}_{3, \,2, \,0}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{2} + \mathit{{x}_1^3}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,
\mathit{{x}_2^3} + \mathit{{x}_1^2}\,\mathit{x_3^3} + \mathit{x}_2
^{3}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 15, sym[3,2,1] =
{x}_1^3*{x}_2^2*x_3+{x}_1^3*{x}_2*x_3^2+{x}_1^2*{x}_2^3*x_3+{x}_1^2*{x}_2*x_3^3+{x}_1*{x}_2^3*x_3^2+{x}_1*{x}_2
^2*x_3^3;}{%
\maplemultiline{
i=15, {\mathit{sym}_{3, \,2, \,1}}= \\
\mathit{{x}_1^3}\,\mathit{{x}_2^2}\,\mathit{x_3} + \mathit{{x}_1^3}\,
\mathit{x}_2\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,
\mathit{x_3} + \mathit{{x}_1^2}\,\mathit{x}_2\,\mathit{x_3^3} + 
\mathit{{x}_1}\,\mathit{{x}_2^3}\,\mathit{x_3^2} + \mathit{{x}_1}\,
\mathit{{x}_2^2}\,\mathit{x_3^3} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 16, sym[3,2,2] = {x}_1^3*{x}_2^2*x_3^2+{x}_1^2*{x}_2^3*x_3^2+{x}_1^2*{x}_2^2*x_3^3;}{%
\[
i=16, \,{\mathit{sym}_{3, \,2, \,2}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{2}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,
\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^2}\,\mathit{x_3^
3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 17, sym[3,3,0] = {x}_1^3*{x}_2^3+{x}_1^3*x_3^3+{x}_2^3*x_3^3;}{%
\[
i=17, \,{\mathit{sym}_{3, \,3, \,0}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{3} + \mathit{{x}_1^3}\,\mathit{x_3^3} + \mathit{{x}_2^3}\,
\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 18, sym[3,3,1] = {x}_1^3*{x}_2^3*x_3+{x}_1^3*{x}_2*x_3^3+{x}_1*{x}_2^3*x_3^3;}{%
\[
i=18, \,{\mathit{sym}_{3, \,3, \,1}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{3}\,\mathit{x_3} + \mathit{{x}_1^3}\,\mathit{x}_2\,\mathit{x_3^3}
 + \mathit{{x}_1}\,\mathit{{x}_2^3}\,\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 19, sym[3,3,2] = {x}_1^3*{x}_2^3*x_3^2+{x}_1^3*{x}_2^2*x_3^3+{x}_1^2*{x}_2^3*x_3^3;}{%
\[
i=19, \,{\mathit{sym}_{3, \,3, \,2}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{3}\,\mathit{x_3^2} + \mathit{{x}_1^3}\,\mathit{{x}_2^2}\,
\mathit{x_3^3} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,\mathit{x_3^
3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 20, sym[3,3,3] = {x}_1^3*{x}_2^3*x_3^3;}{%
\[
i=20, \,{\mathit{sym}_{3, \,3, \,3}}=\mathit{{x}_1^3}\,\mathit{x}_2
^{3}\,\mathit{x_3^3}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 21, sym[4,0,0] = {x}_1^4+{x}_2^4+x_3^4;}{%
\[
i=21, \,{\mathit{sym}_{4, \,0, \,0}}=\mathit{{x}_1^4} + \mathit{{x}_2
^4} + \mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 22, sym[4,1,0] =
{x}_1^4*{x}_2+{x}_1^4*x_3+{x}_1*{x}_2^4+{x}_1*x_3^4+{x}_2^4*x_3+{x}_2*x_3^4;}{%
\[
i=22, \,{\mathit{sym}_{4, \,1, \,0}}=\mathit{{x}_1^4}\,\mathit{x}_2
 + \mathit{{x}_1^4}\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^4} + 
\mathit{{x}_1}\,\mathit{x_3^4} + \mathit{{x}_2^4}\,\mathit{x_3} + 
\mathit{x}_2\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 23, sym[4,1,1] = {x}_1^4*{x}_2*x_3+{x}_1*{x}_2^4*x_3+{x}_1*{x}_2*x_3^4;}{%
\[
i=23, \,{\mathit{sym}_{4, \,1, \,1}}=\mathit{{x}_1^4}\,\mathit{x}_2
\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^4}\,\mathit{x_3} + 
\mathit{{x}_1}\,\mathit{x}_2\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 24, sym[4,2,0] =
{x}_1^4*{x}_2^2+{x}_1^4*x_3^2+{x}_1^2*{x}_2^4+{x}_1^2*x_3^4+{x}_2^4*x_3^2+{x}_2^2*x_3^4;}{%
\[
i=24, \,{\mathit{sym}_{4, \,2, \,0}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{2} + \mathit{{x}_1^4}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,
\mathit{{x}_2^4} + \mathit{{x}_1^2}\,\mathit{x_3^4} + \mathit{x}_2
^{4}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 25, sym[4,2,1] =
{x}_1^4*{x}_2^2*x_3+{x}_1^4*{x}_2*x_3^2+{x}_1^2*{x}_2^4*x_3+{x}_1^2*{x}_2*x_3^4+{x}_1*{x}_2^4*x_3^2+{x}_1*{x}_2
^2*x_3^4;}{%
\maplemultiline{
i=25, {\mathit{sym}_{4, \,2, \,1}}= \\
\mathit{{x}_1^4}\,\mathit{{x}_2^2}\,\mathit{x_3} + \mathit{{x}_1^4}\,
\mathit{x}_2\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,
\mathit{x_3} + \mathit{{x}_1^2}\,\mathit{x}_2\,\mathit{x_3^4} + 
\mathit{{x}_1}\,\mathit{{x}_2^4}\,\mathit{x_3^2} + \mathit{{x}_1}\,
\mathit{{x}_2^2}\,\mathit{x_3^4} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 26, sym[4,2,2] = {x}_1^4*{x}_2^2*x_3^2+{x}_1^2*{x}_2^4*x_3^2+{x}_1^2*{x}_2^2*x_3^4;}{%
\[
i=26, \,{\mathit{sym}_{4, \,2, \,2}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{2}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,
\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^2}\,\mathit{x_3^
4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 27, sym[4,3,0] =
{x}_1^4*{x}_2^3+{x}_1^4*x_3^3+{x}_1^3*{x}_2^4+{x}_1^3*x_3^4+{x}_2^4*x_3^3+{x}_2^3*x_3^4;}{%
\[
i=27, \,{\mathit{sym}_{4, \,3, \,0}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{3} + \mathit{{x}_1^4}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,
\mathit{{x}_2^4} + \mathit{{x}_1^3}\,\mathit{x_3^4} + \mathit{x}_2
^{4}\,\mathit{x_3^3} + \mathit{{x}_2^3}\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 28, sym[4,3,1] =
{x}_1^4*{x}_2^3*x_3+{x}_1^4*{x}_2*x_3^3+{x}_1^3*{x}_2^4*x_3+{x}_1^3*{x}_2*x_3^4+{x}_1*{x}_2^4*x_3^3+{x}_1*{x}_2
^3*x_3^4;}{%
\maplemultiline{
i=28, {\mathit{sym}_{4, \,3, \,1}}= \\
\mathit{{x}_1^4}\,\mathit{{x}_2^3}\,\mathit{x_3} + \mathit{{x}_1^4}\,
\mathit{x}_2\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^4}\,
\mathit{x_3} + \mathit{{x}_1^3}\,\mathit{x}_2\,\mathit{x_3^4} + 
\mathit{{x}_1}\,\mathit{{x}_2^4}\,\mathit{x_3^3} + \mathit{{x}_1}\,
\mathit{{x}_2^3}\,\mathit{x_3^4} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 29, sym[4,3,2] =
{x}_1^4*{x}_2^3*x_3^2+{x}_1^4*{x}_2^2*x_3^3+{x}_1^3*{x}_2^4*x_3^2+{x}_1^3*{x}_2^2*x_3^4+{x}_1^2*{x}_2^4*
x_3^3+{x}_1^2*{x}_2^3*x_3^4;}{%
\maplemultiline{
i=29, {\mathit{sym}_{4, \,3, \,2}}= \\
\mathit{{x}_1^4}\,\mathit{{x}_2^3}\,\mathit{x_3^2} + \mathit{{x}_1^
4}\,\mathit{{x}_2^2}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{
{x}_2^4}\,\mathit{x_3^2} + \mathit{{x}_1^3}\,\mathit{{x}_2^2}\,
\mathit{x_3^4} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,\mathit{x_3^
3} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,\mathit{x_3^4} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 30, sym[4,3,3] = {x}_1^4*{x}_2^3*x_3^3+{x}_1^3*{x}_2^4*x_3^3+{x}_1^3*{x}_2^3*x_3^4;}{%
\[
i=30, \,{\mathit{sym}_{4, \,3, \,3}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{3}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^4}\,
\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^3}\,\mathit{x_3^
4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 31, sym[4,4,0] = {x}_1^4*{x}_2^4+{x}_1^4*x_3^4+{x}_2^4*x_3^4;}{%
\[
i=31, \,{\mathit{sym}_{4, \,4, \,0}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{4} + \mathit{{x}_1^4}\,\mathit{x_3^4} + \mathit{{x}_2^4}\,
\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 32, sym[4,4,1] = {x}_1^4*{x}_2^4*x_3+{x}_1^4*{x}_2*x_3^4+{x}_1*{x}_2^4*x_3^4;}{%
\[
i=32, \,{\mathit{sym}_{4, \,4, \,1}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{4}\,\mathit{x_3} + \mathit{{x}_1^4}\,\mathit{x}_2\,\mathit{x_3^4}
 + \mathit{{x}_1}\,\mathit{{x}_2^4}\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 33, sym[4,4,2] = {x}_1^4*{x}_2^4*x_3^2+{x}_1^4*{x}_2^2*x_3^4+{x}_1^2*{x}_2^4*x_3^4;}{%
\[
i=33, \,{\mathit{sym}_{4, \,4, \,2}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{4}\,\mathit{x_3^2} + \mathit{{x}_1^4}\,\mathit{{x}_2^2}\,
\mathit{x_3^4} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,\mathit{x_3^
4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 34, sym[4,4,3] = {x}_1^4*{x}_2^4*x_3^3+{x}_1^4*{x}_2^3*x_3^4+{x}_1^3*{x}_2^4*x_3^4;}{%
\[
i=34, \,{\mathit{sym}_{4, \,4, \,3}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{4}\,\mathit{x_3^3} + \mathit{{x}_1^4}\,\mathit{{x}_2^3}\,
\mathit{x_3^4} + \mathit{{x}_1^3}\,\mathit{{x}_2^4}\,\mathit{x_3^
4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 35, sym[4,4,4] = {x}_1^4*{x}_2^4*x_3^4;}{%
\[
i=35, \,{\mathit{sym}_{4, \,4, \,4}}=\mathit{{x}_1^4}\,\mathit{x}_2
^{4}\,\mathit{x_3^4}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 36, sym[5,0,0] = {x}_1^5+{x}_2^5+x_3^5;}{%
\[
i=36, \,{\mathit{sym}_{5, \,0, \,0}}=\mathit{{x}_1^5} + \mathit{{x}_2
^5} + \mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 37, sym[5,1,0] =
{x}_1^5*{x}_2+{x}_1^5*x_3+{x}_1*{x}_2^5+{x}_1*x_3^5+{x}_2^5*x_3+{x}_2*x_3^5;}{%
\[
i=37, \,{\mathit{sym}_{5, \,1, \,0}}=\mathit{{x}_1^5}\,\mathit{x}_2
 + \mathit{{x}_1^5}\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^5} + 
\mathit{{x}_1}\,\mathit{x_3^5} + \mathit{{x}_2^5}\,\mathit{x_3} + 
\mathit{x}_2\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 38, sym[5,1,1] = {x}_1^5*{x}_2*x_3+{x}_1*{x}_2^5*x_3+{x}_1*{x}_2*x_3^5;}{%
\[
i=38, \,{\mathit{sym}_{5, \,1, \,1}}=\mathit{{x}_1^5}\,\mathit{x}_2
\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^5}\,\mathit{x_3} + 
\mathit{{x}_1}\,\mathit{x}_2\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 39, sym[5,2,0] =
{x}_1^5*{x}_2^2+{x}_1^5*x_3^2+{x}_1^2*{x}_2^5+{x}_1^2*x_3^5+{x}_2^5*x_3^2+{x}_2^2*x_3^5;}{%
\[
i=39, \,{\mathit{sym}_{5, \,2, \,0}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{2} + \mathit{{x}_1^5}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,
\mathit{{x}_2^5} + \mathit{{x}_1^2}\,\mathit{x_3^5} + \mathit{x}_2
^{5}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 40, sym[5,2,1] =
{x}_1^5*{x}_2^2*x_3+{x}_1^5*{x}_2*x_3^2+{x}_1^2*{x}_2^5*x_3+{x}_1^2*{x}_2*x_3^5+{x}_1*{x}_2^5*x_3^2+{x}_1*{x}_2
^2*x_3^5;}{%
\maplemultiline{
i=40, {\mathit{sym}_{5, \,2, \,1}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^2}\,\mathit{x_3} + \mathit{{x}_1^5}\,
\mathit{x}_2\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,
\mathit{x_3} + \mathit{{x}_1^2}\,\mathit{x}_2\,\mathit{x_3^5} + 
\mathit{{x}_1}\,\mathit{{x}_2^5}\,\mathit{x_3^2} + \mathit{{x}_1}\,
\mathit{{x}_2^2}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 41, sym[5,2,2] = {x}_1^5*{x}_2^2*x_3^2+{x}_1^2*{x}_2^5*x_3^2+{x}_1^2*{x}_2^2*x_3^5;}{%
\[
i=41, \,{\mathit{sym}_{5, \,2, \,2}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{2}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,
\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^2}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 42, sym[5,3,0] =
{x}_1^5*{x}_2^3+{x}_1^5*x_3^3+{x}_1^3*{x}_2^5+{x}_1^3*x_3^5+{x}_2^5*x_3^3+{x}_2^3*x_3^5;}{%
\[
i=42, \,{\mathit{sym}_{5, \,3, \,0}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{3} + \mathit{{x}_1^5}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,
\mathit{{x}_2^5} + \mathit{{x}_1^3}\,\mathit{x_3^5} + \mathit{x}_2
^{5}\,\mathit{x_3^3} + \mathit{{x}_2^3}\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 43, sym[5,3,1] =
{x}_1^5*{x}_2^3*x_3+{x}_1^5*{x}_2*x_3^3+{x}_1^3*{x}_2^5*x_3+{x}_1^3*{x}_2*x_3^5+{x}_1*{x}_2^5*x_3^3+{x}_1*{x}_2
^3*x_3^5;}{%
\maplemultiline{
i=43, {\mathit{sym}_{5, \,3, \,1}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^3}\,\mathit{x_3} + \mathit{{x}_1^5}\,
\mathit{x}_2\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^5}\,
\mathit{x_3} + \mathit{{x}_1^3}\,\mathit{x}_2\,\mathit{x_3^5} + 
\mathit{{x}_1}\,\mathit{{x}_2^5}\,\mathit{x_3^3} + \mathit{{x}_1}\,
\mathit{{x}_2^3}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 44, sym[5,3,2] =
{x}_1^5*{x}_2^3*x_3^2+{x}_1^5*{x}_2^2*x_3^3+{x}_1^3*{x}_2^5*x_3^2+{x}_1^3*{x}_2^2*x_3^5+{x}_1^2*{x}_2^5*
x_3^3+{x}_1^2*{x}_2^3*x_3^5;}{%
\maplemultiline{
i=44, {\mathit{sym}_{5, \,3, \,2}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^3}\,\mathit{x_3^2} + \mathit{{x}_1^
5}\,\mathit{{x}_2^2}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{
{x}_2^5}\,\mathit{x_3^2} + \mathit{{x}_1^3}\,\mathit{{x}_2^2}\,
\mathit{x_3^5} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,\mathit{x_3^
3} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 45, sym[5,3,3] = {x}_1^5*{x}_2^3*x_3^3+{x}_1^3*{x}_2^5*x_3^3+{x}_1^3*{x}_2^3*x_3^5;}{%
\[
i=45, \,{\mathit{sym}_{5, \,3, \,3}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{3}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^5}\,
\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^3}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 46, sym[5,4,0] =
{x}_1^5*{x}_2^4+{x}_1^5*x_3^4+{x}_1^4*{x}_2^5+{x}_1^4*x_3^5+{x}_2^5*x_3^4+{x}_2^4*x_3^5;}{%
\[
i=46, \,{\mathit{sym}_{5, \,4, \,0}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{4} + \mathit{{x}_1^5}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,
\mathit{{x}_2^5} + \mathit{{x}_1^4}\,\mathit{x_3^5} + \mathit{x}_2
^{5}\,\mathit{x_3^4} + \mathit{{x}_2^4}\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 47, sym[5,4,1] =
{x}_1^5*{x}_2^4*x_3+{x}_1^5*{x}_2*x_3^4+{x}_1^4*{x}_2^5*x_3+{x}_1^4*{x}_2*x_3^5+{x}_1*{x}_2^5*x_3^4+{x}_1*{x}_2
^4*x_3^5;}{%
\maplemultiline{
i=47, {\mathit{sym}_{5, \,4, \,1}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^4}\,\mathit{x_3} + \mathit{{x}_1^5}\,
\mathit{x}_2\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^5}\,
\mathit{x_3} + \mathit{{x}_1^4}\,\mathit{x}_2\,\mathit{x_3^5} + 
\mathit{{x}_1}\,\mathit{{x}_2^5}\,\mathit{x_3^4} + \mathit{{x}_1}\,
\mathit{{x}_2^4}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 48, sym[5,4,2] =
{x}_1^5*{x}_2^4*x_3^2+{x}_1^5*{x}_2^2*x_3^4+{x}_1^4*{x}_2^5*x_3^2+{x}_1^4*{x}_2^2*x_3^5+{x}_1^2*{x}_2^5*
x_3^4+{x}_1^2*{x}_2^4*x_3^5;}{%
\maplemultiline{
i=48, {\mathit{sym}_{5, \,4, \,2}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^4}\,\mathit{x_3^2} + \mathit{{x}_1^
5}\,\mathit{{x}_2^2}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{
{x}_2^5}\,\mathit{x_3^2} + \mathit{{x}_1^4}\,\mathit{{x}_2^2}\,
\mathit{x_3^5} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,\mathit{x_3^
4} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 49, sym[5,4,3] =
{x}_1^5*{x}_2^4*x_3^3+{x}_1^5*{x}_2^3*x_3^4+{x}_1^4*{x}_2^5*x_3^3+{x}_1^4*{x}_2^3*x_3^5+{x}_1^3*{x}_2^5*
x_3^4+{x}_1^3*{x}_2^4*x_3^5;}{%
\maplemultiline{
i=49, {\mathit{sym}_{5, \,4, \,3}}= \\
\mathit{{x}_1^5}\,\mathit{{x}_2^4}\,\mathit{x_3^3} + \mathit{{x}_1^
5}\,\mathit{{x}_2^3}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{
{x}_2^5}\,\mathit{x_3^3} + \mathit{{x}_1^4}\,\mathit{{x}_2^3}\,
\mathit{x_3^5} + \mathit{{x}_1^3}\,\mathit{{x}_2^5}\,\mathit{x_3^
4} + \mathit{{x}_1^3}\,\mathit{{x}_2^4}\,\mathit{x_3^5} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 50, sym[5,4,4] = {x}_1^5*{x}_2^4*x_3^4+{x}_1^4*{x}_2^5*x_3^4+{x}_1^4*{x}_2^4*x_3^5;}{%
\[
i=50, \,{\mathit{sym}_{5, \,4, \,4}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{4}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^5}\,
\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^4}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 51, sym[5,5,0] = {x}_1^5*{x}_2^5+{x}_1^5*x_3^5+{x}_2^5*x_3^5;}{%
\[
i=51, \,{\mathit{sym}_{5, \,5, \,0}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5} + \mathit{{x}_1^5}\,\mathit{x_3^5} + \mathit{{x}_2^5}\,
\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 52, sym[5,5,1] = {x}_1^5*{x}_2^5*x_3+{x}_1^5*{x}_2*x_3^5+{x}_1*{x}_2^5*x_3^5;}{%
\[
i=52, \,{\mathit{sym}_{5, \,5, \,1}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5}\,\mathit{x_3} + \mathit{{x}_1^5}\,\mathit{x}_2\,\mathit{x_3^5}
 + \mathit{{x}_1}\,\mathit{{x}_2^5}\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 53, sym[5,5,2] = {x}_1^5*{x}_2^5*x_3^2+{x}_1^5*{x}_2^2*x_3^5+{x}_1^2*{x}_2^5*x_3^5;}{%
\[
i=53, \,{\mathit{sym}_{5, \,5, \,2}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5}\,\mathit{x_3^2} + \mathit{{x}_1^5}\,\mathit{{x}_2^2}\,
\mathit{x_3^5} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 54, sym[5,5,3] = {x}_1^5*{x}_2^5*x_3^3+{x}_1^5*{x}_2^3*x_3^5+{x}_1^3*{x}_2^5*x_3^5;}{%
\[
i=54, \,{\mathit{sym}_{5, \,5, \,3}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5}\,\mathit{x_3^3} + \mathit{{x}_1^5}\,\mathit{{x}_2^3}\,
\mathit{x_3^5} + \mathit{{x}_1^3}\,\mathit{{x}_2^5}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 55, sym[5,5,4] = {x}_1^5*{x}_2^5*x_3^4+{x}_1^5*{x}_2^4*x_3^5+{x}_1^4*{x}_2^5*x_3^5;}{%
\[
i=55, \,{\mathit{sym}_{5, \,5, \,4}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5}\,\mathit{x_3^4} + \mathit{{x}_1^5}\,\mathit{{x}_2^4}\,
\mathit{x_3^5} + \mathit{{x}_1^4}\,\mathit{{x}_2^5}\,\mathit{x_3^
5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 56, sym[5,5,5] = {x}_1^5*{x}_2^5*x_3^5;}{%
\[
i=56, \,{\mathit{sym}_{5, \,5, \,5}}=\mathit{{x}_1^5}\,\mathit{x}_2
^{5}\,\mathit{x_3^5}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 57, sym[6,0,0] = {x}_1^6+{x}_2^6+x_3^6;}{%
\[
i=57, \,{\mathit{sym}_{6, \,0, \,0}}=\mathit{{x}_1^6} + \mathit{{x}_2
^6} + \mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 58, sym[6,1,0] =
{x}_1^6*{x}_2+{x}_1^6*x_3+{x}_1*{x}_2^6+{x}_1*x_3^6+{x}_2^6*x_3+{x}_2*x_3^6;}{%
\[
i=58, \,{\mathit{sym}_{6, \,1, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
 + \mathit{{x}_1^6}\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^6} + 
\mathit{{x}_1}\,\mathit{x_3^6} + \mathit{{x}_2^6}\,\mathit{x_3} + 
\mathit{x}_2\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 59, sym[6,1,1] = {x}_1^6*{x}_2*x_3+{x}_1*{x}_2^6*x_3+{x}_1*{x}_2*x_3^6;}{%
\[
i=59, \,{\mathit{sym}_{6, \,1, \,1}}=\mathit{{x}_1^6}\,\mathit{x}_2
\,\mathit{x_3} + \mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3} + 
\mathit{{x}_1}\,\mathit{x}_2\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 60, sym[6,2,0] =
{x}_1^6*{x}_2^2+{x}_1^6*x_3^2+{x}_1^2*{x}_2^6+{x}_1^2*x_3^6+{x}_2^6*x_3^2+{x}_2^2*x_3^6;}{%
\[
i=60, \,{\mathit{sym}_{6, \,2, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{2} + \mathit{{x}_1^6}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,
\mathit{{x}_2^6} + \mathit{{x}_1^2}\,\mathit{x_3^6} + \mathit{x}_2
^{6}\,\mathit{x_3^2} + \mathit{{x}_2^2}\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 61, sym[6,2,1] =
{x}_1^6*{x}_2^2*x_3+{x}_1^6*{x}_2*x_3^2+{x}_1^2*{x}_2^6*x_3+{x}_1^2*{x}_2*x_3^6+{x}_1*{x}_2^6*x_3^2+{x}_1*{x}_2
^2*x_3^6;}{%
\maplemultiline{
i=61, {\mathit{sym}_{6, \,2, \,1}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^2}\,\mathit{x_3} + \mathit{{x}_1^6}\,
\mathit{x}_2\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,
\mathit{x_3} + \mathit{{x}_1^2}\,\mathit{x}_2\,\mathit{x_3^6} + 
\mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3^2} + \mathit{{x}_1}\,
\mathit{{x}_2^2}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 62, sym[6,2,2] = {x}_1^6*{x}_2^2*x_3^2+{x}_1^2*{x}_2^6*x_3^2+{x}_1^2*{x}_2^2*x_3^6;}{%
\[
i=62, \,{\mathit{sym}_{6, \,2, \,2}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{2}\,\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,
\mathit{x_3^2} + \mathit{{x}_1^2}\,\mathit{{x}_2^2}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 63, sym[6,3,0] =
{x}_1^6*{x}_2^3+{x}_1^6*x_3^3+{x}_1^3*{x}_2^6+{x}_1^3*x_3^6+{x}_2^6*x_3^3+{x}_2^3*x_3^6;}{%
\[
i=63, \,{\mathit{sym}_{6, \,3, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{3} + \mathit{{x}_1^6}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,
\mathit{{x}_2^6} + \mathit{{x}_1^3}\,\mathit{x_3^6} + \mathit{x}_2
^{6}\,\mathit{x_3^3} + \mathit{{x}_2^3}\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 64, sym[6,3,1] =
{x}_1^6*{x}_2^3*x_3+{x}_1^6*{x}_2*x_3^3+{x}_1^3*{x}_2^6*x_3+{x}_1^3*{x}_2*x_3^6+{x}_1*{x}_2^6*x_3^3+{x}_1*{x}_2
^3*x_3^6;}{%
\maplemultiline{
i=64, {\mathit{sym}_{6, \,3, \,1}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^3}\,\mathit{x_3} + \mathit{{x}_1^6}\,
\mathit{x}_2\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^6}\,
\mathit{x_3} + \mathit{{x}_1^3}\,\mathit{x}_2\,\mathit{x_3^6} + 
\mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3^3} + \mathit{{x}_1}\,
\mathit{{x}_2^3}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 65, sym[6,3,2] =
{x}_1^6*{x}_2^3*x_3^2+{x}_1^6*{x}_2^2*x_3^3+{x}_1^3*{x}_2^6*x_3^2+{x}_1^3*{x}_2^2*x_3^6+{x}_1^2*{x}_2^6*
x_3^3+{x}_1^2*{x}_2^3*x_3^6;}{%
\maplemultiline{
i=65, {\mathit{sym}_{6, \,3, \,2}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^3}\,\mathit{x_3^2} + \mathit{{x}_1^
6}\,\mathit{{x}_2^2}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{
{x}_2^6}\,\mathit{x_3^2} + \mathit{{x}_1^3}\,\mathit{{x}_2^2}\,
\mathit{x_3^6} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,\mathit{x_3^
3} + \mathit{{x}_1^2}\,\mathit{{x}_2^3}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 66, sym[6,3,3] = {x}_1^6*{x}_2^3*x_3^3+{x}_1^3*{x}_2^6*x_3^3+{x}_1^3*{x}_2^3*x_3^6;}{%
\[
i=66, \,{\mathit{sym}_{6, \,3, \,3}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{3}\,\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^6}\,
\mathit{x_3^3} + \mathit{{x}_1^3}\,\mathit{{x}_2^3}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 67, sym[6,4,0] =
{x}_1^6*{x}_2^4+{x}_1^6*x_3^4+{x}_1^4*{x}_2^6+{x}_1^4*x_3^6+{x}_2^6*x_3^4+{x}_2^4*x_3^6;}{%
\[
i=67, \,{\mathit{sym}_{6, \,4, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{4} + \mathit{{x}_1^6}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,
\mathit{{x}_2^6} + \mathit{{x}_1^4}\,\mathit{x_3^6} + \mathit{x}_2
^{6}\,\mathit{x_3^4} + \mathit{{x}_2^4}\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 68, sym[6,4,1] =
{x}_1^6*{x}_2^4*x_3+{x}_1^6*{x}_2*x_3^4+{x}_1^4*{x}_2^6*x_3+{x}_1^4*{x}_2*x_3^6+{x}_1*{x}_2^6*x_3^4+{x}_1*{x}_2
^4*x_3^6;}{%
\maplemultiline{
i=68, {\mathit{sym}_{6, \,4, \,1}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^4}\,\mathit{x_3} + \mathit{{x}_1^6}\,
\mathit{x}_2\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^6}\,
\mathit{x_3} + \mathit{{x}_1^4}\,\mathit{x}_2\,\mathit{x_3^6} + 
\mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3^4} + \mathit{{x}_1}\,
\mathit{{x}_2^4}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 69, sym[6,4,2] =
{x}_1^6*{x}_2^4*x_3^2+{x}_1^6*{x}_2^2*x_3^4+{x}_1^4*{x}_2^6*x_3^2+{x}_1^4*{x}_2^2*x_3^6+{x}_1^2*{x}_2^6*
x_3^4+{x}_1^2*{x}_2^4*x_3^6;}{%
\maplemultiline{
i=69, {\mathit{sym}_{6, \,4, \,2}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^4}\,\mathit{x_3^2} + \mathit{{x}_1^
6}\,\mathit{{x}_2^2}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{
{x}_2^6}\,\mathit{x_3^2} + \mathit{{x}_1^4}\,\mathit{{x}_2^2}\,
\mathit{x_3^6} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,\mathit{x_3^
4} + \mathit{{x}_1^2}\,\mathit{{x}_2^4}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 70, sym[6,4,3] =
{x}_1^6*{x}_2^4*x_3^3+{x}_1^6*{x}_2^3*x_3^4+{x}_1^4*{x}_2^6*x_3^3+{x}_1^4*{x}_2^3*x_3^6+{x}_1^3*{x}_2^6*
x_3^4+{x}_1^3*{x}_2^4*x_3^6;}{%
\maplemultiline{
i=70, {\mathit{sym}_{6, \,4, \,3}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^4}\,\mathit{x_3^3} + \mathit{{x}_1^
6}\,\mathit{{x}_2^3}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{
{x}_2^6}\,\mathit{x_3^3} + \mathit{{x}_1^4}\,\mathit{{x}_2^3}\,
\mathit{x_3^6} + \mathit{{x}_1^3}\,\mathit{{x}_2^6}\,\mathit{x_3^
4} + \mathit{{x}_1^3}\,\mathit{{x}_2^4}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 71, sym[6,4,4] = {x}_1^6*{x}_2^4*x_3^4+{x}_1^4*{x}_2^6*x_3^4+{x}_1^4*{x}_2^4*x_3^6;}{%
\[
i=71, \,{\mathit{sym}_{6, \,4, \,4}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{4}\,\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^6}\,
\mathit{x_3^4} + \mathit{{x}_1^4}\,\mathit{{x}_2^4}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 72, sym[6,5,0] =
{x}_1^6*{x}_2^5+{x}_1^6*x_3^5+{x}_1^5*{x}_2^6+{x}_1^5*x_3^6+{x}_2^6*x_3^5+{x}_2^5*x_3^6;}{%
\[
i=72, \,{\mathit{sym}_{6, \,5, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{5} + \mathit{{x}_1^6}\,\mathit{x_3^5} + \mathit{{x}_1^5}\,
\mathit{{x}_2^6} + \mathit{{x}_1^5}\,\mathit{x_3^6} + \mathit{x}_2
^{6}\,\mathit{x_3^5} + \mathit{{x}_2^5}\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 73, sym[6,5,1] =
{x}_1^6*{x}_2^5*x_3+{x}_1^6*{x}_2*x_3^5+{x}_1^5*{x}_2^6*x_3+{x}_1^5*{x}_2*x_3^6+{x}_1*{x}_2^6*x_3^5+{x}_1*{x}_2
^5*x_3^6;}{%
\maplemultiline{
i=73, {\mathit{sym}_{6, \,5, \,1}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^5}\,\mathit{x_3} + \mathit{{x}_1^6}\,
\mathit{x}_2\,\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{{x}_2^6}\,
\mathit{x_3} + \mathit{{x}_1^5}\,\mathit{x}_2\,\mathit{x_3^6} + 
\mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3^5} + \mathit{{x}_1}\,
\mathit{{x}_2^5}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 74, sym[6,5,2] =
{x}_1^6*{x}_2^5*x_3^2+{x}_1^6*{x}_2^2*x_3^5+{x}_1^5*{x}_2^6*x_3^2+{x}_1^5*{x}_2^2*x_3^6+{x}_1^2*{x}_2^6*
x_3^5+{x}_1^2*{x}_2^5*x_3^6;}{%
\maplemultiline{
i=74, {\mathit{sym}_{6, \,5, \,2}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^5}\,\mathit{x_3^2} + \mathit{{x}_1^
6}\,\mathit{{x}_2^2}\,\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{
{x}_2^6}\,\mathit{x_3^2} + \mathit{{x}_1^5}\,\mathit{{x}_2^2}\,
\mathit{x_3^6} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,\mathit{x_3^
5} + \mathit{{x}_1^2}\,\mathit{{x}_2^5}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 75, sym[6,5,3] =
{x}_1^6*{x}_2^5*x_3^3+{x}_1^6*{x}_2^3*x_3^5+{x}_1^5*{x}_2^6*x_3^3+{x}_1^5*{x}_2^3*x_3^6+{x}_1^3*{x}_2^6*
x_3^5+{x}_1^3*{x}_2^5*x_3^6;}{%
\maplemultiline{
i=75, {\mathit{sym}_{6, \,5, \,3}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^5}\,\mathit{x_3^3} + \mathit{{x}_1^
6}\,\mathit{{x}_2^3}\,\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{
{x}_2^6}\,\mathit{x_3^3} + \mathit{{x}_1^5}\,\mathit{{x}_2^3}\,
\mathit{x_3^6} + \mathit{{x}_1^3}\,\mathit{{x}_2^6}\,\mathit{x_3^
5} + \mathit{{x}_1^3}\,\mathit{{x}_2^5}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 76, sym[6,5,4] =
{x}_1^6*{x}_2^5*x_3^4+{x}_1^6*{x}_2^4*x_3^5+{x}_1^5*{x}_2^6*x_3^4+{x}_1^5*{x}_2^4*x_3^6+{x}_1^4*{x}_2^6*
x_3^5+{x}_1^4*{x}_2^5*x_3^6;}{%
\maplemultiline{
i=76, {\mathit{sym}_{6, \,5, \,4}}= \\
\mathit{{x}_1^6}\,\mathit{{x}_2^5}\,\mathit{x_3^4} + \mathit{{x}_1^
6}\,\mathit{{x}_2^4}\,\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{
{x}_2^6}\,\mathit{x_3^4} + \mathit{{x}_1^5}\,\mathit{{x}_2^4}\,
\mathit{x_3^6} + \mathit{{x}_1^4}\,\mathit{{x}_2^6}\,\mathit{x_3^
5} + \mathit{{x}_1^4}\,\mathit{{x}_2^5}\,\mathit{x_3^6} }
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 77, sym[6,5,5] = {x}_1^6*{x}_2^5*x_3^5+{x}_1^5*{x}_2^6*x_3^5+{x}_1^5*{x}_2^5*x_3^6;}{%
\[
i=77, \,{\mathit{sym}_{6, \,5, \,5}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{5}\,\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{{x}_2^6}\,
\mathit{x_3^5} + \mathit{{x}_1^5}\,\mathit{{x}_2^5}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 78, sym[6,6,0] = {x}_1^6*{x}_2^6+{x}_1^6*x_3^6+{x}_2^6*x_3^6;}{%
\[
i=78, \,{\mathit{sym}_{6, \,6, \,0}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6} + \mathit{{x}_1^6}\,\mathit{x_3^6} + \mathit{{x}_2^6}\,
\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 79, sym[6,6,1] = {x}_1^6*{x}_2^6*x_3+{x}_1^6*{x}_2*x_3^6+{x}_1*{x}_2^6*x_3^6;}{%
\[
i=79, \,{\mathit{sym}_{6, \,6, \,1}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x_3} + \mathit{{x}_1^6}\,\mathit{x}_2\,\mathit{x_3^6}
 + \mathit{{x}_1}\,\mathit{{x}_2^6}\,\mathit{x_3^6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 80, sym[6,6,2] = {x}_1^6*{x}_2^6*x_3^2+{x}_1^6*{x}_2^2*x_3^6+{x}_1^2*{x}_2^6*x_3^6;}{%
\[
i=80, \,{\mathit{sym}_{6, \,6, \,2}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x_3^2} + \mathit{{x}_1^6}\,\mathit{{x}_2^2}\,
\mathit{x_3^6} + \mathit{{x}_1^2}\,\mathit{{x}_2^6}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 81, sym[6,6,3] = {x}_1^6*{x}_2^6*x_3^3+{x}_1^6*{x}_2^3*x_3^6+{x}_1^3*{x}_2^6*x_3^6;}{%
\[
i=81, \,{\mathit{sym}_{6, \,6, \,3}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x_3^3} + \mathit{{x}_1^6}\,\mathit{{x}_2^3}\,
\mathit{x_3^6} + \mathit{{x}_1^3}\,\mathit{{x}_2^6}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 82, sym[6,6,4] = {x}_1^6*{x}_2^6*x_3^4+{x}_1^6*{x}_2^4*x_3^6+{x}_1^4*{x}_2^6*x_3^6;}{%
\[
i=82, \,{\mathit{sym}_{6, \,6, \,4}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x_3^4} + \mathit{{x}_1^6}\,\mathit{{x}_2^4}\,
\mathit{x_3^6} + \mathit{{x}_1^4}\,\mathit{{x}_2^6}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 83, sym[6,6,5] = {x}_1^6*{x}_2^6*x_3^5+{x}_1^6*{x}_2^5*x_3^6+{x}_1^5*{x}_2^6*x_3^6;}{%
\[
i=83, \,{\mathit{sym}_{6, \,6, \,5}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x_3^5} + \mathit{{x}_1^6}\,\mathit{{x}_2^5}\,
\mathit{x_3^6} + \mathit{{x}_1^5}\,\mathit{{x}_2^6}\,\mathit{x_3^
6}
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{i = 84, sym[6,6,6] = {x}_1^6*{x}_2^6*x_3^6;}{%
\[
i=84, \,{\mathit{sym}_{6, \,6, \,6}}=\mathit{{x}_1^6}\,\mathit{x}_2
^{6}\,\mathit{x}_3^{6}
\]
%
}
\end{maplelatex}
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A special case}
For some particular values of the Satake parameters $x_0$, $x_1$, $x_2$, $x_3$, the polynomial $P_3$
can be considerably simplified.  For example,
let us substitute $x_0=1$, $x_1=p$, $x_2=p^2$ and $x_3=p^3$ as in Exercise 3.3.40, p.168 of \cite{An87}:
$\mathit{P}_\nu (v) :=\mathit{P}(1, p, p^2, p^3, v)$, where $\nu$ denotes the degree homomorphism
$\nu(x_0)=1$, $\nu(x_1)=p$, $\nu(x_2)=p^2$, $\nu(x_3)=p^3$. Then the polynomial $P$ takes the form

\begin{align*}&
\mathit{P}_\nu (v) = 1 - \left({\displaystyle \frac {p^{7} + p^{8} + p^{9
}}{p}} + (p^{2} + p + 1)\,p^{4} + {\displaystyle \frac {p^{3} + 
p^{4} + p^{5}}{p}} \right)\,v^{2} 
\\ &
\mbox{} + \left((p + 1)\,p^{10} + {\displaystyle \frac {(p + 1)\,(p^{9
} + p^{10} + p^{11})}{p^{2}}}  + {\displaystyle \frac {(p + 1)\,(
p^{7} + p^{8} + p^{9})}{p^{2}}}  + (p + 1)\,p^{4}\right)\,v^{3} \\ &
\mbox{} - \left({\displaystyle \frac {p^{13} + p^{14} + p^{15}}{p^{
2}}} + (p^{2} + p + 1)\,p^{9}+ {\displaystyle \frac {p^{9} + p
^{10} + p^{11}}{p^{2}}} \right)\,v^{4} + p^{15}\,v^{6} \\ &
%\end{maplelatex}
=1 - (p^{8}\,+ p^{7}\,+2\,p^{6}\,+ p^{5}\,+2\,p^{4}\,+ p^{3}+p^{2})v^{2}  \\ &
+(p^{11}+ 2\,p^{10} + 2\,p^{9}+ 3\,p^{8}   + 3\,p^{7} + 2\,p^{6} + 
2\,p^{5} + p^{4})\,v^{3} \\ &
-(p^{13}\,+ p^{12}\,+2\,p^{11}\,+ p^{10}\,+2\,p^{9}\,+ p^{8}+p^{7})v^{4}
+ p^{15}\,v
^{6}. 
%\end{maplelatex}
%\begin{maplelatex}
%\mapleinline{inert}{2d}{Pv =
%(p^3*v-1)*(p^2*v-1)*(p^4*v-1)*(p*v-1)*(p^5*v^2+p^4*v+p^3*v+p^2*v+p*v+1
%);}{%
\end{align*}
This gives the following factorization:
\begin{align*}
\mathit{P}_\nu (v)
=(1-p^{}\,v)\,(1-p^{2}\,v )\,(1-p^{3}\,v )\,(1-p^{4}\,v)\,(1+ p^{}\,v + p^{2}\,v + p^{3}\,v + p^{4}\,v +p^{5}\,v^{2} ).
\end{align*}
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{amsplain}
\begin{thebibliography}{10xxxx}

\bibitem[An67]{An67}
{\sc Andrianov, A.N.}, 
 {\it Shimnra's conjecture for Siegel's modular group of genus 3}, Dokl. Akad. Nauk
SSSR 177 (1967), 755-758 = Soviet Math. Dokl. 8 (1967), 1474-1478.

\bibitem[An68]{An68}
{\sc Andrianov, A.N.}, 
 {\it  Rationality of multiple Hecke series of a complete linear group and Shimura's
hypothesis on Hecke's series of a symplectic group}, Dokl. Akad. Nauk SSSR
183 (1968), 9-11 = Soviet Math. Dokl. 9 (1968), 1295 - 1297. MR



\bibitem[An69]{An69}
{\sc Andrianov, A.N.}, 
{\em Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $SP_n$ over local fields},
Izv. Akad. Nauk SSSR, Ser. Mat., Tom 33 (1969), No. 3, (Math. USSR -- Izvestija, Vol. 3 (1969), No. 3, pp. 439--476).

\bibitem[An70]{An70}
{\sc Andrianov, A.N.}, 
{\em Spherical functions for $GL_n$ over local fields and summation of Hecke series},
Mat. Sbornik, Tom 83 (125) (1970), No 3, (Math. USSR Sbornik, Vol. 12 (1970), No. 3,  pp. 429--452).

\bibitem[An74]{An74}
{\sc Andrianov, A.N.}, 
{\em Euler products corresponding to Siegel modular forms of genus 2},
Russian Math. Surveys, 29:3 (1974), pp. 45--116, (Uspekhi Mat. Nauk 29:3 (1974) pp. 43--110).

\bibitem[An87]{An87}
{\sc Andrianov, A.N.},
{\em Quadratic Forms and Hecke Operators},
Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.

\bibitem[AnZh95]{AnZh95}
{\sc Andrianov, A.N.}, {\sc Zhuravlev, V.G.}, 
{\em Modular Forms and Hecke Operators},
Translations of Mathematical Monographs, Vol. 145, AMS, Providence, Rhode Island, 1995.

\bibitem[CourPa]{CourPa}
{\sc Courtieu,  M.},  {\sc Panchishkin,   A.A.},
{\em Non-Archimedean $L$-Functions and Arithmetical Siegel Modular Forms},
Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

\bibitem[Evd]{Evd}  
{\sc Evdokimov, S. A.},
{\em Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus $3$},
(Russian) Dokl. Akad. Nauk SSSR 277 (1984), no. 1, 25--29. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Ku88]{Ku88}  
{\sc     %Kurokawa, N.}    LNM 1434
 Kurokawa}, Nobushige,  {\em Analyticity of Dirichlet series over prime powers.}  Analytic number theory (Tokyo, 1988),  168--177, Lecture Notes in Math., 1434, Springer, Berlin, 1990.

\bibitem[Maa76]{Maa76}  
{\sc Maass,  H.} 
{\em Indefinite Quadratische Formen  und Eulerprodukte.} 
Comm. on Pure and Appl. Math, 19, 689-699 (1976)  


\bibitem[RhSh]{RhSh}  
{\sc Rhodes, J. A.} and {\sc Shemanske, T. R.} 
{\em Rationality theorems for Hecke operators on $GL_n$}, 
J. Number Theory 102 (2003), no. 2, 278-297.

\bibitem[Sh]{Sh}  
{\sc Shimura}, G., {\it On modular correspondences for Sp(n, Z) and their congruence relations}, Proc. Nat.
Acad. Sci. U.S.A. 49 (1963), 824-828. 

\end{thebibliography}

\end{document}
math0604602.extracted_bib
\begin{thebibliography}{10xxxx}

\bibitem[An67]{An67}
{\sc Andrianov, A.N.}, 
 {\it Shimnra's conjecture for Siegel's modular group of genus 3}, Dokl. Akad. Nauk
SSSR 177 (1967), 755-758 = Soviet Math. Dokl. 8 (1967), 1474-1478.

\bibitem[An68]{An68}
{\sc Andrianov, A.N.}, 
 {\it  Rationality of multiple Hecke series of a complete linear group and Shimura's
hypothesis on Hecke's series of a symplectic group}, Dokl. Akad. Nauk SSSR
183 (1968), 9-11 = Soviet Math. Dokl. 9 (1968), 1295 - 1297. MR



\bibitem[An69]{An69}
{\sc Andrianov, A.N.}, 
{\em Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $SP_n$ over local fields},
Izv. Akad. Nauk SSSR, Ser. Mat., Tom 33 (1969), No. 3, (Math. USSR -- Izvestija, Vol. 3 (1969), No. 3, pp. 439--476).

\bibitem[An70]{An70}
{\sc Andrianov, A.N.}, 
{\em Spherical functions for $GL_n$ over local fields and summation of Hecke series},
Mat. Sbornik, Tom 83 (125) (1970), No 3, (Math. USSR Sbornik, Vol. 12 (1970), No. 3,  pp. 429--452).

\bibitem[An74]{An74}
{\sc Andrianov, A.N.}, 
{\em Euler products corresponding to Siegel modular forms of genus 2},
Russian Math. Surveys, 29:3 (1974), pp. 45--116, (Uspekhi Mat. Nauk 29:3 (1974) pp. 43--110).

\bibitem[An87]{An87}
{\sc Andrianov, A.N.},
{\em Quadratic Forms and Hecke Operators},
Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.

\bibitem[AnZh95]{AnZh95}
{\sc Andrianov, A.N.}, {\sc Zhuravlev, V.G.}, 
{\em Modular Forms and Hecke Operators},
Translations of Mathematical Monographs, Vol. 145, AMS, Providence, Rhode Island, 1995.

\bibitem[CourPa]{CourPa}
{\sc Courtieu,  M.},  {\sc Panchishkin,   A.A.},
{\em Non-Archimedean $L$-Functions and Arithmetical Siegel Modular Forms},
Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

\bibitem[Evd]{Evd}  
{\sc Evdokimov, S. A.},
{\em Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus $3$},
(Russian) Dokl. Akad. Nauk SSSR 277 (1984), no. 1, 25--29. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Ku88]{Ku88}  
{\sc     %Kurokawa, N.}    LNM 1434
 Kurokawa}, Nobushige,  {\em Analyticity of Dirichlet series over prime powers.}  Analytic number theory (Tokyo, 1988),  168--177, Lecture Notes in Math., 1434, Springer, Berlin, 1990.

\bibitem[Maa76]{Maa76}  
{\sc Maass,  H.} 
{\em Indefinite Quadratische Formen  und Eulerprodukte.} 
Comm. on Pure and Appl. Math, 19, 689-699 (1976)  


\bibitem[RhSh]{RhSh}  
{\sc Rhodes, J. A.} and {\sc Shemanske, T. R.} 
{\em Rationality theorems for Hecke operators on $GL_n$}, 
J. Number Theory 102 (2003), no. 2, 278-297.

\bibitem[Sh]{Sh}  
{\sc Shimura}, G., {\it On modular correspondences for Sp(n, Z) and their congruence relations}, Proc. Nat.
Acad. Sci. U.S.A. 49 (1963), 824-828. 

\end{thebibliography}