  We prove that for any generic polynomial $f$ in two variables of degree
$(d_1,d_2)$ over the rationals, for $p$ large enough the Newton slopes of the
character power series $C_f^*(\chi_m,s)$ of $f$ at $p$ is independent of the
choice of the character $\chi_m$ (of conductor $p^m$); and the Newton slopes of
the $L$-function $L_f^*(\chi_m,s)$ of $f$ at $p$ is in weighted arithmetic
progression.
