  We calculate very long low- and high-temperature series for the
susceptibility $\chi$ of the square lattice Ising model as well as very long
series for the five-particle contribution $\chi^{(5)}$ and six-particle
contribution $\chi^{(6)}$. These calculations have been made possible by the
use of highly optimized polynomial time modular algorithms and a total of more
than 150000 CPU hours on computer clusters. For $\chi^{(5)}$ 10000 terms of the
series are calculated {\it modulo} a single prime, and have been used to find
the linear ODE satisfied by $\chi^{(5)}$ {\it modulo} a prime.
  A diff-Pad\'e analysis of 2000 terms series for $\chi^{(5)}$ and $\chi^{(6)}$
confirms to a very high degree of confidence previous conjectures about the
location and strength of the singularities of the $n$-particle components of
the susceptibility, up to a small set of ``additional'' singularities. We find
the presence of singularities at $w=1/2$ for the linear ODE of $\chi^{(5)}$,
and $w^2= 1/8$ for the ODE of $\chi^{(6)}$, which are {\it not} singularities
of the ``physical'' $\chi^{(5)}$ and $\chi^{(6)},$ that is to say the
series-solutions of the ODE's which are analytic at $w =0$.
  Furthermore, analysis of the long series for $\chi^{(5)}$ (and $\chi^{(6)}$)
combined with the corresponding long series for the full susceptibility $\chi$
yields previously conjectured singularities in some $\chi^{(n)}$, $n \ge 7$.
  We also present a mechanism of resummation of the logarithmic singularities
of the $\chi^{(n)}$ leading to the known power-law critical behaviour occurring
in the full $\chi$, and perform a power spectrum analysis giving strong
arguments in favor of the existence of a natural boundary for the full
susceptibility $\chi$.
