All Siegel Hecke eigensystems (mod p) are cuspidal

A. Ghitza
We show that systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms are the same as the systems occurring in the spaces of Siegel cusp forms, of possibly different weight.

Let $n_0(N,k)$ be the number of initial Fourier coefficients necessary to distinguish newforms of level $N$ and even weight $k$. We produce extensive data to support our conjecture that if $N$ is a fixed squarefree positive integer and $k$ is large then $n_0(N,k)$ is the least prime that does not divide $N$.