Congruences Between Systems of Eigenvalues of Modular Forms

We modify and generalize proofs of Tate and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to ${\Gamma _0}(N)\bmod l$. We also summarize results for ${\Gamma _1}(N)$. Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with $k$. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Tate and Serre’s result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators $\bmod l$.