We modify and generalize proofs of Täte and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to T0( N) mod /. We also summarize results for r,(/V). 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

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... per bound. Lastly we use the constructive nature of Täte and Serre's result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators mod /.