A formula for the traces of the Hecke operators on certain spaces of newforms

Abstract.

In this article we compute a formula for the trace of the Hecke operators on the space S ^* _k (N), which is the space generated by newforms of level dividing N, where N is a squarefree positive integer, and of weight k, where k is a positive even integer. This is of interest because, while the computation of the trace of the Hecke operators on the whole space of cusp forms S _k (N) is strongly dependent on the prime factorization of N, the trace on this subspace is only dependent on the magnitude of N and on its residue modulo a number that depends only on n, where n is the number associated to the Hecke operator T _n . An interesting consequence of this calculation is the fact that for any fixed nonsquare n and even k, the trace of the Hecke operator T _k (N) on the space S ^* _k (N) has an absolute bound for all squarefree N.