, Volume 70, Issue 3, pp 204-210

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

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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.

Received: 1.4.1997