A formula for the traces of the Hecke operators on certain spaces of new forms

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 whale space of cusp forms Sk(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 module 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 bet 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.