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

## Authors

DOI: 10.1007/s000130050185

- Cite this article as:
- Hamer, C. Arch. Math. (1998) 70: 204. doi:10.1007/s000130050185

## 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*.