On the trace of Hecke operators for Maass forms for congruence subgroups

Let E-lambda be a Hilbert space, whose elements are functions spanned by the eigenfunctions of the Laplace-Beltrami operator associated with an eigenvalue lambda > 0. The norm of elements in this space is given by the Petersson inner product. In this paper, the trace of Hecke operators T-n acting on the space E-lambda is computed for congruence subgroups Gamma (0)(N) of square free level, which may be considered as the analogue of the Eichler-Selberg trace formula [11] for non-holomorphic cusp forms of weight zero.