On the distribution of eigenvalues of Maass forms on certain moonshine groups

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups Γ0(N), where N > 1 is a square-free integer. After we prove that Γ0(N) has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an “average” Weyl’s law for the distribution of eigenvalues of Maass forms, from which we prove the “classical” Weyl’s law as a special case. The groups corresponding to N = 5 and N = 6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for Γ0(5) than for Γ0(6). We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl’s laws. In addition, we employ Hejhal’s algorithm, together with recently developed refinements from [31], and numerically determine the first 3557 of Γ0(5) and the first 12474 eigenvalues of Γ0(6). With this information, we empirically verify some conjectured distributional properties of the eigenvalues.