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

Jay Jorgenson, Lejla Smajlović, Holger Then

Published 2013-01-08Version 1

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where N>1$ is a square-free integer. After we prove that $\Gamma_0(N)^+$ 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 $\Gamma_0(5)^+$ than for $\Gamma_0(6)^+$. 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 $\Gamma_0(5)^+$ and the first 12474 eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.