  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.
