In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups
Γ
0
(
N
)
+
\Gamma _0(N)^+
, where
N
N
is a positive, square-free integer. After we prove that
Γ
0
(
N
)
+
\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
N=5
and
N
=
6
N=6
have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for
Γ
0
(
5
)
+
\Gamma _0(5)^+
than for
Γ
0
(
6
)
+
\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 [H. Then, Computing large sets of consecutive Maass forms, in preparation], and numerically determine the first
3557
3557
eigenvalues of
Γ
0
(
5
)
+
\Gamma _0(5)^+
and the first
12474
12474
eigenvalues of
Γ
0
(
6
)
+
\Gamma _0(6)^+
. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.