  For prime levels $5 \le p \le 19$, sets of $\Gamma_{0}(p)$-permuted theta
quotients are constructed that generate the graded rings of modular forms of
positive integer weight for $\Gamma_{1}(p)$. An explicit formulation of the
permutation representation and several applications are given, including a new
representation for the number of $t$-core partitions. The
$\Gamma_{0}(p)$-action induces coefficient symmetries within representations
for modular forms and invariance subgroups for coupled systems of differential
equations. The symmetry for levels $p = 5,7,11$ is linked to the Kleinian
automorphism groups.
