Balanced modular parameterizations

For prime levels $5\le p\le 19$, sets of ${\mathrm{\Gamma }}_0(p)$-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for ${\mathrm{\Gamma }}_1(p)$. An explicit formulation of the permutation representation and several applications are given, including a new representation for the number of $t$-core partitions. The ${\mathrm{\Gamma }}_0(p)$-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.