For prime levels 5 ≤ p ≤ 19, sets of Γ 0 (p)-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for Γ 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 Γ 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.

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We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujan's differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujan's original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.

Abstract. Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinatorial generating functions, including the partition function. Further analysis of the quintic theta functions is undertaken by exploring their modular properties and their connection to Eisenstein series. The resulting relations lead to a coupled system of differential equations for the quintic theta functions.

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