Differential equations for septic theta functions

Abstract

We demonstrate that quotients of septic theta functions appearing in S. Ramanujan's Notebooks and in F. Klein's work satisfy a new coupled system of nonlinear differential equations with interesting symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends a technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein's quartic relation induces new symmetric representations for low weight Eisenstein series in terms of weight one modular forms of level seven.