Differential equations for septic theta functions

TL;DR

This paper derives a new coupled system of nonlinear differential equations satisfied by septic theta function quotients, extending Ramanujan's techniques and revealing new symmetric representations related to Klein's quartic and Eisenstein series.

Contribution

It introduces a novel differential system for septic theta functions and connects Klein's quartic relation to symmetric Eisenstein series representations.

Findings

Derived a coupled differential system for septic theta functions.

Connected Klein's quartic relation to Eisenstein series representations.

Extended Ramanujan's differential techniques to new modular forms.

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.