Cubic theta functions and modular forms of level six

The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by modular forms of level six and to construct extensions of the differential field of the cubic theta functions, generalizing the classical Ramanujan and Halphen fields. We treat both modular forms that appear in the literature and others that do not. We find Riccati equations satisfied by level six modular forms, explore applications to number theory, and find new relations among the Eisenstein series.