  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.
