Abstract
This paper concerns the study of the Schwartz differential equation {h, tau} = s E-4(tau), where E-4 is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL2(Z). We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL2(Z). This also leads to the solutions to the Fuchsian differential equation y '' + s E-4 y = 0.