  This paper concerns the study of the Schwarz 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
$\mbox{SL}_2(\mathbb Z)$. 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 $\mbox{SL}_2(\mathbb Z)$. This also leads to the solutions to the
Fuchsian differential equation $y''+s\,E_4\,y=0$.
