  We study differential equations satisfied by modular forms associated to
$\Gamma_1\times\Gamma_2$, where $\Gamma_i (i=1,2)$ are genus zero subgroups of
$SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $\Gamma_0(N)$ or
$\Gamma_0(N)^*$. In some examples, these differential equations are realized as
the Picard--Fuch differential equations of families of K3 surfaces with large
Picard numbers, e.g., $19, 18, 17, 16$. Our method rediscovers some of the
Lian--Yau examples of ``modular relations'' involving power series solutions to
the second and the third order differential equations of Fuchsian type in [14,
15].
