  We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a
2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions
$f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then
$f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the
modular group.
