  Let $\rho: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible
representation of the modular group such that $\rho(T)$ has finite order $N$.
We study holomorphic vector-valued modular forms $F(\tau)$ of integral weight
associated to $\rho$ which have \emph{rational} Fourier coefficients. (These
span the complex space of all integral weight vector-valued modular forms
associated to $\rho$.) As a special case of the main Theorem, we prove that if
$N$ does \emph{not} divide 120 then every nonzero $F(\tau)$ has Fourier
coefficients with \emph{unbounded denominators}.
