On the Fourier coefficients of 2-dimensional vector-valued modular forms

Let $\rho : SL(2, \mathbb{Z}) \rightarrow GL(2, \mathbb{C})$ 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 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 not divide $120$, then every nonzero $F(\tau )$ has Fourier coefficients with unbounded denominators.