Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$

Abstract We prove the following theorem. Suppose that F = ( f 1 , f 2 ) is a 2-dimensional, vectorvalued modular form on SL 2 ( ℤ ) whose component functions f 1 , f 2 have rational Fourier coefficients with bounded denominators . Then f 1 , f 2 are classical modular forms on a congruence subgroup of the modular group.