Fourier coefficients of three-dimensional vector-valued modular forms

We prove that only a finite number of three-dimensional, irreducible representations of the modular group admit vector-valued modular forms with bounded denominators. This provides a verification, in the three-dimensional setting, of a conjecture concerning the Fourier coefficients of noncongruence modular forms, and reinforces the understanding from mathematical physics that when such a representation arises in rational conformal field theory, its kernel should be a congruence subgroup of the modular group.