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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 thedoi:10.4310/cntp.2015.v9.n2.a5 fatcat:jn4nyutb6rcddbsr4n2bhytqdm