  A thorough analysis is made of the Fourier coefficients for vector-valued
modular forms associated to three-dimensional irreducible representations of
the modular group. In particular, the following statement is verified for all
but a finite number of equivalence classes: if a vector-valued modular form
associated to such a representation has rational Fourier coefficients, then
these coefficients have "unbounded denominators", i.e. there is a prime number
p, depending on the representation, which occurs to an arbitrarily high power
in the denominators of the coefficients. This provides a verification in the
three-dimensional setting of a generalization of a long-standing conjecture
about noncongruence modular forms.
