  This paper uses previous results of the authors on vector-valued modular
forms to study certain non-congruence modular forms. We prove that these forms
have unbounded denominators, and in certain cases we verify congruences of
Atkin--Swinnerton-Dyer type satisfied by the Fourier coefficients of these
forms. Our results rest on group-theoretic facts about the modular group, a
detailed study of its imprimitive three-dimensional representations, and the
theory of their associated vector-valued modular forms. For the proof of the
congruences we also make essential use of a result of Katz.
