Automorphic forms for triangle groups: Integrality properties

We classify all primes appearing in the denominators of the Hauptmodul J   and modular forms for non-arithmetic triangle groups with a cusp. These primes have a congruence condition in terms of the order of the generators of the group. As a corollary we show that for the Hecke group of type (2,m,∞)(2,m,∞), the prime p does not appear in the denominator of J   if and only if p≡±1(modm).