  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 $(2,m,\infty)$,
the prime $p$ does not appear in the denominator of $J$ if and only if $p\equiv
\pm 1\pmod m$.

Number Theory;Complex Variables