Principal congruence subgroups of the Hecke groups

Let q be an odd integer > 3 and let G(q) be the Hecke group associated to q . Let (tau) be a prime ideal of Z[lambda (q)] and G(q, tau) the principal c ongruence subgroup of G(q) associated to tau. We give a formula for [G(q) : G(q, tau)], the index of the principal congruence subgroup G(q, tau) in G( q). We also give formulae for the indices [G(1)(q, tau), G(q, tau)] and [G( 0)(q, tau), G(1)(q,tau)]. Finally, we give a formula for the geometric inva riants of G(q, tau) when q is a rational prime. (C) 2000 Academic Press.