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Let r denote the modular group, namely the group of linear fractional transformations az + b T(z) = -, cz + d where a, b, c, d are integers and ad -be = 1. It is well known that the transformations generate T, with defining relations X2 = Y3 = 1. We shall often need to consider the element Z = XY, which is a parabolic transformation. Any parabolic transformation in V is conjugate to a power of Z. Let A be a normal subgroup of finite index p in T. The level n of N is defined as the leastdoi:10.1090/s0002-9939-1966-0199274-4 fatcat:4va6soqanva5xboebaocxwdi2q