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2pac belongs conjugacy class conjugate to H consider contains 3 conjugates contradicting proposition 1.1 cyclic group denote describes all values divides elements of H elements of order F has genus formula 1.1 function field fundamental domain Galois group genus of H Gierster pages GL(L group of order H contains H has order H is conjugate H mod H^Kj I H I identity implies integer isomorphic Klingenberg Lemma level 27 linear fractional transformations LINEAR GROUPS matrix maximal ideal McQuillan mod 2n mod 2n-r non-zero residue mod normal subgroup notation order 12 order 36 order 9 parametric disc possibilities for H Proof quadratic residue mod Riemann surface scalar semi-local rings Sp(L subfield of K(27 subgroup of H subgroup of LF(2,pn subgroup of Sp(R subgroups H subgroups of order Suppose F Sylow Sylow Theorems symplectic symplectic group Theorem thesis values mod 9