INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 768 Minimal number of generators: 129 Number of equivalence classes of cusps: 64 Genus: 33 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -7/1 -6/1 -5/1 -9/2 -4/1 -11/3 -10/3 -3/1 -8/3 -9/4 -2/1 -9/5 -3/2 -10/7 -4/3 -6/5 -1/1 -6/7 -3/4 -2/3 -3/5 -6/11 0/1 1/2 6/11 3/5 2/3 3/4 9/11 6/7 1/1 6/5 5/4 4/3 10/7 3/2 36/23 12/7 7/4 9/5 2/1 24/11 9/4 12/5 5/2 8/3 11/4 3/1 36/11 10/3 24/7 7/2 11/3 4/1 9/2 24/5 5/1 11/2 6/1 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -1/2 1/0 -7/1 -2/1 0/1 -6/1 -1/1 -11/2 -4/5 -5/1 -2/3 -14/3 -1/2 -23/5 -2/3 0/1 -9/2 -2/3 0/1 -13/3 -2/5 0/1 -4/1 -1/1 -15/4 -2/3 -11/3 0/1 -18/5 -1/1 -7/2 -2/3 -24/7 -1/2 -17/5 0/1 -10/3 -1/1 -3/5 -3/1 -2/3 0/1 -14/5 -1/1 -3/5 -25/9 -2/3 -36/13 -1/1 -3/5 -11/4 -2/3 -1/2 -8/3 -1/2 -13/5 -2/3 -18/7 -1/2 -5/2 0/1 -12/5 -1/1 -7/3 -2/3 -23/10 -2/5 -16/7 -1/2 1/0 -25/11 -2/3 0/1 -9/4 -2/3 0/1 -11/5 -2/3 -4/7 -2/1 -1/2 1/0 -13/7 -2/3 -4/7 -24/13 -1/2 -11/6 -2/5 -9/5 0/1 -25/14 -2/3 -16/9 -1/2 1/0 -23/13 -2/3 0/1 -7/4 -1/2 0/1 -12/7 -1/1 -1/3 -5/3 -2/3 0/1 -18/11 -1/2 -13/8 -1/2 -2/5 -8/5 -1/2 -11/7 -2/5 -3/2 0/1 -13/9 -2/1 -36/25 -1/1 -23/16 -1/1 -2/3 -10/7 -1/1 -17/12 -2/3 -1/2 -24/17 -1/2 -7/5 -2/3 0/1 -18/13 -1/2 -11/8 -1/2 0/1 -15/11 -2/5 0/1 -4/3 -1/1 -1/3 -13/10 -2/7 -9/7 0/1 -23/18 -2/1 -14/11 -1/2 -19/15 0/1 -24/19 -1/2 -1/4 -5/4 -1/3 0/1 -11/9 -2/11 0/1 -6/5 0/1 -7/6 0/1 -8/7 -1/2 1/0 -1/1 0/1 -7/8 -1/2 0/1 -6/7 0/1 -5/6 0/1 -14/17 1/1 -23/28 1/1 2/1 -9/11 0/1 2/1 -4/5 -1/1 1/1 -11/14 -2/1 -18/23 -1/1 -7/9 -2/3 0/1 -10/13 -1/2 -3/4 0/1 -14/19 1/4 -25/34 4/13 -36/49 1/3 -11/15 2/5 -8/11 1/2 -13/18 2/3 -5/7 0/1 2/1 -12/17 -1/1 1/1 -7/10 0/1 -16/23 1/2 1/0 -25/36 0/1 1/1 -9/13 0/1 2/1 -11/16 2/1 1/0 -2/3 -1/1 1/1 -13/20 2/1 1/0 -24/37 1/0 -11/17 -4/1 -2/1 -9/14 -2/1 0/1 -25/39 -2/1 0/1 -16/25 -3/2 1/0 -7/11 0/1 -12/19 -1/1 -5/8 -1/1 0/1 -13/21 0/1 -8/13 -1/2 -11/18 0/1 -3/5 0/1 -13/22 0/1 -36/61 1/3 1/1 -23/39 0/1 -10/17 1/4 1/2 -7/12 0/1 1/2 -18/31 1/2 -11/19 0/1 -4/7 1/1 -9/16 0/1 2/1 -23/41 0/1 2/1 -14/25 1/1 -19/34 2/1 -24/43 3/2 1/0 -5/9 2/1 -6/11 1/0 -7/13 -4/1 -2/1 -1/2 0/1 0/1 -1/2 1/0 1/2 0/1 6/11 1/0 5/9 -4/1 -2/1 14/25 -5/2 1/0 9/16 -2/1 4/7 -1/1 11/19 -2/5 0/1 7/12 0/1 1/0 10/17 -1/1 3/5 0/1 14/23 1/1 25/41 0/1 2/1 11/18 0/1 8/13 1/0 5/8 0/1 1/1 7/11 0/1 2/1 16/25 3/2 1/0 9/14 2/1 2/3 1/0 9/13 -4/1 -2/1 16/23 -5/2 1/0 7/10 -2/1 5/7 -2/1 13/18 0/1 8/11 1/0 3/4 -2/1 0/1 10/13 -1/1 1/1 7/9 -2/1 11/14 -2/3 4/5 -1/1 1/1 9/11 -2/1 0/1 14/17 -1/2 1/0 19/23 -2/1 0/1 5/6 0/1 6/7 1/0 7/8 -2/1 1/0 1/1 -2/1 0/1 8/7 -3/2 1/0 7/6 -2/1 6/5 -1/1 11/9 0/1 5/4 -1/1 0/1 14/11 -1/1 1/1 23/18 -2/3 9/7 0/1 13/10 -2/1 4/3 -1/1 1/1 15/11 0/1 2/1 26/19 1/0 11/8 0/1 1/0 18/13 1/0 25/18 -6/1 7/5 -2/1 24/17 1/0 17/12 -2/1 1/0 10/7 -3/2 1/0 3/2 -2/1 0/1 14/9 -3/2 1/0 39/25 -2/1 -4/3 25/16 -4/3 -1/1 36/23 -1/1 11/7 -2/3 0/1 19/12 0/1 1/1 8/5 1/0 13/8 0/1 1/0 18/11 1/0 5/3 -2/1 17/10 -2/1 12/7 -3/1 -1/1 7/4 -2/1 1/0 23/13 -2/1 16/9 -3/2 1/0 25/14 -4/1 9/5 -2/1 11/6 -8/5 2/1 -1/1 13/6 -4/7 24/11 -1/2 11/5 -2/5 9/4 0/1 34/15 1/2 1/0 25/11 0/1 16/7 1/2 1/0 23/10 2/1 7/3 -2/1 0/1 19/8 -2/1 -1/1 12/5 -1/1 5/2 0/1 18/7 0/1 31/12 0/1 1/4 13/5 0/1 2/3 60/23 1/1 47/18 4/3 34/13 1/1 3/1 21/8 0/1 2/1 8/3 1/0 11/4 -2/1 1/0 3/1 -2/1 0/1 13/4 -2/1 1/0 36/11 -3/1 -1/1 59/18 -2/1 23/7 -2/1 0/1 10/3 1/0 17/5 -2/1 0/1 24/7 1/0 7/2 -2/1 18/5 -2/1 11/3 -2/1 -8/5 15/4 -2/1 -4/3 4/1 -1/1 13/3 0/1 48/11 1/0 35/8 -4/1 1/0 22/5 1/0 9/2 -2/1 32/7 -3/2 1/0 23/5 -2/1 14/3 -5/3 -1/1 19/4 -2/1 -1/1 24/5 -3/2 1/0 29/6 -2/1 5/1 -2/1 -4/3 11/2 -6/5 6/1 -1/1 13/2 -6/7 7/1 -2/3 8/1 -1/2 1/0 9/1 -2/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(25,216,-36,-311) (-8/1,1/0) -> (-16/23,-25/36) Hyperbolic Matrix(23,168,36,263) (-8/1,-7/1) -> (7/11,16/25) Hyperbolic Matrix(25,168,-32,-215) (-7/1,-6/1) -> (-18/23,-7/9) Hyperbolic Matrix(47,264,-60,-337) (-6/1,-11/2) -> (-11/14,-18/23) Hyperbolic Matrix(23,120,32,167) (-11/2,-5/1) -> (5/7,13/18) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(119,552,36,167) (-14/3,-23/5) -> (23/7,10/3) Hyperbolic Matrix(95,432,-148,-673) (-23/5,-9/2) -> (-9/14,-25/39) Hyperbolic Matrix(49,216,-76,-335) (-9/2,-13/3) -> (-11/17,-9/14) Hyperbolic Matrix(23,96,40,167) (-13/3,-4/1) -> (4/7,11/19) Hyperbolic Matrix(25,96,44,169) (-4/1,-15/4) -> (9/16,4/7) Hyperbolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(73,264,60,217) (-11/3,-18/5) -> (6/5,11/9) Hyperbolic Matrix(47,168,40,143) (-18/5,-7/2) -> (7/6,6/5) Hyperbolic Matrix(97,336,28,97) (-7/2,-24/7) -> (24/7,7/2) Hyperbolic Matrix(169,576,120,409) (-24/7,-17/5) -> (7/5,24/17) Hyperbolic Matrix(71,240,92,311) (-17/5,-10/3) -> (10/13,7/9) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(623,1728,-1056,-2929) (-25/9,-36/13) -> (-36/61,-23/39) Hyperbolic Matrix(313,864,96,265) (-36/13,-11/4) -> (13/4,36/11) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,-100,-263) (-8/3,-13/5) -> (-11/15,-8/11) Hyperbolic Matrix(167,432,-288,-745) (-13/5,-18/7) -> (-18/31,-11/19) Hyperbolic Matrix(47,120,56,143) (-18/7,-5/2) -> (5/6,6/7) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(145,336,104,241) (-7/3,-23/10) -> (25/18,7/5) Hyperbolic Matrix(335,768,188,431) (-23/10,-16/7) -> (16/9,25/14) Hyperbolic Matrix(95,216,84,191) (-16/7,-25/11) -> (1/1,8/7) Hyperbolic Matrix(191,432,-340,-769) (-25/11,-9/4) -> (-9/16,-23/41) Hyperbolic Matrix(119,264,32,71) (-9/4,-11/5) -> (11/3,15/4) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(311,576,-480,-889) (-13/7,-24/13) -> (-24/37,-11/17) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(119,216,92,167) (-11/6,-9/5) -> (9/7,13/10) Hyperbolic Matrix(241,432,188,337) (-9/5,-25/14) -> (23/18,9/7) Hyperbolic Matrix(431,768,188,335) (-25/14,-16/9) -> (16/7,23/10) Hyperbolic Matrix(433,768,-676,-1199) (-16/9,-23/13) -> (-25/39,-16/25) Hyperbolic Matrix(95,168,108,191) (-23/13,-7/4) -> (7/8,1/1) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(73,120,132,217) (-5/3,-18/11) -> (6/11,5/9) Hyperbolic Matrix(265,432,192,313) (-18/11,-13/8) -> (11/8,18/13) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,-196,-311) (-8/5,-11/7) -> (-13/21,-8/13) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(599,864,-816,-1177) (-13/9,-36/25) -> (-36/49,-11/15) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(385,552,-468,-671) (-23/16,-10/7) -> (-14/17,-23/28) Hyperbolic Matrix(169,240,288,409) (-10/7,-17/12) -> (7/12,10/17) Hyperbolic Matrix(577,816,408,577) (-17/12,-24/17) -> (24/17,17/12) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(121,168,-224,-311) (-7/5,-18/13) -> (-6/11,-7/13) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(193,264,-280,-383) (-11/8,-15/11) -> (-9/13,-11/16) Hyperbolic Matrix(71,96,88,119) (-15/11,-4/3) -> (4/5,9/11) Hyperbolic Matrix(73,96,92,121) (-4/3,-13/10) -> (11/14,4/5) Hyperbolic Matrix(167,216,92,119) (-13/10,-9/7) -> (9/5,11/6) Hyperbolic Matrix(337,432,188,241) (-9/7,-23/18) -> (25/14,9/5) Hyperbolic Matrix(527,672,-716,-913) (-23/18,-14/11) -> (-14/19,-25/34) Hyperbolic Matrix(455,576,-816,-1033) (-19/15,-24/19) -> (-24/43,-5/9) Hyperbolic Matrix(457,576,96,121) (-24/19,-5/4) -> (19/4,24/5) Hyperbolic Matrix(215,264,136,167) (-5/4,-11/9) -> (11/7,19/12) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(145,168,208,241) (-7/6,-8/7) -> (16/23,7/10) Hyperbolic Matrix(191,216,84,95) (-8/7,-1/1) -> (25/11,16/7) Hyperbolic Matrix(191,168,108,95) (-1/1,-7/8) -> (7/4,23/13) Hyperbolic Matrix(361,312,140,121) (-7/8,-6/7) -> (18/7,31/12) Hyperbolic Matrix(143,120,56,47) (-6/7,-5/6) -> (5/2,18/7) Hyperbolic Matrix(407,336,-728,-601) (-5/6,-14/17) -> (-14/25,-19/34) Hyperbolic Matrix(263,216,28,23) (-23/28,-9/11) -> (9/1,1/0) Hyperbolic Matrix(119,96,88,71) (-9/11,-4/5) -> (4/3,15/11) Hyperbolic Matrix(121,96,92,73) (-4/5,-11/14) -> (13/10,4/3) Hyperbolic Matrix(311,240,92,71) (-7/9,-10/13) -> (10/3,17/5) Hyperbolic Matrix(95,72,-128,-97) (-10/13,-3/4) -> (-3/4,-14/19) Parabolic Matrix(4343,3192,1664,1223) (-25/34,-36/49) -> (60/23,47/18) Hyperbolic Matrix(265,192,432,313) (-8/11,-13/18) -> (11/18,8/13) Hyperbolic Matrix(167,120,32,23) (-13/18,-5/7) -> (5/1,11/2) Hyperbolic Matrix(409,288,240,169) (-12/17,-7/10) -> (17/10,12/7) Hyperbolic Matrix(241,168,208,145) (-7/10,-16/23) -> (8/7,7/6) Hyperbolic Matrix(1729,1200,1108,769) (-25/36,-9/13) -> (39/25,25/16) Hyperbolic Matrix(71,48,-108,-73) (-11/16,-2/3) -> (-2/3,-13/20) Parabolic Matrix(2255,1464,516,335) (-13/20,-24/37) -> (48/11,35/8) Hyperbolic Matrix(263,168,36,23) (-16/25,-7/11) -> (7/1,8/1) Hyperbolic Matrix(457,288,192,121) (-12/19,-5/8) -> (19/8,12/5) Hyperbolic Matrix(193,120,156,97) (-5/8,-13/21) -> (11/9,5/4) Hyperbolic Matrix(313,192,432,265) (-8/13,-11/18) -> (13/18,8/11) Hyperbolic Matrix(119,72,-200,-121) (-11/18,-3/5) -> (-3/5,-13/22) Parabolic Matrix(4391,2592,1340,791) (-13/22,-36/61) -> (36/11,59/18) Hyperbolic Matrix(1751,1032,772,455) (-23/39,-10/17) -> (34/15,25/11) Hyperbolic Matrix(409,240,288,169) (-10/17,-7/12) -> (17/12,10/7) Hyperbolic Matrix(289,168,332,193) (-7/12,-18/31) -> (6/7,7/8) Hyperbolic Matrix(167,96,40,23) (-11/19,-4/7) -> (4/1,13/3) Hyperbolic Matrix(169,96,44,25) (-4/7,-9/16) -> (15/4,4/1) Hyperbolic Matrix(1199,672,1968,1103) (-23/41,-14/25) -> (14/23,25/41) Hyperbolic Matrix(2063,1152,428,239) (-19/34,-24/43) -> (24/5,29/6) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(313,168,136,73) (-7/13,-1/2) -> (23/10,7/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(311,-168,224,-121) (1/2,6/11) -> (18/13,25/18) Hyperbolic Matrix(601,-336,728,-407) (5/9,14/25) -> (14/17,19/23) Hyperbolic Matrix(769,-432,340,-191) (14/25,9/16) -> (9/4,34/15) Hyperbolic Matrix(745,-432,288,-167) (11/19,7/12) -> (31/12,13/5) Hyperbolic Matrix(121,-72,200,-119) (10/17,3/5) -> (3/5,14/23) Parabolic Matrix(2833,-1728,864,-527) (25/41,11/18) -> (59/18,23/7) Hyperbolic Matrix(311,-192,196,-121) (8/13,5/8) -> (19/12,8/5) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(673,-432,148,-95) (16/25,9/14) -> (9/2,32/7) Hyperbolic Matrix(335,-216,76,-49) (9/14,2/3) -> (22/5,9/2) Hyperbolic Matrix(383,-264,280,-193) (2/3,9/13) -> (15/11,26/19) Hyperbolic Matrix(311,-216,36,-25) (9/13,16/23) -> (8/1,9/1) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(263,-192,100,-73) (8/11,3/4) -> (21/8,8/3) Hyperbolic Matrix(409,-312,156,-119) (3/4,10/13) -> (34/13,21/8) Hyperbolic Matrix(215,-168,32,-25) (7/9,11/14) -> (13/2,7/1) Hyperbolic Matrix(817,-672,524,-431) (9/11,14/17) -> (14/9,39/25) Hyperbolic Matrix(697,-576,144,-119) (19/23,5/6) -> (29/6,5/1) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(1129,-1440,432,-551) (14/11,23/18) -> (47/18,34/13) Hyperbolic Matrix(841,-1152,192,-263) (26/19,11/8) -> (35/8,22/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(719,-1128,276,-433) (36/23,11/7) -> (13/5,60/23) Hyperbolic Matrix(623,-1104,136,-241) (23/13,16/9) -> (32/7,23/5) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(383,-840,88,-193) (24/11,11/5) -> (13/3,48/11) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-144,4,-23) (11/2,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(25,216,-36,-311) -> Matrix(1,0,2,1) Matrix(23,168,36,263) -> Matrix(1,2,0,1) Matrix(25,168,-32,-215) -> Matrix(1,2,-2,-3) Matrix(47,264,-60,-337) -> Matrix(7,6,-6,-5) Matrix(23,120,32,167) -> Matrix(5,4,-4,-3) Matrix(71,336,-56,-265) -> Matrix(3,2,-8,-5) Matrix(119,552,36,167) -> Matrix(3,2,-2,-1) Matrix(95,432,-148,-673) -> Matrix(3,2,-2,-1) Matrix(49,216,-76,-335) -> Matrix(3,2,-2,-1) Matrix(23,96,40,167) -> Matrix(1,0,0,1) Matrix(25,96,44,169) -> Matrix(5,4,-4,-3) Matrix(71,264,32,119) -> Matrix(3,2,-8,-5) Matrix(73,264,60,217) -> Matrix(1,0,0,1) Matrix(47,168,40,143) -> Matrix(5,4,-4,-3) Matrix(97,336,28,97) -> Matrix(7,4,-2,-1) Matrix(169,576,120,409) -> Matrix(3,2,-2,-1) Matrix(71,240,92,311) -> Matrix(3,2,-2,-1) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(241,672,52,145) -> Matrix(5,4,-4,-3) Matrix(623,1728,-1056,-2929) -> Matrix(3,2,4,3) Matrix(313,864,96,265) -> Matrix(7,4,-2,-1) Matrix(71,192,44,119) -> Matrix(3,2,-2,-1) Matrix(73,192,-100,-263) -> Matrix(1,0,4,1) Matrix(167,432,-288,-745) -> Matrix(3,2,4,3) Matrix(47,120,56,143) -> Matrix(1,0,2,1) Matrix(49,120,20,49) -> Matrix(1,0,0,1) Matrix(71,168,-112,-265) -> Matrix(3,2,-2,-1) Matrix(145,336,104,241) -> Matrix(7,4,-2,-1) Matrix(335,768,188,431) -> Matrix(3,2,-2,-1) Matrix(95,216,84,191) -> Matrix(3,2,-2,-1) Matrix(191,432,-340,-769) -> Matrix(1,0,2,1) Matrix(119,264,32,71) -> Matrix(5,4,-4,-3) Matrix(23,48,-12,-25) -> Matrix(1,0,0,1) Matrix(311,576,-480,-889) -> Matrix(11,6,-2,-1) Matrix(313,576,144,265) -> Matrix(13,6,-24,-11) Matrix(119,216,92,167) -> Matrix(1,0,2,1) Matrix(241,432,188,337) -> Matrix(1,0,0,1) Matrix(431,768,188,335) -> Matrix(1,0,2,1) Matrix(433,768,-676,-1199) -> Matrix(3,2,-2,-1) Matrix(95,168,108,191) -> Matrix(3,2,-2,-1) Matrix(97,168,56,97) -> Matrix(3,2,-2,-1) Matrix(71,120,-100,-169) -> Matrix(1,0,2,1) Matrix(73,120,132,217) -> Matrix(7,4,-2,-1) Matrix(265,432,192,313) -> Matrix(5,2,2,1) Matrix(119,192,44,71) -> Matrix(1,0,2,1) Matrix(121,192,-196,-311) -> Matrix(5,2,-8,-3) Matrix(47,72,-32,-49) -> Matrix(1,0,2,1) Matrix(599,864,-816,-1177) -> Matrix(3,4,8,11) Matrix(1201,1728,768,1105) -> Matrix(7,6,-6,-5) Matrix(385,552,-468,-671) -> Matrix(1,0,2,1) Matrix(169,240,288,409) -> Matrix(3,2,-2,-1) Matrix(577,816,408,577) -> Matrix(7,4,-2,-1) Matrix(409,576,120,169) -> Matrix(3,2,-2,-1) Matrix(121,168,-224,-311) -> Matrix(7,4,-2,-1) Matrix(313,432,192,265) -> Matrix(1,0,2,1) Matrix(193,264,-280,-383) -> Matrix(5,2,2,1) Matrix(71,96,88,119) -> Matrix(1,0,2,1) Matrix(73,96,92,121) -> Matrix(1,0,2,1) Matrix(167,216,92,119) -> Matrix(11,2,-6,-1) Matrix(337,432,188,241) -> Matrix(1,-2,0,1) Matrix(527,672,-716,-913) -> Matrix(3,2,10,7) Matrix(455,576,-816,-1033) -> Matrix(5,2,2,1) Matrix(457,576,96,121) -> Matrix(7,2,-4,-1) Matrix(215,264,136,167) -> Matrix(1,0,4,1) Matrix(217,264,60,73) -> Matrix(15,2,-8,-1) Matrix(143,168,40,47) -> Matrix(1,-2,0,1) Matrix(145,168,208,241) -> Matrix(1,-2,0,1) Matrix(191,216,84,95) -> Matrix(1,0,2,1) Matrix(191,168,108,95) -> Matrix(3,2,-2,-1) Matrix(361,312,140,121) -> Matrix(1,0,6,1) Matrix(143,120,56,47) -> Matrix(1,0,-2,1) Matrix(407,336,-728,-601) -> Matrix(3,-2,2,-1) Matrix(263,216,28,23) -> Matrix(1,-2,0,1) Matrix(119,96,88,71) -> Matrix(1,0,0,1) Matrix(121,96,92,73) -> Matrix(1,0,0,1) Matrix(311,240,92,71) -> Matrix(3,2,-2,-1) Matrix(95,72,-128,-97) -> Matrix(1,0,6,1) Matrix(4343,3192,1664,1223) -> Matrix(25,-8,22,-7) Matrix(265,192,432,313) -> Matrix(3,-2,2,-1) Matrix(167,120,32,23) -> Matrix(3,-4,-2,3) Matrix(409,288,240,169) -> Matrix(1,-2,0,1) Matrix(241,168,208,145) -> Matrix(1,-2,0,1) Matrix(1729,1200,1108,769) -> Matrix(3,-4,-2,3) Matrix(71,48,-108,-73) -> Matrix(1,0,0,1) Matrix(2255,1464,516,335) -> Matrix(1,-6,0,1) Matrix(263,168,36,23) -> Matrix(1,2,-2,-3) Matrix(457,288,192,121) -> Matrix(3,2,-2,-1) Matrix(193,120,156,97) -> Matrix(1,0,0,1) Matrix(313,192,432,265) -> Matrix(1,0,2,1) Matrix(119,72,-200,-121) -> Matrix(1,0,6,1) Matrix(4391,2592,1340,791) -> Matrix(5,-2,-2,1) Matrix(1751,1032,772,455) -> Matrix(1,0,-2,1) Matrix(409,240,288,169) -> Matrix(5,-2,-2,1) Matrix(289,168,332,193) -> Matrix(5,-2,-2,1) Matrix(167,96,40,23) -> Matrix(1,0,-2,1) Matrix(169,96,44,25) -> Matrix(3,-4,-2,3) Matrix(1199,672,1968,1103) -> Matrix(1,0,0,1) Matrix(2063,1152,428,239) -> Matrix(3,-4,-2,3) Matrix(217,120,132,73) -> Matrix(1,-4,0,1) Matrix(313,168,136,73) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(311,-168,224,-121) -> Matrix(1,-6,0,1) Matrix(601,-336,728,-407) -> Matrix(1,2,0,1) Matrix(769,-432,340,-191) -> Matrix(1,2,2,5) Matrix(745,-432,288,-167) -> Matrix(1,0,4,1) Matrix(121,-72,200,-119) -> Matrix(1,0,2,1) Matrix(2833,-1728,864,-527) -> Matrix(1,-2,0,1) Matrix(311,-192,196,-121) -> Matrix(1,0,0,1) Matrix(265,-168,112,-71) -> Matrix(1,-2,0,1) Matrix(673,-432,148,-95) -> Matrix(3,-4,-2,3) Matrix(335,-216,76,-49) -> Matrix(1,-4,0,1) Matrix(383,-264,280,-193) -> Matrix(1,4,0,1) Matrix(311,-216,36,-25) -> Matrix(1,2,0,1) Matrix(169,-120,100,-71) -> Matrix(1,0,0,1) Matrix(263,-192,100,-73) -> Matrix(1,2,0,1) Matrix(409,-312,156,-119) -> Matrix(1,2,0,1) Matrix(215,-168,32,-25) -> Matrix(3,4,-4,-5) Matrix(817,-672,524,-431) -> Matrix(3,2,-2,-1) Matrix(697,-576,144,-119) -> Matrix(3,2,-2,-1) Matrix(265,-336,56,-71) -> Matrix(3,2,-2,-1) Matrix(1129,-1440,432,-551) -> Matrix(1,2,0,1) Matrix(841,-1152,192,-263) -> Matrix(1,-4,0,1) Matrix(49,-72,32,-47) -> Matrix(1,0,0,1) Matrix(719,-1128,276,-433) -> Matrix(3,2,4,3) Matrix(623,-1104,136,-241) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(3,4,-4,-5) Matrix(383,-840,88,-193) -> Matrix(5,2,2,1) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(25,-144,4,-23) -> Matrix(11,12,-12,-13) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 Degree of the the map X: 32 Degree of the the map Y: 128 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 384 Minimal number of generators: 65 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -3/1 -2/1 -9/5 -3/2 -4/3 -6/5 -1/1 -9/11 -3/4 -2/3 -3/5 0/1 1/2 3/5 2/3 3/4 9/11 1/1 6/5 5/4 4/3 3/2 12/7 9/5 2/1 24/11 11/5 7/3 12/5 5/2 3/1 11/3 4/1 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -2/3 -4/1 -1/1 -3/1 -2/3 0/1 -8/3 -1/2 -13/5 -2/3 -18/7 -1/2 -5/2 0/1 -12/5 -1/1 -7/3 -2/3 -9/4 -2/3 0/1 -11/5 -2/3 -4/7 -2/1 -1/2 1/0 -13/7 -2/3 -4/7 -24/13 -1/2 -11/6 -2/5 -9/5 0/1 -7/4 -1/2 0/1 -12/7 -1/1 -1/3 -5/3 -2/3 0/1 -8/5 -1/2 -11/7 -2/5 -3/2 0/1 -13/9 -2/1 -36/25 -1/1 -23/16 -1/1 -2/3 -10/7 -1/1 -7/5 -2/3 0/1 -18/13 -1/2 -11/8 -1/2 0/1 -15/11 -2/5 0/1 -4/3 -1/1 -1/3 -13/10 -2/7 -9/7 0/1 -5/4 -1/3 0/1 -11/9 -2/11 0/1 -6/5 0/1 -1/1 0/1 -6/7 0/1 -5/6 0/1 -9/11 0/1 2/1 -4/5 -1/1 1/1 -11/14 -2/1 -7/9 -2/3 0/1 -3/4 0/1 -11/15 2/5 -8/11 1/2 -5/7 0/1 2/1 -12/17 -1/1 1/1 -7/10 0/1 -9/13 0/1 2/1 -11/16 2/1 1/0 -2/3 -1/1 1/1 -7/11 0/1 -12/19 -1/1 -5/8 -1/1 0/1 -3/5 0/1 -7/12 0/1 1/2 -11/19 0/1 -4/7 1/1 -5/9 2/1 -6/11 1/0 -7/13 -4/1 -2/1 -1/2 0/1 0/1 -1/2 1/0 1/2 0/1 5/9 -4/1 -2/1 4/7 -1/1 3/5 0/1 8/13 1/0 5/8 0/1 1/1 7/11 0/1 2/1 2/3 1/0 9/13 -4/1 -2/1 7/10 -2/1 5/7 -2/1 8/11 1/0 3/4 -2/1 0/1 7/9 -2/1 11/14 -2/3 4/5 -1/1 1/1 9/11 -2/1 0/1 5/6 0/1 1/1 -2/1 0/1 7/6 -2/1 6/5 -1/1 11/9 0/1 5/4 -1/1 0/1 9/7 0/1 13/10 -2/1 4/3 -1/1 1/1 15/11 0/1 2/1 26/19 1/0 11/8 0/1 1/0 7/5 -2/1 10/7 -3/2 1/0 3/2 -2/1 0/1 14/9 -3/2 1/0 25/16 -4/3 -1/1 36/23 -1/1 11/7 -2/3 0/1 19/12 0/1 1/1 8/5 1/0 5/3 -2/1 17/10 -2/1 12/7 -3/1 -1/1 7/4 -2/1 1/0 9/5 -2/1 11/6 -8/5 2/1 -1/1 13/6 -4/7 24/11 -1/2 11/5 -2/5 9/4 0/1 7/3 -2/1 0/1 19/8 -2/1 -1/1 12/5 -1/1 5/2 0/1 3/1 -2/1 0/1 7/2 -2/1 18/5 -2/1 11/3 -2/1 -8/5 15/4 -2/1 -4/3 4/1 -1/1 5/1 -2/1 -4/3 11/2 -6/5 6/1 -1/1 13/2 -6/7 7/1 -2/3 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,84,-8,-61) (-6/1,1/0) -> (-18/13,-11/8) Hyperbolic Matrix(11,60,-20,-109) (-6/1,-5/1) -> (-5/9,-6/11) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(73,192,-100,-263) (-8/3,-13/5) -> (-11/15,-8/11) Hyperbolic Matrix(107,276,88,227) (-13/5,-18/7) -> (6/5,11/9) Hyperbolic Matrix(61,156,52,133) (-18/7,-5/2) -> (7/6,6/5) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,-112,-265) (-12/5,-7/3) -> (-7/11,-12/19) Hyperbolic Matrix(37,84,48,109) (-7/3,-9/4) -> (3/4,7/9) Hyperbolic Matrix(119,264,32,71) (-9/4,-11/5) -> (11/3,15/4) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(395,732,252,467) (-13/7,-24/13) -> (36/23,11/7) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(119,216,92,167) (-11/6,-9/5) -> (9/7,13/10) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(83,132,-144,-229) (-8/5,-11/7) -> (-11/19,-4/7) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(491,708,224,323) (-13/9,-36/25) -> (24/11,11/5) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(493,708,360,517) (-23/16,-10/7) -> (26/19,11/8) Hyperbolic Matrix(59,84,92,131) (-10/7,-7/5) -> (7/11,2/3) Hyperbolic Matrix(121,168,-224,-311) (-7/5,-18/13) -> (-6/11,-7/13) Hyperbolic Matrix(193,264,-280,-383) (-11/8,-15/11) -> (-9/13,-11/16) Hyperbolic Matrix(71,96,88,119) (-15/11,-4/3) -> (4/5,9/11) Hyperbolic Matrix(73,96,92,121) (-4/3,-13/10) -> (11/14,4/5) Hyperbolic Matrix(167,216,92,119) (-13/10,-9/7) -> (9/5,11/6) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(215,264,136,167) (-5/4,-11/9) -> (11/7,19/12) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(157,132,44,37) (-6/7,-5/6) -> (7/2,18/5) Hyperbolic Matrix(131,108,188,155) (-5/6,-9/11) -> (9/13,7/10) Hyperbolic Matrix(119,96,88,71) (-9/11,-4/5) -> (4/3,15/11) Hyperbolic Matrix(121,96,92,73) (-4/5,-11/14) -> (13/10,4/3) Hyperbolic Matrix(107,84,-200,-157) (-11/14,-7/9) -> (-7/13,-1/2) Hyperbolic Matrix(109,84,48,37) (-7/9,-3/4) -> (9/4,7/3) Hyperbolic Matrix(179,132,80,59) (-3/4,-11/15) -> (11/5,9/4) Hyperbolic Matrix(83,60,148,107) (-8/11,-5/7) -> (5/9,4/7) Hyperbolic Matrix(409,288,240,169) (-12/17,-7/10) -> (17/10,12/7) Hyperbolic Matrix(155,108,188,131) (-7/10,-9/13) -> (9/11,5/6) Hyperbolic Matrix(299,204,192,131) (-11/16,-2/3) -> (14/9,25/16) Hyperbolic Matrix(131,84,92,59) (-2/3,-7/11) -> (7/5,10/7) Hyperbolic Matrix(457,288,192,121) (-12/19,-5/8) -> (19/8,12/5) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(227,132,184,107) (-7/12,-11/19) -> (11/9,5/4) Hyperbolic Matrix(107,60,148,83) (-4/7,-5/9) -> (5/7,8/11) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(311,-192,196,-121) (8/13,5/8) -> (19/12,8/5) Hyperbolic Matrix(265,-168,112,-71) (5/8,7/11) -> (7/3,19/8) Hyperbolic Matrix(383,-264,280,-193) (2/3,9/13) -> (15/11,26/19) Hyperbolic Matrix(169,-120,100,-71) (7/10,5/7) -> (5/3,17/10) Hyperbolic Matrix(181,-132,48,-35) (8/11,3/4) -> (15/4,4/1) Hyperbolic Matrix(215,-168,32,-25) (7/9,11/14) -> (13/2,7/1) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(61,-84,8,-11) (11/8,7/5) -> (7/1,1/0) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(25,-144,4,-23) (11/2,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,84,-8,-61) -> Matrix(1,0,-1,1) Matrix(11,60,-20,-109) -> Matrix(5,4,1,1) Matrix(13,60,8,37) -> Matrix(1,0,1,1) Matrix(11,36,-4,-13) -> Matrix(3,2,-5,-3) Matrix(73,192,-100,-263) -> Matrix(1,0,4,1) Matrix(107,276,88,227) -> Matrix(3,2,-5,-3) Matrix(61,156,52,133) -> Matrix(5,2,-3,-1) Matrix(49,120,20,49) -> Matrix(1,0,0,1) Matrix(71,168,-112,-265) -> Matrix(3,2,-2,-1) Matrix(37,84,48,109) -> Matrix(1,0,1,1) Matrix(119,264,32,71) -> Matrix(5,4,-4,-3) Matrix(23,48,-12,-25) -> Matrix(1,0,0,1) Matrix(395,732,252,467) -> Matrix(7,4,-9,-5) Matrix(313,576,144,265) -> Matrix(13,6,-24,-11) Matrix(119,216,92,167) -> Matrix(1,0,2,1) Matrix(61,108,48,85) -> Matrix(1,0,1,1) Matrix(97,168,56,97) -> Matrix(3,2,-2,-1) Matrix(71,120,-100,-169) -> Matrix(1,0,2,1) Matrix(37,60,8,13) -> Matrix(5,2,-3,-1) Matrix(83,132,-144,-229) -> Matrix(5,2,7,3) Matrix(47,72,-32,-49) -> Matrix(1,0,2,1) Matrix(491,708,224,323) -> Matrix(3,4,-7,-9) Matrix(1201,1728,768,1105) -> Matrix(7,6,-6,-5) Matrix(493,708,360,517) -> Matrix(3,2,1,1) Matrix(59,84,92,131) -> Matrix(3,2,1,1) Matrix(121,168,-224,-311) -> Matrix(7,4,-2,-1) Matrix(193,264,-280,-383) -> Matrix(5,2,2,1) Matrix(71,96,88,119) -> Matrix(1,0,2,1) Matrix(73,96,92,121) -> Matrix(1,0,2,1) Matrix(167,216,92,119) -> Matrix(11,2,-6,-1) Matrix(85,108,48,61) -> Matrix(5,2,-3,-1) Matrix(215,264,136,167) -> Matrix(1,0,4,1) Matrix(217,264,60,73) -> Matrix(15,2,-8,-1) Matrix(11,12,-12,-13) -> Matrix(1,0,1,1) Matrix(157,132,44,37) -> Matrix(3,-2,-1,1) Matrix(131,108,188,155) -> Matrix(3,-2,-1,1) Matrix(119,96,88,71) -> Matrix(1,0,0,1) Matrix(121,96,92,73) -> Matrix(1,0,0,1) Matrix(107,84,-200,-157) -> Matrix(1,2,-1,-1) Matrix(109,84,48,37) -> Matrix(1,0,1,1) Matrix(179,132,80,59) -> Matrix(1,0,-5,1) Matrix(83,60,148,107) -> Matrix(3,-2,-1,1) Matrix(409,288,240,169) -> Matrix(1,-2,0,1) Matrix(155,108,188,131) -> Matrix(1,0,-1,1) Matrix(299,204,192,131) -> Matrix(1,2,-1,-1) Matrix(131,84,92,59) -> Matrix(1,2,-1,-1) Matrix(457,288,192,121) -> Matrix(3,2,-2,-1) Matrix(59,36,-100,-61) -> Matrix(1,0,3,1) Matrix(227,132,184,107) -> Matrix(1,0,-3,1) Matrix(107,60,148,83) -> Matrix(1,0,-1,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(109,-60,20,-11) -> Matrix(1,6,-1,-5) Matrix(61,-36,100,-59) -> Matrix(1,0,1,1) Matrix(311,-192,196,-121) -> Matrix(1,0,0,1) Matrix(265,-168,112,-71) -> Matrix(1,-2,0,1) Matrix(383,-264,280,-193) -> Matrix(1,4,0,1) Matrix(169,-120,100,-71) -> Matrix(1,0,0,1) Matrix(181,-132,48,-35) -> Matrix(1,4,-1,-3) Matrix(215,-168,32,-25) -> Matrix(3,4,-4,-5) Matrix(13,-12,12,-11) -> Matrix(1,2,-1,-1) Matrix(61,-84,8,-11) -> Matrix(1,0,-1,1) Matrix(49,-72,32,-47) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(3,4,-4,-5) Matrix(13,-36,4,-11) -> Matrix(1,2,-1,-1) Matrix(25,-144,4,-23) -> Matrix(11,12,-12,-13) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.