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 -5/6 -3/4 -2/3 -23/36 -11/18 -7/12 -1/2 -11/24 -7/16 -5/12 -49/120 -3/8 -17/48 -7/20 -1/3 -11/36 -29/96 -3/10 -7/24 -5/18 -1/4 -11/48 -5/22 -2/9 -3/14 -5/24 -3/16 -1/6 -3/20 -1/8 -1/9 0/1 1/9 1/8 1/7 1/6 2/11 3/16 1/5 3/14 2/9 1/4 3/11 5/18 2/7 3/10 11/36 1/3 7/20 3/8 2/5 5/12 7/16 1/2 11/20 7/12 13/22 11/18 23/36 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/22 -8/9 3/58 -7/8 1/19 -13/15 3/56 -6/7 3/56 -5/6 1/18 -14/17 5/88 -9/11 1/18 -13/16 2/35 -4/5 3/52 -11/14 1/17 4/67 -18/23 13/216 -7/9 1/16 -3/4 1/16 -11/15 1/16 -19/26 0/1 1/15 -8/11 1/16 -13/18 1/16 -18/25 11/172 -5/7 3/46 -7/10 1/15 2/29 -23/33 5/72 -16/23 9/128 -25/36 1/14 -9/13 1/14 -11/16 0/1 -2/3 1/14 -13/20 1/12 -11/17 1/10 -9/14 0/1 1/15 -16/25 7/100 -23/36 1/14 -7/11 1/14 -5/8 1/13 -8/13 1/12 -11/18 1/12 -14/23 5/56 -17/28 1/10 -3/5 1/14 -7/12 1/12 -11/19 3/34 -26/45 3/34 -41/71 11/122 -15/26 0/1 1/11 -19/33 1/12 -4/7 1/12 -13/23 3/26 -9/16 0/1 -14/25 3/44 -5/9 1/12 -21/38 0/1 1/9 -16/29 1/16 -11/20 1/12 -6/11 1/12 -1/2 0/1 1/11 -6/13 1/12 -11/24 1/11 -5/11 1/10 -14/31 1/8 -9/20 1/10 -13/29 1/6 -4/9 1/10 -11/25 3/22 -7/16 0/1 -10/23 3/40 -3/7 1/10 -5/12 1/10 -7/17 1/10 -9/22 2/19 1/9 -29/71 13/118 -49/120 1/9 -20/49 9/80 -11/27 1/8 -2/5 1/8 -11/28 1/12 -20/51 3/34 -9/23 5/54 -7/18 1/10 -5/13 1/10 -3/8 1/9 -4/11 1/8 -13/36 1/8 -9/25 7/54 -5/14 0/1 1/7 -11/31 1/2 -17/48 0/1 -6/17 1/12 -7/20 1/10 -8/23 7/64 -1/3 1/8 -6/19 1/8 -5/16 0/1 -4/13 1/8 -11/36 1/8 -7/23 9/70 -10/33 5/38 -13/43 13/98 -29/96 2/15 -16/53 15/112 -3/10 2/15 1/7 -5/17 7/50 -7/24 1/7 -2/7 3/20 -7/25 11/70 -5/18 1/6 -3/11 1/6 -1/4 1/6 -3/13 1/6 -11/48 2/11 -8/35 3/16 -5/22 2/11 1/5 -7/31 1/6 -2/9 1/6 -5/23 13/70 -3/14 4/21 1/5 -4/19 11/56 -5/24 1/5 -1/5 3/14 -4/21 3/14 -3/16 2/9 -2/11 1/4 -1/6 1/4 -2/13 1/4 -3/20 1/4 -1/7 3/10 -2/15 3/10 -1/8 1/3 -1/9 3/8 0/1 1/0 1/9 -3/8 1/8 -1/3 2/15 -3/10 1/7 -3/10 1/6 -1/4 3/17 -5/22 2/11 -1/4 3/16 -2/9 1/5 -3/14 3/14 -1/5 -4/21 5/23 -13/70 2/9 -1/6 1/4 -1/6 4/15 -1/6 7/26 -1/7 0/1 3/11 -1/6 5/18 -1/6 7/25 -11/70 2/7 -3/20 3/10 -1/7 -2/15 10/33 -5/38 7/23 -9/70 11/36 -1/8 4/13 -1/8 5/16 0/1 1/3 -1/8 7/20 -1/10 6/17 -1/12 5/14 -1/7 0/1 9/25 -7/54 13/36 -1/8 4/11 -1/8 3/8 -1/9 5/13 -1/10 7/18 -1/10 9/23 -5/54 11/28 -1/12 2/5 -1/8 5/12 -1/10 8/19 -3/32 19/45 -3/32 30/71 -11/120 11/26 -1/11 0/1 14/33 -1/10 3/7 -1/10 10/23 -3/40 7/16 0/1 11/25 -3/22 4/9 -1/10 17/38 -1/13 0/1 13/29 -1/6 9/20 -1/10 5/11 -1/10 1/2 -1/11 0/1 7/13 -1/10 13/24 -1/11 6/11 -1/12 17/31 -1/14 11/20 -1/12 16/29 -1/16 5/9 -1/12 14/25 -3/44 9/16 0/1 13/23 -3/26 4/7 -1/12 7/12 -1/12 10/17 -1/12 13/22 -2/25 -1/13 42/71 -13/168 71/120 -1/13 29/49 -9/118 16/27 -1/14 3/5 -1/14 17/28 -1/10 31/51 -3/32 14/23 -5/56 11/18 -1/12 8/13 -1/12 5/8 -1/13 7/11 -1/14 23/36 -1/14 16/25 -7/100 9/14 -1/15 0/1 20/31 -1/20 31/48 0/1 11/17 -1/10 13/20 -1/12 15/23 -7/90 2/3 -1/14 13/19 -1/14 11/16 0/1 9/13 -1/14 25/36 -1/14 16/23 -9/128 23/33 -5/72 30/43 -13/188 67/96 -2/29 37/53 -15/218 7/10 -2/29 -1/15 12/17 -7/104 17/24 -1/15 5/7 -3/46 18/25 -11/172 13/18 -1/16 8/11 -1/16 3/4 -1/16 10/13 -1/16 37/48 -2/33 27/35 -3/50 17/22 -2/33 -1/17 24/31 -1/16 7/9 -1/16 18/23 -13/216 11/14 -4/67 -1/17 15/19 -11/186 19/24 -1/17 4/5 -3/52 17/21 -3/52 13/16 -2/35 9/11 -1/18 5/6 -1/18 11/13 -1/18 17/20 -1/18 6/7 -3/56 13/15 -3/56 7/8 -1/19 8/9 -3/58 1/1 -1/22 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(73,66,240,217) (-1/1,-8/9) -> (10/33,7/23) Hyperbolic Matrix(25,22,192,169) (-8/9,-7/8) -> (1/8,2/15) Hyperbolic Matrix(23,20,192,167) (-7/8,-13/15) -> (1/9,1/8) Hyperbolic Matrix(409,354,528,457) (-13/15,-6/7) -> (24/31,7/9) Hyperbolic Matrix(119,100,-144,-121) (-6/7,-5/6) -> (-5/6,-14/17) Parabolic Matrix(239,196,-528,-433) (-14/17,-9/11) -> (-5/11,-14/31) Hyperbolic Matrix(265,216,384,313) (-9/11,-13/16) -> (11/16,9/13) Hyperbolic Matrix(121,98,-384,-311) (-13/16,-4/5) -> (-6/19,-5/16) Hyperbolic Matrix(71,56,-336,-265) (-4/5,-11/14) -> (-3/14,-4/19) Hyperbolic Matrix(431,338,672,527) (-11/14,-18/23) -> (16/25,9/14) Hyperbolic Matrix(241,188,432,337) (-18/23,-7/9) -> (5/9,14/25) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(719,526,-1248,-913) (-11/15,-19/26) -> (-15/26,-19/33) Hyperbolic Matrix(263,192,-1152,-841) (-19/26,-8/11) -> (-8/35,-5/22) Hyperbolic Matrix(265,192,432,313) (-8/11,-13/18) -> (11/18,8/13) Hyperbolic Matrix(527,380,864,623) (-13/18,-18/25) -> (14/23,11/18) Hyperbolic Matrix(145,104,336,241) (-18/25,-5/7) -> (3/7,10/23) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(743,518,-1344,-937) (-7/10,-23/33) -> (-5/9,-21/38) Hyperbolic Matrix(23,16,240,167) (-23/33,-16/23) -> (0/1,1/9) Hyperbolic Matrix(1105,768,1728,1201) (-16/23,-25/36) -> (23/36,16/25) Hyperbolic Matrix(551,382,864,599) (-25/36,-9/13) -> (7/11,23/36) Hyperbolic Matrix(313,216,384,265) (-9/13,-11/16) -> (13/16,9/11) Hyperbolic Matrix(73,50,-384,-263) (-11/16,-2/3) -> (-4/21,-3/16) Hyperbolic Matrix(433,282,-1104,-719) (-2/3,-13/20) -> (-11/28,-20/51) Hyperbolic Matrix(527,342,960,623) (-13/20,-11/17) -> (17/31,11/20) Hyperbolic Matrix(239,154,-672,-433) (-11/17,-9/14) -> (-5/14,-11/31) Hyperbolic Matrix(527,338,672,431) (-9/14,-16/25) -> (18/23,11/14) Hyperbolic Matrix(1201,768,1728,1105) (-16/25,-23/36) -> (25/36,16/23) Hyperbolic Matrix(599,382,864,551) (-23/36,-7/11) -> (9/13,25/36) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(313,192,432,265) (-8/13,-11/18) -> (13/18,8/11) Hyperbolic Matrix(623,380,864,527) (-11/18,-14/23) -> (18/25,13/18) Hyperbolic Matrix(385,234,-1104,-671) (-14/23,-17/28) -> (-7/20,-8/23) Hyperbolic Matrix(409,248,912,553) (-17/28,-3/5) -> (13/29,9/20) Hyperbolic Matrix(167,98,-288,-169) (-3/5,-7/12) -> (-7/12,-11/19) Parabolic Matrix(623,360,912,527) (-11/19,-26/45) -> (2/3,13/19) Hyperbolic Matrix(817,472,1248,721) (-26/45,-41/71) -> (15/23,2/3) Hyperbolic Matrix(1393,804,-3408,-1967) (-41/71,-15/26) -> (-9/22,-29/71) Hyperbolic Matrix(289,166,336,193) (-19/33,-4/7) -> (6/7,13/15) Hyperbolic Matrix(95,54,336,191) (-4/7,-13/23) -> (7/25,2/7) Hyperbolic Matrix(337,190,768,433) (-13/23,-9/16) -> (7/16,11/25) Hyperbolic Matrix(335,188,768,431) (-9/16,-14/25) -> (10/23,7/16) Hyperbolic Matrix(337,188,432,241) (-14/25,-5/9) -> (7/9,18/23) Hyperbolic Matrix(695,384,-2304,-1273) (-21/38,-16/29) -> (-16/53,-3/10) Hyperbolic Matrix(359,198,912,503) (-16/29,-11/20) -> (11/28,2/5) Hyperbolic Matrix(73,40,-480,-263) (-11/20,-6/11) -> (-2/13,-3/20) Hyperbolic Matrix(23,12,-48,-25) (-6/11,-1/2) -> (-1/2,-6/13) Parabolic Matrix(313,144,576,265) (-6/13,-11/24) -> (13/24,6/11) Hyperbolic Matrix(311,142,576,263) (-11/24,-5/11) -> (7/13,13/24) Hyperbolic Matrix(337,152,960,433) (-14/31,-9/20) -> (7/20,6/17) Hyperbolic Matrix(553,248,912,409) (-9/20,-13/29) -> (3/5,17/28) Hyperbolic Matrix(407,182,-1344,-601) (-13/29,-4/9) -> (-10/33,-13/43) Hyperbolic Matrix(95,42,432,191) (-4/9,-11/25) -> (5/23,2/9) Hyperbolic Matrix(433,190,768,337) (-11/25,-7/16) -> (9/16,13/23) Hyperbolic Matrix(431,188,768,335) (-7/16,-10/23) -> (14/25,9/16) Hyperbolic Matrix(241,104,336,145) (-10/23,-3/7) -> (5/7,18/25) Hyperbolic Matrix(119,50,-288,-121) (-3/7,-5/12) -> (-5/12,-7/17) Parabolic Matrix(239,98,-1056,-433) (-7/17,-9/22) -> (-5/22,-7/31) Hyperbolic Matrix(8521,3480,14400,5881) (-29/71,-49/120) -> (71/120,29/49) Hyperbolic Matrix(8519,3478,14400,5879) (-49/120,-20/49) -> (42/71,71/120) Hyperbolic Matrix(1897,774,3120,1273) (-20/49,-11/27) -> (31/51,14/23) Hyperbolic Matrix(193,78,240,97) (-11/27,-2/5) -> (4/5,17/21) Hyperbolic Matrix(503,198,912,359) (-2/5,-11/28) -> (11/20,16/29) Hyperbolic Matrix(1847,724,3120,1223) (-20/51,-9/23) -> (29/49,16/27) Hyperbolic Matrix(241,94,864,337) (-9/23,-7/18) -> (5/18,7/25) Hyperbolic Matrix(119,46,432,167) (-7/18,-5/13) -> (3/11,5/18) Hyperbolic Matrix(121,46,192,73) (-5/13,-3/8) -> (5/8,7/11) Hyperbolic Matrix(119,44,192,71) (-3/8,-4/11) -> (8/13,5/8) Hyperbolic Matrix(265,96,864,313) (-4/11,-13/36) -> (11/36,4/13) Hyperbolic Matrix(527,190,1728,623) (-13/36,-9/25) -> (7/23,11/36) Hyperbolic Matrix(145,52,672,241) (-9/25,-5/14) -> (3/14,5/23) Hyperbolic Matrix(1489,528,2304,817) (-11/31,-17/48) -> (31/48,11/17) Hyperbolic Matrix(1487,526,2304,815) (-17/48,-6/17) -> (20/31,31/48) Hyperbolic Matrix(409,144,480,169) (-6/17,-7/20) -> (17/20,6/7) Hyperbolic Matrix(527,182,1248,431) (-8/23,-1/3) -> (19/45,30/71) Hyperbolic Matrix(385,122,912,289) (-1/3,-6/19) -> (8/19,19/45) Hyperbolic Matrix(71,22,384,119) (-5/16,-4/13) -> (2/11,3/16) Hyperbolic Matrix(313,96,864,265) (-4/13,-11/36) -> (13/36,4/11) Hyperbolic Matrix(623,190,1728,527) (-11/36,-7/23) -> (9/25,13/36) Hyperbolic Matrix(217,66,240,73) (-7/23,-10/33) -> (8/9,1/1) Hyperbolic Matrix(6433,1944,9216,2785) (-13/43,-29/96) -> (67/96,37/53) Hyperbolic Matrix(6431,1942,9216,2783) (-29/96,-16/53) -> (30/43,67/96) Hyperbolic Matrix(409,120,576,169) (-5/17,-7/24) -> (17/24,5/7) Hyperbolic Matrix(407,118,576,167) (-7/24,-2/7) -> (12/17,17/24) Hyperbolic Matrix(191,54,336,95) (-2/7,-7/25) -> (13/23,4/7) Hyperbolic Matrix(337,94,864,241) (-7/25,-5/18) -> (7/18,9/23) Hyperbolic Matrix(167,46,432,119) (-5/18,-3/11) -> (5/13,7/18) Hyperbolic Matrix(23,6,-96,-25) (-3/11,-1/4) -> (-1/4,-3/13) Parabolic Matrix(1777,408,2304,529) (-3/13,-11/48) -> (37/48,27/35) Hyperbolic Matrix(1775,406,2304,527) (-11/48,-8/35) -> (10/13,37/48) Hyperbolic Matrix(71,16,528,119) (-7/31,-2/9) -> (2/15,1/7) Hyperbolic Matrix(191,42,432,95) (-2/9,-5/23) -> (11/25,4/9) Hyperbolic Matrix(241,52,672,145) (-5/23,-3/14) -> (5/14,9/25) Hyperbolic Matrix(457,96,576,121) (-4/19,-5/24) -> (19/24,4/5) Hyperbolic Matrix(455,94,576,119) (-5/24,-1/5) -> (15/19,19/24) Hyperbolic Matrix(143,28,240,47) (-1/5,-4/21) -> (16/27,3/5) Hyperbolic Matrix(119,22,384,71) (-3/16,-2/11) -> (4/13,5/16) Hyperbolic Matrix(23,4,-144,-25) (-2/11,-1/6) -> (-1/6,-2/13) Parabolic Matrix(311,46,480,71) (-3/20,-1/7) -> (11/17,13/20) Hyperbolic Matrix(143,20,336,47) (-1/7,-2/15) -> (14/33,3/7) Hyperbolic Matrix(169,22,192,25) (-2/15,-1/8) -> (7/8,8/9) Hyperbolic Matrix(167,20,192,23) (-1/8,-1/9) -> (13/15,7/8) Hyperbolic Matrix(167,16,240,23) (-1/9,0/1) -> (16/23,23/33) Hyperbolic Matrix(25,-4,144,-23) (1/7,1/6) -> (1/6,3/17) Parabolic Matrix(289,-52,528,-95) (3/17,2/11) -> (6/11,17/31) Hyperbolic Matrix(263,-50,384,-73) (3/16,1/5) -> (13/19,11/16) Hyperbolic Matrix(265,-56,336,-71) (1/5,3/14) -> (11/14,15/19) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(529,-142,1248,-335) (4/15,7/26) -> (11/26,14/33) Hyperbolic Matrix(889,-240,1152,-311) (7/26,3/11) -> (27/35,17/22) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(601,-182,1344,-407) (3/10,10/33) -> (4/9,17/38) Hyperbolic Matrix(311,-98,384,-121) (5/16,1/3) -> (17/21,13/16) Hyperbolic Matrix(671,-234,1104,-385) (1/3,7/20) -> (17/28,31/51) Hyperbolic Matrix(433,-154,672,-239) (6/17,5/14) -> (9/14,20/31) Hyperbolic Matrix(719,-282,1104,-433) (9/23,11/28) -> (13/20,15/23) Hyperbolic Matrix(121,-50,288,-119) (2/5,5/12) -> (5/12,8/19) Parabolic Matrix(2015,-852,3408,-1441) (30/71,11/26) -> (13/22,42/71) Hyperbolic Matrix(1609,-720,2304,-1031) (17/38,13/29) -> (37/53,7/10) Hyperbolic Matrix(407,-184,480,-217) (9/20,5/11) -> (11/13,17/20) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(937,-518,1344,-743) (16/29,5/9) -> (23/33,30/43) Hyperbolic Matrix(169,-98,288,-167) (4/7,7/12) -> (7/12,10/17) Parabolic Matrix(817,-482,1056,-623) (10/17,13/22) -> (17/22,24/31) Hyperbolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(121,-100,144,-119) (9/11,5/6) -> (5/6,11/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-44,1) Matrix(73,66,240,217) -> Matrix(79,-4,-612,31) Matrix(25,22,192,169) -> Matrix(115,-6,-364,19) Matrix(23,20,192,167) -> Matrix(113,-6,-320,17) Matrix(409,354,528,457) -> Matrix(75,-4,-1256,67) Matrix(119,100,-144,-121) -> Matrix(73,-4,1296,-71) Matrix(239,196,-528,-433) -> Matrix(35,-2,368,-21) Matrix(265,216,384,313) -> Matrix(35,-2,-472,27) Matrix(121,98,-384,-311) -> Matrix(35,-2,228,-13) Matrix(71,56,-336,-265) -> Matrix(135,-8,692,-41) Matrix(431,338,672,527) -> Matrix(67,-4,-988,59) Matrix(241,188,432,337) -> Matrix(33,-2,-412,25) Matrix(71,54,-96,-73) -> Matrix(33,-2,512,-31) Matrix(719,526,-1248,-913) -> Matrix(1,0,-4,1) Matrix(263,192,-1152,-841) -> Matrix(29,-2,160,-11) Matrix(265,192,432,313) -> Matrix(1,0,-28,1) Matrix(527,380,864,623) -> Matrix(63,-4,-740,47) Matrix(145,104,336,241) -> Matrix(31,-2,-356,23) Matrix(71,50,-240,-169) -> Matrix(59,-4,428,-29) Matrix(743,518,-1344,-937) -> Matrix(29,-2,276,-19) Matrix(23,16,240,167) -> Matrix(57,-4,-128,9) Matrix(1105,768,1728,1201) -> Matrix(113,-8,-1596,113) Matrix(551,382,864,599) -> Matrix(27,-2,-364,27) Matrix(313,216,384,265) -> Matrix(27,-2,-472,35) Matrix(73,50,-384,-263) -> Matrix(31,-2,140,-9) Matrix(433,282,-1104,-719) -> Matrix(25,-2,288,-23) Matrix(527,342,960,623) -> Matrix(1,0,-24,1) Matrix(239,154,-672,-433) -> Matrix(1,0,-8,1) Matrix(527,338,672,431) -> Matrix(59,-4,-988,67) Matrix(1201,768,1728,1105) -> Matrix(113,-8,-1596,113) Matrix(599,382,864,551) -> Matrix(27,-2,-364,27) Matrix(73,46,192,121) -> Matrix(27,-2,-256,19) Matrix(71,44,192,119) -> Matrix(25,-2,-212,17) Matrix(313,192,432,265) -> Matrix(1,0,-28,1) Matrix(623,380,864,527) -> Matrix(47,-4,-740,63) Matrix(385,234,-1104,-671) -> Matrix(21,-2,200,-19) Matrix(409,248,912,553) -> Matrix(1,0,-20,1) Matrix(167,98,-288,-169) -> Matrix(25,-2,288,-23) Matrix(623,360,912,527) -> Matrix(23,-2,-356,31) Matrix(817,472,1248,721) -> Matrix(45,-4,-596,53) Matrix(1393,804,-3408,-1967) -> Matrix(21,-2,200,-19) Matrix(289,166,336,193) -> Matrix(21,-2,-388,37) Matrix(95,54,336,191) -> Matrix(21,-2,-136,13) Matrix(337,190,768,433) -> Matrix(1,0,-16,1) Matrix(335,188,768,431) -> Matrix(1,0,-28,1) Matrix(337,188,432,241) -> Matrix(25,-2,-412,33) Matrix(695,384,-2304,-1273) -> Matrix(17,-2,128,-15) Matrix(359,198,912,503) -> Matrix(1,0,-24,1) Matrix(73,40,-480,-263) -> Matrix(25,-2,88,-7) Matrix(23,12,-48,-25) -> Matrix(1,0,0,1) Matrix(313,144,576,265) -> Matrix(23,-2,-264,23) Matrix(311,142,576,263) -> Matrix(21,-2,-220,21) Matrix(337,152,960,433) -> Matrix(1,0,-20,1) Matrix(553,248,912,409) -> Matrix(1,0,-20,1) Matrix(407,182,-1344,-601) -> Matrix(25,-2,188,-15) Matrix(95,42,432,191) -> Matrix(19,-2,-104,11) Matrix(433,190,768,337) -> Matrix(1,0,-16,1) Matrix(431,188,768,335) -> Matrix(1,0,-28,1) Matrix(241,104,336,145) -> Matrix(23,-2,-356,31) Matrix(119,50,-288,-121) -> Matrix(21,-2,200,-19) Matrix(239,98,-1056,-433) -> Matrix(1,0,-4,1) Matrix(8521,3480,14400,5881) -> Matrix(199,-22,-2596,287) Matrix(8519,3478,14400,5879) -> Matrix(197,-22,-2552,285) Matrix(1897,774,3120,1273) -> Matrix(35,-4,-376,43) Matrix(193,78,240,97) -> Matrix(19,-2,-332,35) Matrix(503,198,912,359) -> Matrix(1,0,-24,1) Matrix(1847,724,3120,1223) -> Matrix(45,-4,-596,53) Matrix(241,94,864,337) -> Matrix(41,-4,-256,25) Matrix(119,46,432,167) -> Matrix(1,0,-16,1) Matrix(121,46,192,73) -> Matrix(19,-2,-256,27) Matrix(119,44,192,71) -> Matrix(17,-2,-212,25) Matrix(265,96,864,313) -> Matrix(17,-2,-144,17) Matrix(527,190,1728,623) -> Matrix(63,-8,-496,63) Matrix(145,52,672,241) -> Matrix(29,-4,-152,21) Matrix(1489,528,2304,817) -> Matrix(1,0,-12,1) Matrix(1487,526,2304,815) -> Matrix(1,0,-32,1) Matrix(409,144,480,169) -> Matrix(21,-2,-388,37) Matrix(527,182,1248,431) -> Matrix(35,-4,-376,43) Matrix(385,122,912,289) -> Matrix(13,-2,-136,21) Matrix(71,22,384,119) -> Matrix(17,-2,-76,9) Matrix(313,96,864,265) -> Matrix(17,-2,-144,17) Matrix(623,190,1728,527) -> Matrix(63,-8,-496,63) Matrix(217,66,240,73) -> Matrix(31,-4,-612,79) Matrix(6433,1944,9216,2785) -> Matrix(421,-56,-6112,813) Matrix(6431,1942,9216,2783) -> Matrix(419,-56,-6068,811) Matrix(409,120,576,169) -> Matrix(71,-10,-1072,151) Matrix(407,118,576,167) -> Matrix(69,-10,-1028,149) Matrix(191,54,336,95) -> Matrix(13,-2,-136,21) Matrix(337,94,864,241) -> Matrix(25,-4,-256,41) Matrix(167,46,432,119) -> Matrix(1,0,-16,1) Matrix(23,6,-96,-25) -> Matrix(13,-2,72,-11) Matrix(1777,408,2304,529) -> Matrix(45,-8,-748,133) Matrix(1775,406,2304,527) -> Matrix(43,-8,-704,131) Matrix(71,16,528,119) -> Matrix(21,-4,-68,13) Matrix(191,42,432,95) -> Matrix(11,-2,-104,19) Matrix(241,52,672,145) -> Matrix(21,-4,-152,29) Matrix(457,96,576,121) -> Matrix(71,-14,-1212,239) Matrix(455,94,576,119) -> Matrix(69,-14,-1168,237) Matrix(143,28,240,47) -> Matrix(9,-2,-112,25) Matrix(119,22,384,71) -> Matrix(9,-2,-76,17) Matrix(23,4,-144,-25) -> Matrix(17,-4,64,-15) Matrix(311,46,480,71) -> Matrix(7,-2,-80,23) Matrix(143,20,336,47) -> Matrix(7,-2,-80,23) Matrix(169,22,192,25) -> Matrix(19,-6,-364,115) Matrix(167,20,192,23) -> Matrix(17,-6,-320,113) Matrix(167,16,240,23) -> Matrix(9,-4,-128,57) Matrix(25,-4,144,-23) -> Matrix(15,4,-64,-17) Matrix(289,-52,528,-95) -> Matrix(9,2,-104,-23) Matrix(263,-50,384,-73) -> Matrix(9,2,-140,-31) Matrix(265,-56,336,-71) -> Matrix(41,8,-692,-135) Matrix(25,-6,96,-23) -> Matrix(11,2,-72,-13) Matrix(529,-142,1248,-335) -> Matrix(1,0,-4,1) Matrix(889,-240,1152,-311) -> Matrix(15,2,-248,-33) Matrix(169,-50,240,-71) -> Matrix(29,4,-428,-59) Matrix(601,-182,1344,-407) -> Matrix(15,2,-188,-25) Matrix(311,-98,384,-121) -> Matrix(13,2,-228,-35) Matrix(671,-234,1104,-385) -> Matrix(19,2,-200,-21) Matrix(433,-154,672,-239) -> Matrix(1,0,-8,1) Matrix(719,-282,1104,-433) -> Matrix(23,2,-288,-25) Matrix(121,-50,288,-119) -> Matrix(19,2,-200,-21) Matrix(2015,-852,3408,-1441) -> Matrix(23,2,-288,-25) Matrix(1609,-720,2304,-1031) -> Matrix(27,2,-392,-29) Matrix(407,-184,480,-217) -> Matrix(19,2,-352,-37) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(937,-518,1344,-743) -> Matrix(19,2,-276,-29) Matrix(169,-98,288,-167) -> Matrix(23,2,-288,-25) Matrix(817,-482,1056,-623) -> Matrix(1,0,-4,1) Matrix(73,-54,96,-71) -> Matrix(31,2,-512,-33) Matrix(121,-100,144,-119) -> Matrix(71,4,-1296,-73) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 Degree of the the map X: 64 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/7 1/6 1/5 2/9 1/4 3/11 3/10 1/3 13/36 3/8 5/12 7/16 4/9 1/2 13/24 5/8 2/3 17/24 3/4 19/24 5/6 7/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/7 -3/10 1/6 -1/4 1/5 -3/14 3/14 -1/5 -4/21 5/23 -13/70 2/9 -1/6 1/4 -1/6 4/15 -1/6 3/11 -1/6 5/18 -1/6 2/7 -3/20 3/10 -1/7 -2/15 1/3 -1/8 5/14 -1/7 0/1 9/25 -7/54 13/36 -1/8 4/11 -1/8 3/8 -1/9 5/13 -1/10 2/5 -1/8 5/12 -1/10 3/7 -1/10 10/23 -3/40 7/16 0/1 11/25 -3/22 4/9 -1/10 5/11 -1/10 1/2 -1/11 0/1 7/13 -1/10 13/24 -1/11 6/11 -1/12 5/9 -1/12 14/25 -3/44 9/16 0/1 4/7 -1/12 7/12 -1/12 3/5 -1/14 11/18 -1/12 8/13 -1/12 5/8 -1/13 7/11 -1/14 2/3 -1/14 9/13 -1/14 25/36 -1/14 16/23 -9/128 7/10 -2/29 -1/15 12/17 -7/104 17/24 -1/15 5/7 -3/46 13/18 -1/16 8/11 -1/16 3/4 -1/16 10/13 -1/16 7/9 -1/16 18/23 -13/216 11/14 -4/67 -1/17 15/19 -11/186 19/24 -1/17 4/5 -3/52 9/11 -1/18 5/6 -1/18 6/7 -3/56 7/8 -1/19 1/1 -1/22 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(73,-10,168,-23) (0/1,1/7) -> (3/7,10/23) Hyperbolic Matrix(47,-7,168,-25) (1/7,1/6) -> (5/18,2/7) Hyperbolic Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(265,-56,336,-71) (1/5,3/14) -> (11/14,15/19) Hyperbolic Matrix(385,-83,552,-119) (3/14,5/23) -> (16/23,7/10) Hyperbolic Matrix(241,-53,432,-95) (5/23,2/9) -> (5/9,14/25) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(145,-39,264,-71) (4/15,3/11) -> (6/11,5/9) Hyperbolic Matrix(265,-73,432,-119) (3/11,5/18) -> (11/18,8/13) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(527,-189,672,-241) (5/14,9/25) -> (18/23,11/14) Hyperbolic Matrix(1201,-433,1728,-623) (9/25,13/36) -> (25/36,16/23) Hyperbolic Matrix(599,-217,864,-313) (13/36,4/11) -> (9/13,25/36) Hyperbolic Matrix(73,-27,192,-71) (4/11,3/8) -> (3/8,5/13) Parabolic Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(71,-29,120,-49) (2/5,5/12) -> (7/12,3/5) Hyperbolic Matrix(97,-41,168,-71) (5/12,3/7) -> (4/7,7/12) Hyperbolic Matrix(337,-147,768,-335) (10/23,7/16) -> (7/16,11/25) Parabolic Matrix(337,-149,432,-191) (11/25,4/9) -> (7/9,18/23) Hyperbolic Matrix(167,-75,216,-97) (4/9,5/11) -> (10/13,7/9) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(313,-169,576,-311) (7/13,13/24) -> (13/24,6/11) Parabolic Matrix(191,-107,216,-121) (14/25,9/16) -> (7/8,1/1) Hyperbolic Matrix(145,-82,168,-95) (9/16,4/7) -> (6/7,7/8) Hyperbolic Matrix(121,-75,192,-119) (8/13,5/8) -> (5/8,7/11) Parabolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(409,-289,576,-407) (12/17,17/24) -> (17/24,5/7) Parabolic Matrix(143,-103,168,-121) (5/7,13/18) -> (5/6,6/7) Hyperbolic Matrix(217,-157,264,-191) (13/18,8/11) -> (9/11,5/6) Hyperbolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(457,-361,576,-455) (15/19,19/24) -> (19/24,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-22,1) Matrix(73,-10,168,-23) -> Matrix(3,1,-40,-13) Matrix(47,-7,168,-25) -> Matrix(11,3,-70,-19) Matrix(73,-14,120,-23) -> Matrix(5,1,-56,-11) Matrix(265,-56,336,-71) -> Matrix(41,8,-692,-135) Matrix(385,-83,552,-119) -> Matrix(37,7,-534,-101) Matrix(241,-53,432,-95) -> Matrix(11,2,-138,-25) Matrix(25,-6,96,-23) -> Matrix(11,2,-72,-13) Matrix(145,-39,264,-71) -> Matrix(7,1,-78,-11) Matrix(265,-73,432,-119) -> Matrix(1,0,-6,1) Matrix(169,-50,240,-71) -> Matrix(29,4,-428,-59) Matrix(25,-8,72,-23) -> Matrix(7,1,-64,-9) Matrix(527,-189,672,-241) -> Matrix(29,4,-486,-67) Matrix(1201,-433,1728,-623) -> Matrix(63,8,-890,-113) Matrix(599,-217,864,-313) -> Matrix(17,2,-230,-27) Matrix(73,-27,192,-71) -> Matrix(17,2,-162,-19) Matrix(97,-38,120,-47) -> Matrix(11,1,-188,-17) Matrix(71,-29,120,-49) -> Matrix(9,1,-118,-13) Matrix(97,-41,168,-71) -> Matrix(11,1,-122,-11) Matrix(337,-147,768,-335) -> Matrix(1,0,6,1) Matrix(337,-149,432,-191) -> Matrix(19,2,-314,-33) Matrix(167,-75,216,-97) -> Matrix(9,1,-154,-17) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(313,-169,576,-311) -> Matrix(21,2,-242,-23) Matrix(191,-107,216,-121) -> Matrix(15,1,-286,-19) Matrix(145,-82,168,-95) -> Matrix(9,1,-172,-19) Matrix(121,-75,192,-119) -> Matrix(25,2,-338,-27) Matrix(49,-32,72,-47) -> Matrix(13,1,-196,-15) Matrix(409,-289,576,-407) -> Matrix(149,10,-2250,-151) Matrix(143,-103,168,-121) -> Matrix(47,3,-862,-55) Matrix(217,-157,264,-191) -> Matrix(17,1,-290,-17) Matrix(73,-54,96,-71) -> Matrix(31,2,-512,-33) Matrix(457,-361,576,-455) -> Matrix(237,14,-4046,-239) 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 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 1/0 1 24 1/7 -3/10 1 24 1/6 -1/4 4 4 1/5 -3/14 1 24 5/24 -1/5 14 1 3/14 (-1/5,-4/21) 0 12 5/23 -13/70 1 24 2/9 -1/6 1 8 1/4 -1/6 2 6 4/15 -1/6 1 8 3/11 -1/6 1 24 5/18 -1/6 4 4 2/7 -3/20 1 24 7/24 -1/7 10 1 3/10 (-1/7,-2/15) 0 12 1/3 -1/8 1 8 5/14 (-1/7,0/1) 0 12 9/25 -7/54 1 24 13/36 -1/8 10 2 4/11 -1/8 1 24 3/8 -1/9 2 3 5/13 -1/10 1 24 7/18 -1/10 4 4 2/5 -1/8 1 24 5/12 -1/10 2 2 3/7 -1/10 1 24 10/23 -3/40 1 24 7/16 0/1 6 3 11/25 -3/22 1 24 4/9 -1/10 1 8 5/11 -1/10 1 24 11/24 -1/11 2 1 1/2 (-1/11,0/1) 0 12 1/0 0/1 22 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(73,-10,168,-23) (0/1,1/7) -> (3/7,10/23) Hyperbolic Matrix(47,-7,168,-25) (1/7,1/6) -> (5/18,2/7) Hyperbolic Matrix(47,-9,120,-23) (1/6,1/5) -> (7/18,2/5) Glide Reflection Matrix(49,-10,240,-49) (1/5,5/24) -> (1/5,5/24) Reflection Matrix(71,-15,336,-71) (5/24,3/14) -> (5/24,3/14) Reflection Matrix(241,-52,672,-145) (3/14,5/23) -> (5/14,9/25) Glide Reflection Matrix(191,-42,432,-95) (5/23,2/9) -> (11/25,4/9) Glide Reflection Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(119,-32,264,-71) (4/15,3/11) -> (4/9,5/11) Glide Reflection Matrix(167,-46,432,-119) (3/11,5/18) -> (5/13,7/18) Glide Reflection Matrix(97,-28,336,-97) (2/7,7/24) -> (2/7,7/24) Reflection Matrix(71,-21,240,-71) (7/24,3/10) -> (7/24,3/10) Reflection Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(649,-234,1800,-649) (9/25,13/36) -> (9/25,13/36) Reflection Matrix(287,-104,792,-287) (13/36,4/11) -> (13/36,4/11) Reflection Matrix(73,-27,192,-71) (4/11,3/8) -> (3/8,5/13) Parabolic Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(337,-147,768,-335) (10/23,7/16) -> (7/16,11/25) Parabolic Matrix(241,-110,528,-241) (5/11,11/24) -> (5/11,11/24) Reflection Matrix(23,-11,48,-23) (11/24,1/2) -> (11/24,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(73,-10,168,-23) -> Matrix(3,1,-40,-13) Matrix(47,-7,168,-25) -> Matrix(11,3,-70,-19) Matrix(47,-9,120,-23) -> Matrix(5,1,-54,-11) Matrix(49,-10,240,-49) -> Matrix(29,6,-140,-29) (1/5,5/24) -> (-3/14,-1/5) Matrix(71,-15,336,-71) -> Matrix(41,8,-210,-41) (5/24,3/14) -> (-1/5,-4/21) Matrix(241,-52,672,-145) -> Matrix(21,4,-152,-29) Matrix(191,-42,432,-95) -> Matrix(11,2,-104,-19) Matrix(25,-6,96,-23) -> Matrix(11,2,-72,-13) -1/6 Matrix(119,-32,264,-71) -> Matrix(7,1,-76,-11) Matrix(167,-46,432,-119) -> Matrix(-1,0,16,1) *** -> (-1/8,0/1) Matrix(97,-28,336,-97) -> Matrix(41,6,-280,-41) (2/7,7/24) -> (-3/20,-1/7) Matrix(71,-21,240,-71) -> Matrix(29,4,-210,-29) (7/24,3/10) -> (-1/7,-2/15) Matrix(25,-8,72,-23) -> Matrix(7,1,-64,-9) -1/8 Matrix(649,-234,1800,-649) -> Matrix(55,7,-432,-55) (9/25,13/36) -> (-7/54,-1/8) Matrix(287,-104,792,-287) -> Matrix(25,3,-208,-25) (13/36,4/11) -> (-1/8,-3/26) Matrix(73,-27,192,-71) -> Matrix(17,2,-162,-19) -1/9 Matrix(49,-20,120,-49) -> Matrix(9,1,-80,-9) (2/5,5/12) -> (-1/8,-1/10) Matrix(71,-30,168,-71) -> Matrix(11,1,-120,-11) (5/12,3/7) -> (-1/10,-1/12) Matrix(337,-147,768,-335) -> Matrix(1,0,6,1) 0/1 Matrix(241,-110,528,-241) -> Matrix(21,2,-220,-21) (5/11,11/24) -> (-1/10,-1/11) Matrix(23,-11,48,-23) -> Matrix(-1,0,22,1) (11/24,1/2) -> (-1/11,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,22,1) (1/2,1/0) -> (-1/11,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.