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 -1/1 0/1 -8/9 -1/2 -7/8 -1/3 -13/15 -1/4 -6/7 -1/5 0/1 -5/6 -1/4 -14/17 -3/13 -2/9 -9/11 -1/5 -2/11 -13/16 0/1 -4/5 -2/9 -1/5 -11/14 -3/16 -18/23 -1/5 -2/11 -7/9 -1/6 -3/4 -1/6 -11/15 -1/6 -19/26 -1/6 -8/11 -2/13 -1/7 -13/18 -1/6 -18/25 -3/19 -2/13 -5/7 -4/27 -1/7 -7/10 -3/22 -23/33 -1/8 -16/23 -2/15 -3/23 -25/36 -1/8 -9/13 -1/7 0/1 -11/16 -2/15 -2/3 -1/8 -13/20 -1/8 -11/17 -7/59 -2/17 -9/14 -3/26 -16/25 -2/17 -1/9 -23/36 -3/26 -7/11 -2/17 -1/9 -5/8 -1/9 -8/13 -1/9 -4/37 -11/18 -3/28 -14/23 -1/9 -2/19 -17/28 -3/28 -3/5 -2/19 -3/29 -7/12 -1/10 -11/19 -8/81 -7/71 -26/45 -1/10 -41/71 -6/61 -5/51 -15/26 -1/10 -19/33 -1/10 -4/7 -4/41 -3/31 -13/23 -7/73 -2/21 -9/16 -2/21 -14/25 -2/21 -5/53 -5/9 -3/32 -21/38 -11/118 -16/29 -4/43 -17/183 -11/20 -5/54 -6/11 -10/109 -1/11 -1/2 -1/12 -6/13 -12/155 -1/13 -11/24 -1/13 -5/11 -1/13 -10/131 -14/31 -13/171 -6/79 -9/20 -5/66 -13/29 -17/225 -4/53 -4/9 -3/40 -11/25 -5/67 -2/27 -7/16 -2/27 -10/23 -2/27 -7/95 -3/7 -3/41 -4/55 -5/12 -1/14 -7/17 -7/99 -6/85 -9/22 -1/14 -29/71 -6/85 -5/71 -49/120 -5/71 -20/49 -5/71 -4/57 -11/27 -1/14 -2/5 -3/43 -2/29 -11/28 -3/44 -20/51 -3/44 -9/23 -2/29 -1/15 -7/18 -3/44 -5/13 -4/59 -1/15 -3/8 -1/15 -4/11 -1/15 -2/31 -13/36 -3/46 -9/25 -1/15 -2/31 -5/14 -3/46 -11/31 -13/201 -2/31 -17/48 -2/31 -6/17 -2/31 -7/109 -7/20 -1/16 -8/23 -2/31 -3/47 -1/3 -1/16 -6/19 -4/65 -3/49 -5/16 -2/33 -4/13 -1/17 0/1 -11/36 -1/16 -7/23 -3/49 -2/33 -10/33 -1/16 -13/43 -11/181 -2/33 -29/96 -2/33 -16/53 -2/33 -17/281 -3/10 -3/50 -5/17 -10/169 -1/17 -7/24 -1/17 -2/7 -1/17 -4/69 -7/25 -2/35 -3/53 -5/18 -1/18 -3/11 -1/17 -2/35 -1/4 -1/18 -3/13 -1/19 0/1 -11/48 0/1 -8/35 -1/17 0/1 -5/22 -1/18 -7/31 -3/55 -2/37 -2/9 -1/18 -5/23 -2/37 -1/19 -3/14 -3/56 -4/19 -8/151 -1/19 -5/24 -1/19 -1/5 -1/19 -2/39 -4/21 -1/20 -3/16 0/1 -2/11 -2/37 -1/19 -1/6 -1/20 -2/13 -4/83 -1/21 -3/20 -1/22 -1/7 -1/19 0/1 -2/15 -1/20 -1/8 -1/21 -1/9 -1/22 0/1 -1/23 0/1 1/9 -1/24 1/8 -1/25 2/15 -1/26 1/7 -1/27 0/1 1/6 -1/26 3/17 -3/79 -2/53 2/11 -1/27 -2/55 3/16 0/1 1/5 -2/53 -1/27 3/14 -3/82 5/23 -1/27 -2/55 2/9 -1/28 1/4 -1/28 4/15 -1/28 7/26 -1/28 3/11 -2/57 -1/29 5/18 -1/28 7/25 -3/85 -2/57 2/7 -4/115 -1/29 3/10 -3/88 10/33 -1/30 7/23 -2/59 -3/89 11/36 -1/30 4/13 -1/29 0/1 5/16 -2/59 1/3 -1/30 7/20 -1/30 6/17 -7/213 -2/61 5/14 -3/92 9/25 -2/61 -1/31 13/36 -3/92 4/11 -2/61 -1/31 3/8 -1/31 5/13 -1/31 -4/125 7/18 -3/94 9/23 -1/31 -2/63 11/28 -3/94 2/5 -2/63 -3/95 5/12 -1/32 8/19 -8/257 -7/225 19/45 -1/32 30/71 -6/193 -5/161 11/26 -1/32 14/33 -1/32 3/7 -4/129 -3/97 10/23 -7/227 -2/65 7/16 -2/65 11/25 -2/65 -5/163 4/9 -3/98 17/38 -11/360 13/29 -4/131 -17/557 9/20 -5/164 5/11 -10/329 -1/33 1/2 -1/34 7/13 -12/419 -1/35 13/24 -1/35 6/11 -1/35 -10/351 17/31 -13/457 -6/211 11/20 -5/176 16/29 -17/599 -4/141 5/9 -3/106 14/25 -5/177 -2/71 9/16 -2/71 13/23 -2/71 -7/249 4/7 -3/107 -4/143 7/12 -1/36 10/17 -7/253 -6/217 13/22 -1/36 42/71 -6/217 -5/181 71/120 -5/181 29/49 -5/181 -4/145 16/27 -1/36 3/5 -3/109 -2/73 17/28 -3/110 31/51 -3/110 14/23 -2/73 -1/37 11/18 -3/110 8/13 -4/147 -1/37 5/8 -1/37 7/11 -1/37 -2/75 23/36 -3/112 16/25 -1/37 -2/75 9/14 -3/112 20/31 -13/487 -2/75 31/48 -2/75 11/17 -2/75 -7/263 13/20 -1/38 15/23 -2/75 -3/113 2/3 -1/38 13/19 -4/153 -3/115 11/16 -2/77 9/13 -1/39 0/1 25/36 -1/38 16/23 -3/115 -2/77 23/33 -1/38 30/43 -11/423 -2/77 67/96 -2/77 37/53 -2/77 -17/655 7/10 -3/116 12/17 -10/389 -1/39 17/24 -1/39 5/7 -1/39 -4/157 18/25 -2/79 -3/119 13/18 -1/40 8/11 -1/39 -2/79 3/4 -1/40 10/13 -1/41 0/1 37/48 0/1 27/35 -1/39 0/1 17/22 -1/40 24/31 -3/121 -2/81 7/9 -1/40 18/23 -2/81 -1/41 11/14 -3/122 15/19 -8/327 -1/41 19/24 -1/41 4/5 -1/41 -2/83 17/21 -1/42 13/16 0/1 9/11 -2/81 -1/41 5/6 -1/42 11/13 -4/171 -1/43 17/20 -1/44 6/7 -1/41 0/1 13/15 -1/42 7/8 -1/43 8/9 -1/44 1/1 -1/45 0/1 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(5,2,-148,-59) Matrix(25,22,192,169) -> Matrix(5,2,-128,-51) Matrix(23,20,192,167) -> Matrix(7,2,-172,-49) Matrix(409,354,528,457) -> Matrix(7,2,-284,-81) Matrix(119,100,-144,-121) -> Matrix(7,2,-32,-9) Matrix(239,196,-528,-433) -> Matrix(39,8,-512,-105) Matrix(265,216,384,313) -> Matrix(11,2,-424,-77) Matrix(121,98,-384,-311) -> Matrix(7,2,-116,-33) Matrix(71,56,-336,-265) -> Matrix(31,6,-584,-113) Matrix(431,338,672,527) -> Matrix(1,0,-32,1) Matrix(241,188,432,337) -> Matrix(45,8,-1592,-283) Matrix(71,54,-96,-73) -> Matrix(11,2,-72,-13) Matrix(719,526,-1248,-913) -> Matrix(23,4,-236,-41) Matrix(263,192,-1152,-841) -> Matrix(13,2,-228,-35) Matrix(265,192,432,313) -> Matrix(15,2,-548,-73) Matrix(527,380,864,623) -> Matrix(51,8,-1868,-293) Matrix(145,104,336,241) -> Matrix(53,8,-1716,-259) Matrix(71,50,-240,-169) -> Matrix(43,6,-724,-101) Matrix(743,518,-1344,-937) -> Matrix(77,10,-824,-107) Matrix(23,16,240,167) -> Matrix(15,2,-368,-49) Matrix(1105,768,1728,1201) -> Matrix(61,8,-2280,-299) Matrix(551,382,864,599) -> Matrix(13,2,-488,-75) Matrix(313,216,384,265) -> Matrix(15,2,-608,-81) Matrix(73,50,-384,-263) -> Matrix(15,2,-308,-41) Matrix(433,282,-1104,-719) -> Matrix(67,8,-980,-117) Matrix(527,342,960,623) -> Matrix(133,16,-4680,-563) Matrix(239,154,-672,-433) -> Matrix(103,12,-1588,-185) Matrix(527,338,672,431) -> Matrix(1,0,-32,1) Matrix(1201,768,1728,1105) -> Matrix(69,8,-2648,-307) Matrix(599,382,864,551) -> Matrix(17,2,-672,-79) Matrix(73,46,192,121) -> Matrix(53,6,-1652,-187) Matrix(71,44,192,119) -> Matrix(55,6,-1696,-185) Matrix(313,192,432,265) -> Matrix(19,2,-732,-77) Matrix(623,380,864,527) -> Matrix(75,8,-2972,-317) Matrix(385,234,-1104,-671) -> Matrix(75,8,-1172,-125) Matrix(409,248,912,553) -> Matrix(207,22,-6784,-721) Matrix(167,98,-288,-169) -> Matrix(99,10,-1000,-101) Matrix(623,360,912,527) -> Matrix(41,4,-1548,-151) Matrix(817,472,1248,721) -> Matrix(81,8,-3068,-303) Matrix(1393,804,-3408,-1967) -> Matrix(1,0,-4,1) Matrix(289,166,336,193) -> Matrix(41,4,-1712,-167) Matrix(95,54,336,191) -> Matrix(83,8,-2376,-229) Matrix(337,190,768,433) -> Matrix(251,24,-8168,-781) Matrix(335,188,768,431) -> Matrix(253,24,-8212,-779) Matrix(337,188,432,241) -> Matrix(85,8,-3432,-323) Matrix(695,384,-2304,-1273) -> Matrix(365,34,-6044,-563) Matrix(359,198,912,503) -> Matrix(237,22,-7444,-691) Matrix(73,40,-480,-263) -> Matrix(65,6,-1376,-127) Matrix(23,12,-48,-25) -> Matrix(23,2,-288,-25) Matrix(313,144,576,265) -> Matrix(285,22,-9988,-771) Matrix(311,142,576,263) -> Matrix(287,22,-10032,-769) Matrix(337,152,960,433) -> Matrix(211,16,-6396,-485) Matrix(553,248,912,409) -> Matrix(291,22,-10648,-805) Matrix(407,182,-1344,-601) -> Matrix(133,10,-2168,-163) Matrix(95,42,432,191) -> Matrix(107,8,-2956,-221) Matrix(433,190,768,337) -> Matrix(323,24,-11480,-853) Matrix(431,188,768,335) -> Matrix(325,24,-11524,-851) Matrix(241,104,336,145) -> Matrix(109,8,-4292,-315) Matrix(119,50,-288,-121) -> Matrix(139,10,-1960,-141) Matrix(239,98,-1056,-433) -> Matrix(57,4,-1012,-71) Matrix(8521,3480,14400,5881) -> Matrix(709,50,-25680,-1811) Matrix(8519,3478,14400,5879) -> Matrix(711,50,-25724,-1809) Matrix(1897,774,3120,1273) -> Matrix(199,14,-7292,-513) Matrix(193,78,240,97) -> Matrix(57,4,-2380,-167) Matrix(503,198,912,359) -> Matrix(321,22,-11308,-775) Matrix(1847,724,3120,1223) -> Matrix(205,14,-7424,-507) Matrix(241,94,864,337) -> Matrix(117,8,-3320,-227) Matrix(119,46,432,167) -> Matrix(29,2,-856,-59) Matrix(121,46,192,73) -> Matrix(89,6,-3308,-223) Matrix(119,44,192,71) -> Matrix(91,6,-3352,-221) Matrix(265,96,864,313) -> Matrix(31,2,-884,-57) Matrix(527,190,1728,623) -> Matrix(123,8,-3644,-237) Matrix(145,52,672,241) -> Matrix(1,0,-12,1) Matrix(1489,528,2304,817) -> Matrix(619,40,-23228,-1501) Matrix(1487,526,2304,815) -> Matrix(621,40,-23272,-1499) Matrix(409,144,480,169) -> Matrix(31,2,-1380,-89) Matrix(527,182,1248,431) -> Matrix(127,8,-4080,-257) Matrix(385,122,912,289) -> Matrix(63,4,-2032,-129) Matrix(71,22,384,119) -> Matrix(33,2,-908,-55) Matrix(313,96,864,265) -> Matrix(35,2,-1068,-61) Matrix(623,190,1728,527) -> Matrix(131,8,-4012,-245) Matrix(217,66,240,73) -> Matrix(33,2,-1436,-87) Matrix(6433,1944,9216,2785) -> Matrix(923,56,-35552,-2157) Matrix(6431,1942,9216,2783) -> Matrix(925,56,-35596,-2155) Matrix(409,120,576,169) -> Matrix(237,14,-9260,-547) Matrix(407,118,576,167) -> Matrix(239,14,-9304,-545) Matrix(191,54,336,95) -> Matrix(139,8,-4952,-285) Matrix(337,94,864,241) -> Matrix(141,8,-4424,-251) Matrix(167,46,432,119) -> Matrix(33,2,-1040,-63) Matrix(23,6,-96,-25) -> Matrix(35,2,-648,-37) Matrix(1777,408,2304,529) -> Matrix(1,0,-20,1) Matrix(1775,406,2304,527) -> Matrix(1,0,-24,1) Matrix(71,16,528,119) -> Matrix(37,2,-944,-51) Matrix(191,42,432,95) -> Matrix(147,8,-4796,-261) Matrix(241,52,672,145) -> Matrix(1,0,-12,1) Matrix(457,96,576,121) -> Matrix(189,10,-7768,-411) Matrix(455,94,576,119) -> Matrix(191,10,-7812,-409) Matrix(143,28,240,47) -> Matrix(79,4,-2864,-145) Matrix(119,22,384,71) -> Matrix(37,2,-1092,-59) Matrix(23,4,-144,-25) -> Matrix(39,2,-800,-41) Matrix(311,46,480,71) -> Matrix(45,2,-1688,-75) Matrix(143,20,336,47) -> Matrix(79,4,-2548,-129) Matrix(169,22,192,25) -> Matrix(41,2,-1784,-87) Matrix(167,20,192,23) -> Matrix(43,2,-1828,-85) Matrix(167,16,240,23) -> Matrix(43,2,-1656,-77) Matrix(25,-4,144,-23) -> Matrix(51,2,-1352,-53) Matrix(289,-52,528,-95) -> Matrix(215,8,-7552,-281) Matrix(263,-50,384,-73) -> Matrix(51,2,-1964,-77) Matrix(265,-56,336,-71) -> Matrix(163,6,-6656,-245) Matrix(25,-6,96,-23) -> Matrix(55,2,-1568,-57) Matrix(529,-142,1248,-335) -> Matrix(111,4,-3580,-129) Matrix(889,-240,1152,-311) -> Matrix(57,2,-2252,-79) Matrix(169,-50,240,-71) -> Matrix(175,6,-6796,-233) Matrix(601,-182,1344,-407) -> Matrix(297,10,-9712,-327) Matrix(311,-98,384,-121) -> Matrix(59,2,-2508,-85) Matrix(671,-234,1104,-385) -> Matrix(243,8,-8900,-293) Matrix(433,-154,672,-239) -> Matrix(367,12,-13732,-449) Matrix(719,-282,1104,-433) -> Matrix(251,8,-9444,-301) Matrix(121,-50,288,-119) -> Matrix(319,10,-10240,-321) Matrix(2015,-852,3408,-1441) -> Matrix(1,0,-4,1) Matrix(1609,-720,2304,-1031) -> Matrix(1113,34,-42916,-1311) Matrix(407,-184,480,-217) -> Matrix(197,6,-8504,-259) Matrix(25,-12,48,-23) -> Matrix(67,2,-2312,-69) Matrix(937,-518,1344,-743) -> Matrix(353,10,-13520,-383) Matrix(169,-98,288,-167) -> Matrix(359,10,-12960,-361) Matrix(817,-482,1056,-623) -> Matrix(145,4,-5764,-159) Matrix(73,-54,96,-71) -> Matrix(79,2,-3200,-81) Matrix(121,-100,144,-119) -> Matrix(83,2,-3528,-85) 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 1/6 3/16 1/5 3/14 1/4 5/18 3/10 1/3 3/8 7/18 1/2 7/12 13/22 5/8 23/36 2/3 17/24 3/4 37/48 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 -1/23 0/1 1/6 -1/26 2/11 -1/27 -2/55 3/16 0/1 1/5 -2/53 -1/27 3/14 -3/82 2/9 -1/28 1/4 -1/28 4/15 -1/28 7/26 -1/28 3/11 -2/57 -1/29 5/18 -1/28 2/7 -4/115 -1/29 3/10 -3/88 4/13 -1/29 0/1 5/16 -2/59 1/3 -1/30 5/14 -3/92 4/11 -2/61 -1/31 3/8 -1/31 5/13 -1/31 -4/125 7/18 -3/94 2/5 -2/63 -3/95 1/2 -1/34 4/7 -3/107 -4/143 7/12 -1/36 10/17 -7/253 -6/217 13/22 -1/36 16/27 -1/36 3/5 -3/109 -2/73 11/18 -3/110 8/13 -4/147 -1/37 5/8 -1/37 7/11 -1/37 -2/75 23/36 -3/112 16/25 -1/37 -2/75 9/14 -3/112 2/3 -1/38 13/19 -4/153 -3/115 11/16 -2/77 9/13 -1/39 0/1 25/36 -1/38 16/23 -3/115 -2/77 7/10 -3/116 12/17 -10/389 -1/39 17/24 -1/39 5/7 -1/39 -4/157 13/18 -1/40 8/11 -1/39 -2/79 3/4 -1/40 10/13 -1/41 0/1 37/48 0/1 27/35 -1/39 0/1 17/22 -1/40 24/31 -3/121 -2/81 7/9 -1/40 11/14 -3/122 15/19 -8/327 -1/41 19/24 -1/41 4/5 -1/41 -2/83 5/6 -1/42 6/7 -1/41 0/1 13/15 -1/42 7/8 -1/43 1/1 -1/45 0/1 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(13,-2,72,-11) (0/1,1/6) -> (1/6,2/11) Parabolic Matrix(265,-49,384,-71) (2/11,3/16) -> (11/16,9/13) 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(59,-13,168,-37) (3/14,2/9) -> (1/3,5/14) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(611,-164,1032,-277) (4/15,7/26) -> (13/22,16/27) Hyperbolic Matrix(889,-240,1152,-311) (7/26,3/11) -> (27/35,17/22) Hyperbolic Matrix(265,-73,432,-119) (3/11,5/18) -> (11/18,8/13) Hyperbolic Matrix(131,-37,216,-61) (5/18,2/7) -> (3/5,11/18) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(251,-76,360,-109) (3/10,4/13) -> (16/23,7/10) Hyperbolic Matrix(107,-33,120,-37) (4/13,5/16) -> (7/8,1/1) Hyperbolic Matrix(229,-72,264,-83) (5/16,1/3) -> (13/15,7/8) Hyperbolic Matrix(323,-116,504,-181) (5/14,4/11) -> (16/25,9/14) Hyperbolic Matrix(73,-27,192,-71) (4/11,3/8) -> (3/8,5/13) Parabolic Matrix(313,-121,432,-167) (5/13,7/18) -> (13/18,8/11) Hyperbolic Matrix(155,-61,216,-85) (7/18,2/5) -> (5/7,13/18) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic 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(227,-135,264,-157) (16/27,3/5) -> (6/7,13/15) Hyperbolic Matrix(121,-75,192,-119) (8/13,5/8) -> (5/8,7/11) Parabolic Matrix(851,-543,1224,-781) (7/11,23/36) -> (25/36,16/23) Hyperbolic Matrix(949,-607,1368,-875) (23/36,16/25) -> (9/13,25/36) Hyperbolic Matrix(131,-85,168,-109) (9/14,2/3) -> (7/9,11/14) Hyperbolic Matrix(205,-139,264,-179) (2/3,13/19) -> (24/31,7/9) Hyperbolic Matrix(409,-289,576,-407) (12/17,17/24) -> (17/24,5/7) Parabolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(1777,-1369,2304,-1775) (10/13,37/48) -> (37/48,27/35) Parabolic Matrix(457,-361,576,-455) (15/19,19/24) -> (19/24,4/5) Parabolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-22,1) Matrix(13,-2,72,-11) -> Matrix(25,1,-676,-27) Matrix(265,-49,384,-71) -> Matrix(55,2,-2118,-77) Matrix(263,-50,384,-73) -> Matrix(51,2,-1964,-77) Matrix(265,-56,336,-71) -> Matrix(163,6,-6656,-245) Matrix(59,-13,168,-37) -> Matrix(83,3,-2518,-91) Matrix(25,-6,96,-23) -> Matrix(55,2,-1568,-57) Matrix(611,-164,1032,-277) -> Matrix(83,3,-3016,-109) Matrix(889,-240,1152,-311) -> Matrix(57,2,-2252,-79) Matrix(265,-73,432,-119) -> Matrix(59,2,-2154,-73) Matrix(131,-37,216,-61) -> Matrix(143,5,-5234,-183) Matrix(169,-50,240,-71) -> Matrix(175,6,-6796,-233) Matrix(251,-76,360,-109) -> Matrix(89,3,-3412,-115) Matrix(107,-33,120,-37) -> Matrix(29,1,-1306,-45) Matrix(229,-72,264,-83) -> Matrix(89,3,-3768,-127) Matrix(323,-116,504,-181) -> Matrix(91,3,-3428,-113) Matrix(73,-27,192,-71) -> Matrix(185,6,-5766,-187) Matrix(313,-121,432,-167) -> Matrix(63,2,-2426,-77) Matrix(155,-61,216,-85) -> Matrix(157,5,-6186,-197) Matrix(13,-6,24,-11) -> Matrix(33,1,-1156,-35) Matrix(169,-98,288,-167) -> Matrix(359,10,-12960,-361) Matrix(817,-482,1056,-623) -> Matrix(145,4,-5764,-159) Matrix(227,-135,264,-157) -> Matrix(109,3,-4542,-125) Matrix(121,-75,192,-119) -> Matrix(221,6,-8214,-223) Matrix(851,-543,1224,-781) -> Matrix(261,7,-10030,-269) Matrix(949,-607,1368,-875) -> Matrix(37,1,-1518,-41) Matrix(131,-85,168,-109) -> Matrix(113,3,-4558,-121) Matrix(205,-139,264,-179) -> Matrix(39,1,-1522,-39) Matrix(409,-289,576,-407) -> Matrix(545,14,-21294,-547) Matrix(73,-54,96,-71) -> Matrix(79,2,-3200,-81) Matrix(1777,-1369,2304,-1775) -> Matrix(1,0,2,1) Matrix(457,-361,576,-455) -> Matrix(409,10,-16810,-411) Matrix(61,-50,72,-59) -> Matrix(41,1,-1764,-43) 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 (-1/23,0/1) 0 24 1/6 -1/26 2 4 2/11 (-1/27,-2/55) 0 24 3/16 0/1 2 3 1/5 (-2/53,-1/27) 0 24 5/24 -1/27 10 1 3/14 -3/82 2 12 2/9 -1/28 2 8 1/4 -1/28 2 6 4/15 -1/28 2 8 7/26 -1/28 2 12 13/48 -2/57 2 1 3/11 (-2/57,-1/29) 0 24 5/18 -1/28 2 4 2/7 (-4/115,-1/29) 0 24 7/24 -1/29 14 1 3/10 -3/88 2 12 11/36 -1/30 2 2 4/13 (-1/29,0/1) 0 24 5/16 -2/59 2 3 6/19 (-3/89,-4/119) 0 24 1/3 -1/30 2 8 5/14 -3/92 2 12 13/36 -3/92 2 2 4/11 (-2/61,-1/31) 0 24 3/8 -1/31 6 3 5/13 (-1/31,-4/125) 0 24 7/18 -3/94 2 4 2/5 (-2/63,-3/95) 0 24 11/27 -1/32 2 8 9/22 -1/32 2 12 5/12 -1/32 10 2 1/2 -1/34 2 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(13,-2,72,-11) (0/1,1/6) -> (1/6,2/11) Parabolic Matrix(119,-22,384,-71) (2/11,3/16) -> (4/13,5/16) Glide Reflection Matrix(121,-23,384,-73) (3/16,1/5) -> (5/16,6/19) 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(59,-13,168,-37) (3/14,2/9) -> (1/3,5/14) Hyperbolic Matrix(25,-6,96,-23) (2/9,1/4) -> (1/4,4/15) Parabolic Matrix(421,-113,1032,-277) (4/15,7/26) -> (11/27,9/22) Glide Reflection Matrix(337,-91,1248,-337) (7/26,13/48) -> (7/26,13/48) Reflection Matrix(287,-78,1056,-287) (13/48,3/11) -> (13/48,3/11) Reflection Matrix(167,-46,432,-119) (3/11,5/18) -> (5/13,7/18) Glide Reflection Matrix(85,-24,216,-61) (5/18,2/7) -> (7/18,2/5) 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(109,-33,360,-109) (3/10,11/36) -> (3/10,11/36) Reflection Matrix(313,-96,864,-265) (11/36,4/13) -> (13/36,4/11) Glide Reflection Matrix(215,-68,528,-167) (6/19,1/3) -> (2/5,11/27) Glide Reflection Matrix(181,-65,504,-181) (5/14,13/36) -> (5/14,13/36) Reflection Matrix(73,-27,192,-71) (4/11,3/8) -> (3/8,5/13) Parabolic Matrix(109,-45,264,-109) (9/22,5/12) -> (9/22,5/12) Reflection Matrix(11,-5,24,-11) (5/12,1/2) -> (5/12,1/2) 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(-1,0,46,1) (0/1,1/0) -> (-1/23,0/1) Matrix(13,-2,72,-11) -> Matrix(25,1,-676,-27) -1/26 Matrix(119,-22,384,-71) -> Matrix(55,2,-1622,-59) Matrix(121,-23,384,-73) -> Matrix(51,2,-1504,-59) Matrix(49,-10,240,-49) -> Matrix(107,4,-2862,-107) (1/5,5/24) -> (-2/53,-1/27) Matrix(71,-15,336,-71) -> Matrix(163,6,-4428,-163) (5/24,3/14) -> (-1/27,-3/82) Matrix(59,-13,168,-37) -> Matrix(83,3,-2518,-91) Matrix(25,-6,96,-23) -> Matrix(55,2,-1568,-57) -1/28 Matrix(421,-113,1032,-277) -> Matrix(83,3,-2628,-95) Matrix(337,-91,1248,-337) -> Matrix(113,4,-3192,-113) (7/26,13/48) -> (-1/28,-2/57) Matrix(287,-78,1056,-287) -> Matrix(115,4,-3306,-115) (13/48,3/11) -> (-2/57,-1/29) Matrix(167,-46,432,-119) -> Matrix(59,2,-1858,-63) Matrix(85,-24,216,-61) -> Matrix(143,5,-4490,-157) Matrix(97,-28,336,-97) -> Matrix(231,8,-6670,-231) (2/7,7/24) -> (-4/115,-1/29) Matrix(71,-21,240,-71) -> Matrix(175,6,-5104,-175) (7/24,3/10) -> (-1/29,-3/88) Matrix(109,-33,360,-109) -> Matrix(89,3,-2640,-89) (3/10,11/36) -> (-3/88,-1/30) Matrix(313,-96,864,-265) -> Matrix(57,2,-1738,-61) Matrix(215,-68,528,-167) -> Matrix(179,6,-5698,-191) Matrix(181,-65,504,-181) -> Matrix(91,3,-2760,-91) (5/14,13/36) -> (-1/30,-3/92) Matrix(73,-27,192,-71) -> Matrix(185,6,-5766,-187) -1/31 Matrix(109,-45,264,-109) -> Matrix(287,9,-9152,-287) (9/22,5/12) -> (-9/286,-1/32) Matrix(11,-5,24,-11) -> Matrix(33,1,-1088,-33) (5/12,1/2) -> (-1/32,-1/34) Matrix(-1,1,0,1) -> Matrix(-1,0,68,1) (1/2,1/0) -> (-1/34,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.