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): 864 Minimal number of generators: 145 Number of equivalence classes of cusps: 48 Genus: 49 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/15 3/14 4/13 5/12 1/2 6/11 7/10 8/9 1/1 9/8 19/15 7/5 10/7 3/2 52/33 21/13 11/6 2/1 12/5 5/2 8/3 25/9 3/1 13/4 10/3 105/31 7/2 11/3 67/18 27/7 4/1 13/3 9/2 14/3 5/1 57/11 16/3 11/2 29/5 6/1 13/2 7/1 15/2 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/30 -6/13 3/83 -17/37 2/55 -11/24 1/28 -5/11 4/109 -9/20 7/188 -4/9 1/27 -19/43 2/53 -15/34 1/26 -26/59 5/129 -11/25 0/1 -7/16 5/132 -17/39 4/105 -27/62 19/498 -10/23 7/183 -3/7 2/51 -8/19 1/27 -5/12 3/76 -7/17 4/99 -9/22 3/74 -2/5 1/25 -9/23 4/95 -7/18 3/70 -5/13 2/45 -8/21 1/27 -11/29 4/99 -3/8 1/24 -7/19 2/45 -25/68 1/24 -18/49 3/67 -11/30 1/22 -4/11 1/23 -5/14 1/26 -6/17 1/21 -1/3 0/1 -5/16 1/24 -4/13 1/19 -3/10 1/22 -8/27 1/23 -13/44 3/68 -5/17 2/45 -2/7 1/21 -7/25 4/81 -26/93 11/219 -19/68 1/20 -12/43 1/19 -5/18 1/18 -8/29 1/27 -3/11 2/45 -4/15 3/65 -9/34 7/150 -5/19 4/85 -1/4 1/20 -6/25 1/21 -5/21 6/125 -4/17 3/61 -3/13 4/81 -2/9 3/59 -3/14 1/18 -1/5 2/39 -3/16 7/132 -11/59 6/113 -8/43 19/357 -5/27 4/75 -7/38 5/94 -2/11 5/93 -5/28 5/92 -3/17 0/1 -7/40 5/96 -11/63 10/189 -4/23 1/19 -5/29 2/37 -6/35 5/93 -1/6 1/18 -2/13 7/127 -1/7 4/71 -2/15 1/17 -5/38 13/230 -3/23 2/35 -4/31 11/193 -1/8 3/52 0/1 1/15 1/8 7/92 2/15 1/13 1/7 6/77 1/6 3/38 3/17 2/25 2/11 1/13 3/16 1/12 4/21 1/11 5/26 3/38 1/5 4/49 3/14 1/12 5/23 18/215 2/9 7/83 1/4 1/12 5/19 2/23 9/34 1/10 4/15 1/11 7/26 5/54 3/11 0/1 2/7 5/57 5/17 4/45 8/27 19/213 3/10 7/78 4/13 1/11 5/16 11/120 1/3 2/21 4/11 1/11 11/30 1/10 7/19 4/43 3/8 1/12 5/13 8/85 7/18 9/94 2/5 3/31 5/12 1/10 8/19 11/109 3/7 4/39 13/30 13/126 10/23 13/125 17/39 2/19 7/16 3/28 11/25 2/19 15/34 1/10 4/9 3/29 1/2 1/10 6/11 1/9 11/20 9/80 5/9 4/35 4/7 3/25 11/19 0/1 7/12 1/8 3/5 2/15 5/8 1/12 7/11 4/39 2/3 1/9 7/10 1/8 12/17 5/39 5/7 2/15 18/25 1/9 31/43 4/31 13/18 3/22 8/11 1/7 3/4 1/8 10/13 3/19 17/22 1/6 7/9 2/9 4/5 1/11 5/6 1/6 21/25 0/1 16/19 3/29 11/13 2/17 6/7 1/9 13/15 2/15 7/8 1/8 8/9 1/7 9/10 1/6 1/1 0/1 9/8 1/8 8/7 1/7 15/13 2/15 7/6 1/6 13/11 2/13 19/16 3/16 25/21 0/1 31/26 -1/2 6/5 1/9 5/4 1/4 19/15 0/1 14/11 1/17 9/7 2/21 22/17 1/9 13/10 3/26 4/3 1/7 19/14 5/42 34/25 9/73 15/11 4/31 11/8 1/8 18/13 3/23 7/5 2/15 10/7 1/7 13/9 4/27 3/2 1/6 11/7 4/21 52/33 1/5 93/59 26/129 41/26 11/54 30/19 1/5 19/12 1/4 8/5 1/3 21/13 0/1 34/21 1/21 13/8 1/12 5/3 2/15 32/19 1/7 27/16 11/76 49/29 8/55 22/13 9/61 17/10 1/6 12/7 1/7 31/18 1/6 19/11 0/1 7/4 3/20 16/9 7/45 9/5 4/25 11/6 1/6 13/7 6/35 2/1 1/5 13/6 1/6 11/5 6/35 9/4 3/16 25/11 2/11 16/7 3/17 23/10 13/70 7/3 4/21 12/5 1/5 17/7 10/49 5/2 3/14 18/7 9/41 49/19 2/9 31/12 17/76 13/5 8/35 21/8 1/4 8/3 1/3 19/7 4/17 11/4 1/4 25/9 0/1 39/14 1/6 14/5 1/5 3/1 2/9 13/4 1/4 23/7 16/63 10/3 7/27 37/11 6/23 27/8 19/72 44/13 21/79 105/31 4/15 166/49 67/251 61/18 23/86 17/5 4/15 24/7 5/19 79/23 4/15 55/16 15/56 31/9 10/37 38/11 3/11 7/2 5/18 18/5 5/17 11/3 0/1 26/7 5/21 67/18 1/4 108/29 25/99 41/11 10/39 15/4 1/4 49/13 4/15 34/9 1/5 53/14 11/42 19/5 2/7 23/6 5/18 27/7 2/7 31/8 7/24 4/1 1/3 13/3 10/33 9/2 7/22 14/3 1/3 19/4 15/44 5/1 4/11 31/6 7/18 57/11 2/5 26/5 3/7 21/4 1/4 37/7 2/5 16/3 1/3 43/8 3/8 27/5 2/5 11/2 1/2 28/5 13/35 17/3 2/5 23/4 11/28 29/5 2/5 35/6 17/42 6/1 3/7 13/2 1/2 7/1 6/13 15/2 1/2 23/3 20/39 8/1 7/13 17/2 1/2 9/1 4/7 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(255,118,188,87) (-1/2,-6/13) -> (4/3,19/14) Hyperbolic Matrix(187,86,-1446,-665) (-6/13,-17/37) -> (-3/23,-4/31) Hyperbolic Matrix(375,172,-2008,-921) (-17/37,-11/24) -> (-3/16,-11/59) Hyperbolic Matrix(437,200,378,173) (-11/24,-5/11) -> (15/13,7/6) Hyperbolic Matrix(309,140,64,29) (-5/11,-9/20) -> (19/4,5/1) Hyperbolic Matrix(557,250,430,193) (-9/20,-4/9) -> (22/17,13/10) Hyperbolic Matrix(303,134,-1766,-781) (-4/9,-19/43) -> (-5/29,-6/35) Hyperbolic Matrix(1037,458,120,53) (-19/43,-15/34) -> (17/2,9/1) Hyperbolic Matrix(2005,884,-7174,-3163) (-15/34,-26/59) -> (-26/93,-19/68) Hyperbolic Matrix(849,374,-4556,-2007) (-26/59,-11/25) -> (-11/59,-8/43) Hyperbolic Matrix(305,134,-1154,-507) (-11/25,-7/16) -> (-9/34,-5/19) Hyperbolic Matrix(1087,474,-2956,-1289) (-7/16,-17/39) -> (-7/19,-25/68) Hyperbolic Matrix(2291,998,1926,839) (-17/39,-27/62) -> (19/16,25/21) Hyperbolic Matrix(239,104,-1804,-785) (-27/62,-10/23) -> (-2/15,-5/38) Hyperbolic Matrix(599,260,182,79) (-10/23,-3/7) -> (23/7,10/3) Hyperbolic Matrix(237,100,410,173) (-3/7,-8/19) -> (4/7,11/19) Hyperbolic Matrix(177,74,232,97) (-8/19,-5/12) -> (3/4,10/13) Hyperbolic Matrix(289,120,118,49) (-5/12,-7/17) -> (17/7,5/2) Hyperbolic Matrix(405,166,344,141) (-7/17,-9/22) -> (7/6,13/11) Hyperbolic Matrix(289,118,-1036,-423) (-9/22,-2/5) -> (-12/43,-5/18) Hyperbolic Matrix(111,44,58,23) (-2/5,-9/23) -> (13/7,2/1) Hyperbolic Matrix(169,66,-950,-371) (-9/23,-7/18) -> (-5/28,-3/17) Hyperbolic Matrix(165,64,446,173) (-7/18,-5/13) -> (7/19,3/8) Hyperbolic Matrix(167,64,-608,-233) (-5/13,-8/21) -> (-8/29,-3/11) Hyperbolic Matrix(221,84,-934,-355) (-8/21,-11/29) -> (-5/21,-4/17) Hyperbolic Matrix(549,208,710,269) (-11/29,-3/8) -> (17/22,7/9) Hyperbolic Matrix(161,60,110,41) (-3/8,-7/19) -> (13/9,3/2) Hyperbolic Matrix(2797,1028,536,197) (-25/68,-18/49) -> (26/5,21/4) Hyperbolic Matrix(1291,474,-4622,-1697) (-18/49,-11/30) -> (-19/68,-12/43) Hyperbolic Matrix(323,118,52,19) (-11/30,-4/11) -> (6/1,13/2) Hyperbolic Matrix(105,38,268,97) (-4/11,-5/14) -> (7/18,2/5) Hyperbolic Matrix(209,74,-788,-279) (-5/14,-6/17) -> (-4/15,-9/34) Hyperbolic Matrix(473,166,208,73) (-6/17,-1/3) -> (25/11,16/7) Hyperbolic Matrix(197,62,448,141) (-1/3,-5/16) -> (7/16,11/25) Hyperbolic Matrix(97,30,-540,-167) (-5/16,-4/13) -> (-2/11,-5/28) Hyperbolic Matrix(245,74,96,29) (-4/13,-3/10) -> (5/2,18/7) Hyperbolic Matrix(337,100,920,273) (-3/10,-8/27) -> (4/11,11/30) Hyperbolic Matrix(291,86,-1208,-357) (-8/27,-13/44) -> (-1/4,-6/25) Hyperbolic Matrix(339,100,-1834,-541) (-13/44,-5/17) -> (-5/27,-7/38) Hyperbolic Matrix(239,70,338,99) (-5/17,-2/7) -> (12/17,5/7) Hyperbolic Matrix(613,172,474,133) (-2/7,-7/25) -> (9/7,22/17) Hyperbolic Matrix(6513,1822,1748,489) (-7/25,-26/93) -> (108/29,41/11) Hyperbolic Matrix(989,274,610,169) (-5/18,-8/29) -> (34/21,13/8) Hyperbolic Matrix(373,100,138,37) (-3/11,-4/15) -> (8/3,19/7) Hyperbolic Matrix(229,60,416,109) (-5/19,-1/4) -> (11/20,5/9) Hyperbolic Matrix(359,86,-2058,-493) (-6/25,-5/21) -> (-11/63,-4/23) Hyperbolic Matrix(265,62,312,73) (-4/17,-3/13) -> (11/13,6/7) Hyperbolic Matrix(131,30,310,71) (-3/13,-2/9) -> (8/19,3/7) Hyperbolic Matrix(173,38,132,29) (-2/9,-3/14) -> (13/10,4/3) Hyperbolic Matrix(301,64,174,37) (-3/14,-1/5) -> (19/11,7/4) Hyperbolic Matrix(41,8,128,25) (-1/5,-3/16) -> (5/16,1/3) Hyperbolic Matrix(1335,248,1588,295) (-8/43,-5/27) -> (21/25,16/19) Hyperbolic Matrix(207,38,1084,199) (-7/38,-2/11) -> (4/21,5/26) Hyperbolic Matrix(205,36,-1566,-275) (-3/17,-7/40) -> (-5/38,-3/23) Hyperbolic Matrix(6303,1102,3998,699) (-7/40,-11/63) -> (93/59,41/26) Hyperbolic Matrix(1315,228,248,43) (-4/23,-5/29) -> (37/7,16/3) Hyperbolic Matrix(1109,190,286,49) (-6/35,-1/6) -> (31/8,4/1) Hyperbolic Matrix(281,44,364,57) (-1/6,-2/13) -> (10/13,17/22) Hyperbolic Matrix(79,12,362,55) (-2/13,-1/7) -> (5/23,2/9) Hyperbolic Matrix(237,32,274,37) (-1/7,-2/15) -> (6/7,13/15) Hyperbolic Matrix(1133,146,194,25) (-4/31,-1/8) -> (35/6,6/1) Hyperbolic Matrix(193,22,114,13) (-1/8,0/1) -> (22/13,17/10) Hyperbolic Matrix(93,-10,214,-23) (0/1,1/8) -> (13/30,10/23) Hyperbolic Matrix(661,-86,392,-51) (1/8,2/15) -> (32/19,27/16) Hyperbolic Matrix(299,-42,178,-25) (2/15,1/7) -> (5/3,32/19) Hyperbolic Matrix(119,-18,324,-49) (1/7,1/6) -> (11/30,7/19) Hyperbolic Matrix(493,-86,86,-15) (1/6,3/17) -> (17/3,23/4) Hyperbolic Matrix(577,-104,172,-31) (3/17,2/11) -> (10/3,37/11) Hyperbolic Matrix(229,-42,518,-95) (2/11,3/16) -> (15/34,4/9) Hyperbolic Matrix(485,-92,58,-11) (3/16,4/21) -> (8/1,17/2) Hyperbolic Matrix(285,-56,56,-11) (5/26,1/5) -> (5/1,31/6) Hyperbolic Matrix(85,-18,392,-83) (1/5,3/14) -> (3/14,5/23) Parabolic Matrix(221,-50,84,-19) (2/9,1/4) -> (21/8,8/3) Hyperbolic Matrix(513,-134,134,-35) (1/4,5/19) -> (19/5,23/6) Hyperbolic Matrix(2179,-576,1290,-341) (5/19,9/34) -> (27/16,49/29) Hyperbolic Matrix(1397,-370,404,-107) (9/34,4/15) -> (38/11,7/2) Hyperbolic Matrix(1395,-374,884,-237) (4/15,7/26) -> (41/26,30/19) Hyperbolic Matrix(1259,-340,374,-101) (7/26,3/11) -> (37/11,27/8) Hyperbolic Matrix(239,-66,134,-37) (3/11,2/7) -> (16/9,9/5) Hyperbolic Matrix(341,-100,474,-139) (2/7,5/17) -> (5/7,18/25) Hyperbolic Matrix(629,-186,1444,-427) (5/17,8/27) -> (10/23,17/39) Hyperbolic Matrix(577,-172,104,-31) (8/27,3/10) -> (11/2,28/5) Hyperbolic Matrix(105,-32,338,-103) (3/10,4/13) -> (4/13,5/16) Parabolic Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(221,-84,50,-19) (3/8,5/13) -> (13/3,9/2) Hyperbolic Matrix(637,-246,246,-95) (5/13,7/18) -> (31/12,13/5) Hyperbolic Matrix(121,-50,288,-119) (2/5,5/12) -> (5/12,8/19) Parabolic Matrix(1257,-544,238,-103) (3/7,13/30) -> (21/4,37/7) Hyperbolic Matrix(2651,-1156,782,-341) (17/39,7/16) -> (61/18,17/5) Hyperbolic Matrix(259,-114,284,-125) (11/25,15/34) -> (9/10,1/1) Hyperbolic Matrix(187,-84,118,-53) (4/9,1/2) -> (19/12,8/5) Hyperbolic Matrix(133,-72,242,-131) (1/2,6/11) -> (6/11,11/20) Parabolic Matrix(239,-134,66,-37) (5/9,4/7) -> (18/5,11/3) Hyperbolic Matrix(817,-474,474,-275) (11/19,7/12) -> (31/18,19/11) Hyperbolic Matrix(235,-138,172,-101) (7/12,3/5) -> (15/11,11/8) Hyperbolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(187,-118,84,-53) (5/8,7/11) -> (11/5,9/4) Hyperbolic Matrix(83,-54,20,-13) (7/11,2/3) -> (4/1,13/3) Hyperbolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(2687,-1936,712,-513) (18/25,31/43) -> (49/13,34/9) Hyperbolic Matrix(2923,-2108,850,-613) (31/43,13/18) -> (55/16,31/9) Hyperbolic Matrix(749,-542,474,-343) (13/18,8/11) -> (30/19,19/12) Hyperbolic Matrix(235,-172,138,-101) (8/11,3/4) -> (17/10,12/7) Hyperbolic Matrix(171,-134,134,-105) (7/9,4/5) -> (14/11,9/7) Hyperbolic Matrix(131,-108,74,-61) (4/5,5/6) -> (7/4,16/9) Hyperbolic Matrix(925,-776,776,-651) (5/6,21/25) -> (25/21,31/26) Hyperbolic Matrix(1217,-1026,720,-607) (16/19,11/13) -> (49/29,22/13) Hyperbolic Matrix(617,-538,164,-143) (13/15,7/8) -> (15/4,49/13) Hyperbolic Matrix(145,-128,162,-143) (7/8,8/9) -> (8/9,9/10) Parabolic Matrix(259,-286,48,-53) (1/1,9/8) -> (43/8,27/5) Hyperbolic Matrix(429,-488,80,-91) (9/8,8/7) -> (16/3,43/8) Hyperbolic Matrix(711,-818,206,-237) (8/7,15/13) -> (31/9,38/11) Hyperbolic Matrix(887,-1050,234,-277) (13/11,19/16) -> (53/14,19/5) Hyperbolic Matrix(4621,-5512,1364,-1627) (31/26,6/5) -> (166/49,61/18) Hyperbolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(661,-834,256,-323) (5/4,19/15) -> (49/19,31/12) Hyperbolic Matrix(809,-1028,314,-399) (19/15,14/11) -> (18/7,49/19) Hyperbolic Matrix(1801,-2448,476,-647) (19/14,34/25) -> (34/9,53/14) Hyperbolic Matrix(663,-902,86,-117) (34/25,15/11) -> (23/3,8/1) Hyperbolic Matrix(185,-256,86,-119) (11/8,18/13) -> (2/1,13/6) Hyperbolic Matrix(341,-474,100,-139) (18/13,7/5) -> (17/5,24/7) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(173,-270,66,-103) (3/2,11/7) -> (13/5,21/8) Hyperbolic Matrix(3433,-5408,2178,-3431) (11/7,52/33) -> (52/33,93/59) Parabolic Matrix(547,-882,338,-545) (8/5,21/13) -> (21/13,34/21) Parabolic Matrix(525,-902,188,-323) (12/7,31/18) -> (39/14,14/5) Hyperbolic Matrix(133,-242,72,-131) (9/5,11/6) -> (11/6,13/7) Parabolic Matrix(321,-698,86,-187) (13/6,11/5) -> (41/11,15/4) Hyperbolic Matrix(229,-518,42,-95) (9/4,25/11) -> (27/5,11/2) Hyperbolic Matrix(629,-1444,186,-427) (16/7,23/10) -> (27/8,44/13) Hyperbolic Matrix(93,-214,10,-23) (23/10,7/3) -> (9/1,1/0) Hyperbolic Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(119,-324,18,-49) (19/7,11/4) -> (13/2,7/1) Hyperbolic Matrix(451,-1250,162,-449) (11/4,25/9) -> (25/9,39/14) Parabolic Matrix(105,-338,32,-103) (3/1,13/4) -> (13/4,23/7) Parabolic Matrix(6511,-22050,1922,-6509) (44/13,105/31) -> (105/31,166/49) Parabolic Matrix(997,-3422,192,-659) (24/7,79/23) -> (57/11,26/5) Hyperbolic Matrix(1625,-5584,314,-1079) (79/23,55/16) -> (31/6,57/11) Hyperbolic Matrix(203,-750,36,-133) (11/3,26/7) -> (28/5,17/3) Hyperbolic Matrix(2413,-8978,648,-2411) (26/7,67/18) -> (67/18,108/29) Parabolic Matrix(379,-1458,98,-377) (23/6,27/7) -> (27/7,31/8) Parabolic Matrix(85,-392,18,-83) (9/2,14/3) -> (14/3,19/4) Parabolic Matrix(291,-1682,50,-289) (23/4,29/5) -> (29/5,35/6) Parabolic Matrix(61,-450,8,-59) (7/1,15/2) -> (15/2,23/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,30,1) Matrix(255,118,188,87) -> Matrix(55,-2,468,-17) Matrix(187,86,-1446,-665) -> Matrix(439,-16,7710,-281) Matrix(375,172,-2008,-921) -> Matrix(217,-8,4096,-151) Matrix(437,200,378,173) -> Matrix(55,-2,358,-13) Matrix(309,140,64,29) -> Matrix(217,-8,624,-23) Matrix(557,250,430,193) -> Matrix(107,-4,990,-37) Matrix(303,134,-1766,-781) -> Matrix(211,-8,3930,-149) Matrix(1037,458,120,53) -> Matrix(157,-6,288,-11) Matrix(2005,884,-7174,-3163) -> Matrix(157,-6,3114,-119) Matrix(849,374,-4556,-2007) -> Matrix(151,-6,2844,-113) Matrix(305,134,-1154,-507) -> Matrix(107,-4,2274,-85) Matrix(1087,474,-2956,-1289) -> Matrix(53,-2,1140,-43) Matrix(2291,998,1926,839) -> Matrix(105,-4,394,-15) Matrix(239,104,-1804,-785) -> Matrix(209,-8,3736,-143) Matrix(599,260,182,79) -> Matrix(365,-14,1434,-55) Matrix(237,100,410,173) -> Matrix(51,-2,434,-17) Matrix(177,74,232,97) -> Matrix(51,-2,332,-13) Matrix(289,120,118,49) -> Matrix(151,-6,730,-29) Matrix(405,166,344,141) -> Matrix(49,-2,368,-15) Matrix(289,118,-1036,-423) -> Matrix(49,-2,956,-39) Matrix(111,44,58,23) -> Matrix(49,-2,270,-11) Matrix(169,66,-950,-371) -> Matrix(95,-4,1734,-73) Matrix(165,64,446,173) -> Matrix(47,-2,494,-21) Matrix(167,64,-608,-233) -> Matrix(1,0,0,1) Matrix(221,84,-934,-355) -> Matrix(51,-2,1046,-41) Matrix(549,208,710,269) -> Matrix(49,-2,270,-11) Matrix(161,60,110,41) -> Matrix(47,-2,306,-13) Matrix(2797,1028,536,197) -> Matrix(1,0,-20,1) Matrix(1291,474,-4622,-1697) -> Matrix(89,-4,1758,-79) Matrix(323,118,52,19) -> Matrix(43,-2,108,-5) Matrix(105,38,268,97) -> Matrix(43,-2,452,-21) Matrix(209,74,-788,-279) -> Matrix(45,-2,968,-43) Matrix(473,166,208,73) -> Matrix(45,-2,248,-11) Matrix(197,62,448,141) -> Matrix(45,-2,428,-19) Matrix(97,30,-540,-167) -> Matrix(43,-2,796,-37) Matrix(245,74,96,29) -> Matrix(47,-2,212,-9) Matrix(337,100,920,273) -> Matrix(1,0,-12,1) Matrix(291,86,-1208,-357) -> Matrix(45,-2,968,-43) Matrix(339,100,-1834,-541) -> Matrix(47,-2,870,-37) Matrix(239,70,338,99) -> Matrix(89,-4,690,-31) Matrix(613,172,474,133) -> Matrix(41,-2,390,-19) Matrix(6513,1822,1748,489) -> Matrix(281,-14,1104,-55) Matrix(989,274,610,169) -> Matrix(1,0,-6,1) Matrix(373,100,138,37) -> Matrix(43,-2,194,-9) Matrix(229,60,416,109) -> Matrix(169,-8,1500,-71) Matrix(359,86,-2058,-493) -> Matrix(85,-4,1594,-75) Matrix(265,62,312,73) -> Matrix(41,-2,308,-15) Matrix(131,30,310,71) -> Matrix(161,-8,1590,-79) Matrix(173,38,132,29) -> Matrix(39,-2,332,-17) Matrix(301,64,174,37) -> Matrix(39,-2,254,-13) Matrix(41,8,128,25) -> Matrix(77,-4,828,-43) Matrix(1335,248,1588,295) -> Matrix(75,-4,844,-45) Matrix(207,38,1084,199) -> Matrix(37,-2,500,-27) Matrix(205,36,-1566,-275) -> Matrix(41,-2,718,-35) Matrix(6303,1102,3998,699) -> Matrix(305,-16,1506,-79) Matrix(1315,228,248,43) -> Matrix(1,0,-16,1) Matrix(1109,190,286,49) -> Matrix(223,-12,762,-41) Matrix(281,44,364,57) -> Matrix(73,-4,420,-23) Matrix(79,12,362,55) -> Matrix(253,-14,3018,-167) Matrix(237,32,274,37) -> Matrix(35,-2,298,-17) Matrix(1133,146,194,25) -> Matrix(491,-28,1210,-69) Matrix(193,22,114,13) -> Matrix(69,-4,466,-27) Matrix(93,-10,214,-23) -> Matrix(107,-8,1030,-77) Matrix(661,-86,392,-51) -> Matrix(235,-18,1632,-125) Matrix(299,-42,178,-25) -> Matrix(103,-8,734,-57) Matrix(119,-18,324,-49) -> Matrix(25,-2,288,-23) Matrix(493,-86,86,-15) -> Matrix(199,-16,510,-41) Matrix(577,-104,172,-31) -> Matrix(97,-8,376,-31) Matrix(229,-42,518,-95) -> Matrix(49,-4,478,-39) Matrix(485,-92,58,-11) -> Matrix(73,-6,134,-11) Matrix(285,-56,56,-11) -> Matrix(99,-8,260,-21) Matrix(85,-18,392,-83) -> Matrix(265,-22,3168,-263) Matrix(221,-50,84,-19) -> Matrix(71,-6,296,-25) Matrix(513,-134,134,-35) -> Matrix(91,-8,330,-29) Matrix(2179,-576,1290,-341) -> Matrix(111,-10,766,-69) Matrix(1397,-370,404,-107) -> Matrix(25,-2,88,-7) Matrix(1395,-374,884,-237) -> Matrix(67,-6,324,-29) Matrix(1259,-340,374,-101) -> Matrix(61,-6,234,-23) Matrix(239,-66,134,-37) -> Matrix(47,-4,294,-25) Matrix(341,-100,474,-139) -> Matrix(23,-2,150,-13) Matrix(629,-186,1444,-427) -> Matrix(337,-30,3224,-287) Matrix(577,-172,104,-31) -> Matrix(89,-8,256,-23) Matrix(105,-32,338,-103) -> Matrix(199,-18,2178,-197) Matrix(75,-26,26,-9) -> Matrix(1,0,-6,1) Matrix(221,-84,50,-19) -> Matrix(65,-6,206,-19) Matrix(637,-246,246,-95) -> Matrix(169,-16,750,-71) Matrix(121,-50,288,-119) -> Matrix(141,-14,1400,-139) Matrix(1257,-544,238,-103) -> Matrix(97,-10,262,-27) Matrix(2651,-1156,782,-341) -> Matrix(325,-34,1214,-127) Matrix(259,-114,284,-125) -> Matrix(19,-2,124,-13) Matrix(187,-84,118,-53) -> Matrix(19,-2,86,-9) Matrix(133,-72,242,-131) -> Matrix(91,-10,810,-89) Matrix(239,-134,66,-37) -> Matrix(35,-4,114,-13) Matrix(817,-474,474,-275) -> Matrix(1,0,-2,1) Matrix(235,-138,172,-101) -> Matrix(17,-2,128,-15) Matrix(105,-64,64,-39) -> Matrix(1,0,0,1) Matrix(187,-118,84,-53) -> Matrix(21,-2,116,-11) Matrix(83,-54,20,-13) -> Matrix(17,-2,60,-7) Matrix(141,-98,200,-139) -> Matrix(49,-6,384,-47) Matrix(2687,-1936,712,-513) -> Matrix(1,0,-4,1) Matrix(2923,-2108,850,-613) -> Matrix(137,-18,510,-67) Matrix(749,-542,474,-343) -> Matrix(29,-4,138,-19) Matrix(235,-172,138,-101) -> Matrix(15,-2,98,-13) Matrix(171,-134,134,-105) -> Matrix(1,0,6,1) Matrix(131,-108,74,-61) -> Matrix(15,-2,98,-13) Matrix(925,-776,776,-651) -> Matrix(1,0,-8,1) Matrix(1217,-1026,720,-607) -> Matrix(55,-6,376,-41) Matrix(617,-538,164,-143) -> Matrix(17,-2,60,-7) Matrix(145,-128,162,-143) -> Matrix(15,-2,98,-13) Matrix(259,-286,48,-53) -> Matrix(19,-2,48,-5) Matrix(429,-488,80,-91) -> Matrix(29,-4,80,-11) Matrix(711,-818,206,-237) -> Matrix(25,-4,94,-15) Matrix(887,-1050,234,-277) -> Matrix(25,-4,94,-15) Matrix(4621,-5512,1364,-1627) -> Matrix(31,4,116,15) Matrix(107,-132,30,-37) -> Matrix(13,-2,46,-7) Matrix(661,-834,256,-323) -> Matrix(9,2,40,9) Matrix(809,-1028,314,-399) -> Matrix(43,-2,194,-9) Matrix(1801,-2448,476,-647) -> Matrix(65,-8,252,-31) Matrix(663,-902,86,-117) -> Matrix(129,-16,250,-31) Matrix(185,-256,86,-119) -> Matrix(15,-2,98,-13) Matrix(341,-474,100,-139) -> Matrix(17,-2,60,-7) Matrix(141,-200,98,-139) -> Matrix(43,-6,294,-41) Matrix(173,-270,66,-103) -> Matrix(23,-4,98,-17) Matrix(3433,-5408,2178,-3431) -> Matrix(151,-30,750,-149) Matrix(547,-882,338,-545) -> Matrix(1,0,18,1) Matrix(525,-902,188,-323) -> Matrix(13,-2,72,-11) Matrix(133,-242,72,-131) -> Matrix(61,-10,360,-59) Matrix(321,-698,86,-187) -> Matrix(25,-4,94,-15) Matrix(229,-518,42,-95) -> Matrix(21,-4,58,-11) Matrix(629,-1444,186,-427) -> Matrix(163,-30,614,-113) Matrix(93,-214,10,-23) -> Matrix(43,-8,70,-13) Matrix(121,-288,50,-119) -> Matrix(71,-14,350,-69) Matrix(119,-324,18,-49) -> Matrix(7,-2,18,-5) Matrix(451,-1250,162,-449) -> Matrix(1,0,2,1) Matrix(105,-338,32,-103) -> Matrix(73,-18,288,-71) Matrix(6511,-22050,1922,-6509) -> Matrix(1321,-352,4950,-1319) Matrix(997,-3422,192,-659) -> Matrix(83,-22,200,-53) Matrix(1625,-5584,314,-1079) -> Matrix(217,-58,550,-147) Matrix(203,-750,36,-133) -> Matrix(11,-2,28,-5) Matrix(2413,-8978,648,-2411) -> Matrix(121,-30,480,-119) Matrix(379,-1458,98,-377) -> Matrix(85,-24,294,-83) Matrix(85,-392,18,-83) -> Matrix(67,-22,198,-65) Matrix(291,-1682,50,-289) -> Matrix(141,-56,350,-139) Matrix(61,-450,8,-59) -> Matrix(53,-26,104,-51) 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): 72 Degree of the the map X: 72 Degree of the the map Y: 144 ----------------------------------------------------------------------- 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): 144 Minimal number of generators: 25 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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/1 19/15 10/7 11/6 2/1 12/5 25/9 3/1 13/4 7/2 4/1 14/3 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/15 1/5 4/49 1/4 1/12 1/3 2/21 4/11 1/11 3/8 1/12 5/13 8/85 2/5 3/31 1/2 1/10 4/7 3/25 3/5 2/15 5/8 1/12 2/3 1/9 3/4 1/8 7/9 2/9 4/5 1/11 5/6 1/6 1/1 0/1 5/4 1/4 19/15 0/1 14/11 1/17 9/7 2/21 4/3 1/7 7/5 2/15 10/7 1/7 3/2 1/6 5/3 2/15 12/7 1/7 7/4 3/20 9/5 4/25 11/6 1/6 2/1 1/5 7/3 4/21 12/5 1/5 5/2 3/14 8/3 1/3 11/4 1/4 25/9 0/1 14/5 1/5 3/1 2/9 13/4 1/4 10/3 7/27 7/2 5/18 4/1 1/3 9/2 7/22 14/3 1/3 5/1 4/11 6/1 3/7 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(28,-5,73,-13) (0/1,1/5) -> (3/8,5/13) Hyperbolic Matrix(56,-13,13,-3) (1/5,1/4) -> (4/1,9/2) Hyperbolic Matrix(26,-7,41,-11) (1/4,1/3) -> (5/8,2/3) Hyperbolic Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(63,-23,74,-27) (4/11,3/8) -> (5/6,1/1) Hyperbolic Matrix(74,-29,97,-38) (5/13,2/5) -> (3/4,7/9) Hyperbolic Matrix(24,-11,11,-5) (2/5,1/2) -> (2/1,7/3) Hyperbolic Matrix(77,-43,43,-24) (1/2,4/7) -> (7/4,9/5) Hyperbolic Matrix(53,-31,65,-38) (4/7,3/5) -> (4/5,5/6) Hyperbolic Matrix(106,-65,31,-19) (3/5,5/8) -> (10/3,7/2) Hyperbolic Matrix(40,-29,29,-21) (2/3,3/4) -> (4/3,7/5) Hyperbolic Matrix(171,-134,134,-105) (7/9,4/5) -> (14/11,9/7) Hyperbolic Matrix(53,-65,31,-38) (1/1,5/4) -> (5/3,12/7) Hyperbolic Matrix(286,-361,225,-284) (5/4,19/15) -> (19/15,14/11) Parabolic Matrix(45,-58,7,-9) (9/7,4/3) -> (6/1,1/0) Hyperbolic Matrix(71,-100,49,-69) (7/5,10/7) -> (10/7,3/2) Parabolic Matrix(26,-41,7,-11) (3/2,5/3) -> (7/2,4/1) Hyperbolic Matrix(100,-173,37,-64) (12/7,7/4) -> (8/3,11/4) Hyperbolic Matrix(67,-121,36,-65) (9/5,11/6) -> (11/6,2/1) Parabolic Matrix(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(28,-73,5,-13) (5/2,8/3) -> (5/1,6/1) Hyperbolic Matrix(226,-625,81,-224) (11/4,25/9) -> (25/9,14/5) Parabolic Matrix(53,-169,16,-51) (3/1,13/4) -> (13/4,10/3) Parabolic Matrix(43,-196,9,-41) (9/2,14/3) -> (14/3,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,15,1) Matrix(28,-5,73,-13) -> Matrix(37,-3,395,-32) Matrix(56,-13,13,-3) -> Matrix(35,-3,117,-10) Matrix(26,-7,41,-11) -> Matrix(11,-1,111,-10) Matrix(75,-26,26,-9) -> Matrix(1,0,-6,1) Matrix(63,-23,74,-27) -> Matrix(11,-1,78,-7) Matrix(74,-29,97,-38) -> Matrix(21,-2,137,-13) Matrix(24,-11,11,-5) -> Matrix(9,-1,55,-6) Matrix(77,-43,43,-24) -> Matrix(26,-3,165,-19) Matrix(53,-31,65,-38) -> Matrix(8,-1,73,-9) Matrix(106,-65,31,-19) -> Matrix(5,-1,21,-4) Matrix(40,-29,29,-21) -> Matrix(7,-1,57,-8) Matrix(171,-134,134,-105) -> Matrix(1,0,6,1) Matrix(53,-65,31,-38) -> Matrix(6,-1,43,-7) Matrix(286,-361,225,-284) -> Matrix(1,0,13,1) Matrix(45,-58,7,-9) -> Matrix(10,-1,21,-2) Matrix(71,-100,49,-69) -> Matrix(22,-3,147,-20) Matrix(26,-41,7,-11) -> Matrix(5,-1,21,-4) Matrix(100,-173,37,-64) -> Matrix(13,-2,59,-9) Matrix(67,-121,36,-65) -> Matrix(31,-5,180,-29) Matrix(61,-144,25,-59) -> Matrix(36,-7,175,-34) Matrix(28,-73,5,-13) -> Matrix(13,-3,35,-8) Matrix(226,-625,81,-224) -> Matrix(1,0,1,1) Matrix(53,-169,16,-51) -> Matrix(37,-9,144,-35) Matrix(43,-196,9,-41) -> Matrix(34,-11,99,-32) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 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: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 72 ----------------------------------------------------------------------- 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 -1/1 0/1 15 1 1/1 0/1 1 17 5/4 1/4 1 17 19/15 0/1 13 1 9/7 2/21 1 17 4/3 1/7 1 17 10/7 1/7 3 1 3/2 1/6 1 17 5/3 2/15 1 17 12/7 1/7 1 17 7/4 3/20 1 17 11/6 1/6 5 1 2/1 1/5 1 17 12/5 1/5 7 1 5/2 3/14 1 17 8/3 1/3 1 17 11/4 1/4 1 17 25/9 0/1 1 1 3/1 2/9 1 17 13/4 1/4 9 1 7/2 5/18 1 17 4/1 1/3 1 17 14/3 1/3 11 1 5/1 4/11 1 17 6/1 3/7 1 17 1/0 1/0 1 17 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(53,-65,31,-38) (1/1,5/4) -> (5/3,12/7) Hyperbolic Matrix(151,-190,120,-151) (5/4,19/15) -> (5/4,19/15) Reflection Matrix(134,-171,105,-134) (19/15,9/7) -> (19/15,9/7) Reflection Matrix(45,-58,7,-9) (9/7,4/3) -> (6/1,1/0) Hyperbolic Matrix(29,-40,21,-29) (4/3,10/7) -> (4/3,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(26,-41,7,-11) (3/2,5/3) -> (7/2,4/1) Hyperbolic Matrix(100,-173,37,-64) (12/7,7/4) -> (8/3,11/4) Hyperbolic Matrix(43,-77,24,-43) (7/4,11/6) -> (7/4,11/6) Reflection Matrix(23,-44,12,-23) (11/6,2/1) -> (11/6,2/1) Reflection Matrix(11,-24,5,-11) (2/1,12/5) -> (2/1,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(28,-73,5,-13) (5/2,8/3) -> (5/1,6/1) Hyperbolic Matrix(199,-550,72,-199) (11/4,25/9) -> (11/4,25/9) Reflection Matrix(26,-75,9,-26) (25/9,3/1) -> (25/9,3/1) Reflection Matrix(25,-78,8,-25) (3/1,13/4) -> (3/1,13/4) Reflection Matrix(27,-91,8,-27) (13/4,7/2) -> (13/4,7/2) Reflection Matrix(13,-56,3,-13) (4/1,14/3) -> (4/1,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,15,-1) (-1/1,1/1) -> (0/1,2/15) Matrix(53,-65,31,-38) -> Matrix(6,-1,43,-7) (1/7,1/5).(0/1,2/13).(1/8,1/6) Matrix(151,-190,120,-151) -> Matrix(1,0,8,-1) (5/4,19/15) -> (0/1,1/4) Matrix(134,-171,105,-134) -> Matrix(1,0,21,-1) (19/15,9/7) -> (0/1,2/21) Matrix(45,-58,7,-9) -> Matrix(10,-1,21,-2) Matrix(29,-40,21,-29) -> Matrix(8,-1,63,-8) (4/3,10/7) -> (1/9,1/7) Matrix(41,-60,28,-41) -> Matrix(13,-2,84,-13) (10/7,3/2) -> (1/7,1/6) Matrix(26,-41,7,-11) -> Matrix(5,-1,21,-4) (1/5,1/3).(0/1,2/9).(1/6,1/4) Matrix(100,-173,37,-64) -> Matrix(13,-2,59,-9) Matrix(43,-77,24,-43) -> Matrix(19,-3,120,-19) (7/4,11/6) -> (3/20,1/6) Matrix(23,-44,12,-23) -> Matrix(11,-2,60,-11) (11/6,2/1) -> (1/6,1/5) Matrix(11,-24,5,-11) -> Matrix(6,-1,35,-6) (2/1,12/5) -> (1/7,1/5) Matrix(49,-120,20,-49) -> Matrix(29,-6,140,-29) (12/5,5/2) -> (1/5,3/14) Matrix(28,-73,5,-13) -> Matrix(13,-3,35,-8) Matrix(199,-550,72,-199) -> Matrix(1,0,8,-1) (11/4,25/9) -> (0/1,1/4) Matrix(26,-75,9,-26) -> Matrix(1,0,9,-1) (25/9,3/1) -> (0/1,2/9) Matrix(25,-78,8,-25) -> Matrix(17,-4,72,-17) (3/1,13/4) -> (2/9,1/4) Matrix(27,-91,8,-27) -> Matrix(19,-5,72,-19) (13/4,7/2) -> (1/4,5/18) Matrix(13,-56,3,-13) -> Matrix(10,-3,33,-10) (4/1,14/3) -> (3/11,1/3) Matrix(29,-140,6,-29) -> Matrix(23,-8,66,-23) (14/3,5/1) -> (1/3,4/11) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.