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: 56 Genus: 37 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/8 -3/4 -5/8 -7/12 -1/2 -7/15 -5/12 -3/8 -17/48 -3/10 -19/64 -9/32 -1/4 -15/64 -29/128 -7/32 -3/14 -11/64 -1/6 -5/32 -2/13 -1/8 0/1 1/9 1/8 1/7 1/6 2/11 3/16 1/5 3/14 2/9 3/13 1/4 4/15 3/11 2/7 3/10 5/16 1/3 3/8 2/5 5/12 7/16 1/2 9/16 7/12 5/8 31/48 2/3 11/16 3/4 13/16 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -8/9 -1/2 -7/8 0/1 -6/7 1/4 -5/6 1/3 1/1 -9/11 2/3 -13/16 1/1 -4/5 1/0 -11/14 -1/1 1/1 -7/9 0/1 -10/13 -1/2 -3/4 0/1 -14/19 1/2 -11/15 0/1 -8/11 1/2 -13/18 1/3 1/1 -5/7 0/1 -17/24 1/2 -29/41 6/11 -12/17 1/2 -7/10 3/5 1/1 -9/13 4/5 -11/16 1/1 -2/3 1/0 -11/17 0/1 -9/14 -1/1 1/1 -7/11 0/1 -5/8 0/1 -13/21 0/1 -8/13 1/2 -19/31 0/1 -11/18 1/3 1/1 -3/5 0/1 -13/22 1/3 1/1 -10/17 1/2 -17/29 0/1 -7/12 1/2 -18/31 1/2 -11/19 2/3 -15/26 3/5 1/1 -4/7 3/4 -9/16 1/1 -5/9 4/3 -11/20 2/1 -6/11 1/0 -1/2 -1/1 1/1 -7/15 0/1 -6/13 1/0 -5/11 0/1 -14/31 1/2 -9/20 1/2 -13/29 2/3 -4/9 3/4 -7/16 1/1 -3/7 4/3 -8/19 3/2 -13/31 2/1 -18/43 3/2 -5/12 2/1 -17/41 4/1 -12/29 1/0 -7/17 2/1 -2/5 1/0 -7/18 1/1 3/1 -19/49 2/1 -31/80 3/1 -12/31 1/0 -5/13 2/1 -3/8 1/0 -7/19 -2/1 -11/30 -3/1 -1/1 -4/11 1/0 -5/14 -1/1 1/1 -11/31 0/1 -17/48 1/1 -6/17 1/0 -1/3 0/1 -5/16 1/1 -4/13 5/4 -7/23 4/3 -10/33 3/2 -13/43 2/1 -3/10 1/1 5/3 -11/37 2/1 -19/64 1/1 5/3 -8/27 3/2 -5/17 2/1 -12/41 11/6 -7/24 2/1 -2/7 1/0 -9/32 1/1 3/1 -7/25 2/1 -5/18 1/1 3/1 -3/11 2/1 -7/26 3/1 5/1 -18/67 1/0 -11/41 6/1 -4/15 1/0 -1/4 1/0 -4/17 1/0 -15/64 -3/1 -1/1 -11/47 -2/1 -7/30 -3/1 -1/1 -3/13 -2/1 -5/22 -1/1 -1/3 -17/75 0/1 -29/128 -1/1 -1/3 -12/53 -1/2 -7/31 0/1 -2/9 1/0 -7/32 -1/1 1/1 -5/23 0/1 -3/14 -1/1 1/1 -1/5 0/1 -3/16 1/1 -2/11 3/2 -3/17 2/1 -4/23 5/2 -5/29 2/1 -11/64 1/1 3/1 -6/35 1/0 -1/6 1/1 3/1 -3/19 2/1 -5/32 1/1 3/1 -2/13 1/0 -1/7 4/1 -1/8 1/0 -1/9 -2/1 0/1 1/0 1/9 2/1 1/8 1/0 1/7 -4/1 1/6 -3/1 -1/1 2/11 -3/2 3/16 -1/1 1/5 0/1 3/14 -1/1 1/1 2/9 1/0 3/13 2/1 1/4 1/0 5/19 -2/1 4/15 1/0 3/11 -2/1 5/18 -3/1 -1/1 2/7 1/0 7/24 -2/1 12/41 -11/6 5/17 -2/1 3/10 -5/3 -1/1 4/13 -5/4 5/16 -1/1 1/3 0/1 6/17 1/0 5/14 -1/1 1/1 4/11 1/0 3/8 1/0 8/21 1/0 5/13 -2/1 12/31 1/0 7/18 -3/1 -1/1 2/5 1/0 9/22 -3/1 -1/1 7/17 -2/1 12/29 1/0 5/12 -2/1 13/31 -2/1 8/19 -3/2 11/26 -5/3 -1/1 3/7 -4/3 7/16 -1/1 4/9 -3/4 9/20 -1/2 5/11 0/1 1/2 -1/1 1/1 8/15 1/0 7/13 0/1 6/11 1/0 17/31 -2/1 11/20 -2/1 16/29 -3/2 5/9 -4/3 9/16 -1/1 4/7 -3/4 11/19 -2/3 18/31 -1/2 25/43 -2/3 7/12 -1/2 24/41 -1/4 17/29 0/1 10/17 -1/2 3/5 0/1 11/18 -1/1 -1/3 30/49 -1/2 49/80 -1/3 19/31 0/1 8/13 -1/2 5/8 0/1 12/19 1/2 19/30 1/3 1/1 7/11 0/1 9/14 -1/1 1/1 20/31 1/0 31/48 -1/1 11/17 0/1 2/3 1/0 11/16 -1/1 9/13 -4/5 16/23 -3/4 23/33 -2/3 30/43 -1/2 7/10 -1/1 -3/5 26/37 -1/2 45/64 -1/1 -3/5 19/27 -2/3 12/17 -1/2 29/41 -6/11 17/24 -1/2 5/7 0/1 23/32 -1/1 -1/3 18/25 -1/2 13/18 -1/1 -1/3 8/11 -1/2 19/26 -1/3 -1/5 49/67 0/1 30/41 -1/6 11/15 0/1 3/4 0/1 13/17 0/1 49/64 1/3 1/1 36/47 1/2 23/30 1/3 1/1 10/13 1/2 17/22 1/1 3/1 58/75 1/0 99/128 1/1 3/1 41/53 2/1 24/31 1/0 7/9 0/1 25/32 -1/1 1/1 18/23 1/0 11/14 -1/1 1/1 4/5 1/0 13/16 -1/1 9/11 -2/3 14/17 -1/2 19/23 -2/5 24/29 -1/2 53/64 -1/1 -1/3 29/35 0/1 5/6 -1/1 -1/3 16/19 -1/2 27/32 -1/1 -1/3 11/13 0/1 6/7 -1/4 7/8 0/1 8/9 1/2 1/1 0/1 1/0 -1/1 1/1 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(223,200,320,287) (-1/1,-8/9) -> (16/23,23/33) Hyperbolic Matrix(159,140,544,479) (-8/9,-7/8) -> (7/24,12/41) Hyperbolic Matrix(65,56,224,193) (-7/8,-6/7) -> (2/7,7/24) Hyperbolic Matrix(129,110,-224,-191) (-6/7,-5/6) -> (-15/26,-4/7) Hyperbolic Matrix(95,78,-352,-289) (-5/6,-9/11) -> (-3/11,-7/26) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(63,50,160,127) (-4/5,-11/14) -> (7/18,2/5) Hyperbolic Matrix(97,76,-448,-351) (-11/14,-7/9) -> (-5/23,-3/14) Hyperbolic Matrix(127,98,-416,-321) (-7/9,-10/13) -> (-4/13,-7/23) Hyperbolic Matrix(95,72,-128,-97) (-10/13,-3/4) -> (-3/4,-14/19) Parabolic Matrix(223,164,-480,-353) (-14/19,-11/15) -> (-7/15,-6/13) Hyperbolic Matrix(63,46,-352,-257) (-11/15,-8/11) -> (-2/11,-3/17) Hyperbolic Matrix(127,92,352,255) (-8/11,-13/18) -> (5/14,4/11) Hyperbolic Matrix(161,116,-576,-415) (-13/18,-5/7) -> (-7/25,-5/18) Hyperbolic Matrix(31,22,224,159) (-5/7,-17/24) -> (1/8,1/7) Hyperbolic Matrix(65,46,544,385) (-17/24,-29/41) -> (1/9,1/8) Hyperbolic Matrix(897,634,1088,769) (-29/41,-12/17) -> (14/17,19/23) Hyperbolic Matrix(321,226,-544,-383) (-12/17,-7/10) -> (-13/22,-10/17) Hyperbolic Matrix(95,66,-416,-289) (-7/10,-9/13) -> (-3/13,-5/22) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) Hyperbolic Matrix(65,44,96,65) (-11/16,-2/3) -> (2/3,11/16) Hyperbolic Matrix(225,146,544,353) (-2/3,-11/17) -> (7/17,12/29) Hyperbolic Matrix(385,248,-992,-639) (-11/17,-9/14) -> (-7/18,-19/49) Hyperbolic Matrix(97,62,352,225) (-9/14,-7/11) -> (3/11,5/18) Hyperbolic Matrix(159,100,-256,-161) (-7/11,-5/8) -> (-5/8,-13/21) Parabolic Matrix(191,118,-416,-257) (-13/21,-8/13) -> (-6/13,-5/11) Hyperbolic Matrix(417,256,992,609) (-8/13,-19/31) -> (13/31,8/19) Hyperbolic Matrix(353,216,-992,-607) (-19/31,-11/18) -> (-5/14,-11/31) Hyperbolic Matrix(33,20,160,97) (-11/18,-3/5) -> (1/5,3/14) Hyperbolic Matrix(257,152,-864,-511) (-3/5,-13/22) -> (-3/10,-11/37) Hyperbolic Matrix(191,112,544,319) (-10/17,-17/29) -> (1/3,6/17) Hyperbolic Matrix(417,244,-928,-543) (-17/29,-7/12) -> (-9/20,-13/29) Hyperbolic Matrix(447,260,-992,-577) (-7/12,-18/31) -> (-14/31,-9/20) Hyperbolic Matrix(383,222,992,575) (-18/31,-11/19) -> (5/13,12/31) Hyperbolic Matrix(97,56,-608,-351) (-11/19,-15/26) -> (-1/6,-3/19) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(385,212,-928,-511) (-5/9,-11/20) -> (-5/12,-17/41) Hyperbolic Matrix(415,228,-992,-545) (-11/20,-6/11) -> (-18/43,-5/12) Hyperbolic Matrix(129,70,-352,-191) (-6/11,-1/2) -> (-11/30,-4/11) Hyperbolic Matrix(127,60,-544,-257) (-1/2,-7/15) -> (-11/47,-7/30) Hyperbolic Matrix(513,232,-1696,-767) (-5/11,-14/31) -> (-10/33,-13/43) Hyperbolic Matrix(161,72,-928,-415) (-13/29,-4/9) -> (-4/23,-5/29) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(33,14,-224,-95) (-3/7,-8/19) -> (-2/13,-1/7) Hyperbolic Matrix(609,256,992,417) (-8/19,-13/31) -> (19/31,8/13) Hyperbolic Matrix(673,282,-2976,-1247) (-13/31,-18/43) -> (-12/53,-7/31) Hyperbolic Matrix(1057,438,-3936,-1631) (-17/41,-12/29) -> (-18/67,-11/41) Hyperbolic Matrix(353,146,544,225) (-12/29,-7/17) -> (11/17,2/3) Hyperbolic Matrix(161,66,-544,-223) (-7/17,-2/5) -> (-8/27,-5/17) Hyperbolic Matrix(127,50,160,63) (-2/5,-7/18) -> (11/14,4/5) Hyperbolic Matrix(2399,930,3712,1439) (-19/49,-31/80) -> (31/48,11/17) Hyperbolic Matrix(2561,992,3968,1537) (-31/80,-12/31) -> (20/31,31/48) Hyperbolic Matrix(575,222,992,383) (-12/31,-5/13) -> (11/19,18/31) Hyperbolic Matrix(95,36,-256,-97) (-5/13,-3/8) -> (-3/8,-7/19) Parabolic Matrix(223,82,-960,-353) (-7/19,-11/30) -> (-7/30,-3/13) Hyperbolic Matrix(255,92,352,127) (-4/11,-5/14) -> (13/18,8/11) Hyperbolic Matrix(2431,862,3968,1407) (-11/31,-17/48) -> (49/80,19/31) Hyperbolic Matrix(2273,804,3712,1313) (-17/48,-6/17) -> (30/49,49/80) Hyperbolic Matrix(319,112,544,191) (-6/17,-1/3) -> (17/29,10/17) Hyperbolic Matrix(31,10,96,31) (-1/3,-5/16) -> (5/16,1/3) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(33,10,320,97) (-7/23,-10/33) -> (0/1,1/9) Hyperbolic Matrix(385,116,-1696,-511) (-13/43,-3/10) -> (-5/22,-17/75) Hyperbolic Matrix(2881,856,4096,1217) (-11/37,-19/64) -> (45/64,19/27) Hyperbolic Matrix(2879,854,4096,1215) (-19/64,-8/27) -> (26/37,45/64) Hyperbolic Matrix(961,282,1312,385) (-5/17,-12/41) -> (30/41,11/15) Hyperbolic Matrix(479,140,544,159) (-12/41,-7/24) -> (7/8,8/9) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(737,208,1024,289) (-2/7,-9/32) -> (23/32,18/25) Hyperbolic Matrix(735,206,1024,287) (-9/32,-7/25) -> (5/7,23/32) Hyperbolic Matrix(225,62,352,97) (-5/18,-3/11) -> (7/11,9/14) Hyperbolic Matrix(223,60,-1312,-353) (-7/26,-18/67) -> (-6/35,-1/6) Hyperbolic Matrix(927,248,1312,351) (-11/41,-4/15) -> (12/17,29/41) Hyperbolic Matrix(31,8,-128,-33) (-4/15,-1/4) -> (-1/4,-4/17) Parabolic Matrix(3137,736,4096,961) (-4/17,-15/64) -> (49/64,36/47) Hyperbolic Matrix(3135,734,4096,959) (-15/64,-11/47) -> (13/17,49/64) Hyperbolic Matrix(12673,2872,16384,3713) (-17/75,-29/128) -> (99/128,41/53) Hyperbolic Matrix(12671,2870,16384,3711) (-29/128,-12/53) -> (58/75,99/128) Hyperbolic Matrix(257,58,288,65) (-7/31,-2/9) -> (8/9,1/1) Hyperbolic Matrix(801,176,1024,225) (-2/9,-7/32) -> (25/32,18/23) Hyperbolic Matrix(799,174,1024,223) (-7/32,-5/23) -> (7/9,25/32) Hyperbolic Matrix(97,20,160,33) (-3/14,-1/5) -> (3/5,11/18) Hyperbolic Matrix(31,6,160,31) (-1/5,-3/16) -> (3/16,1/5) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(319,56,1088,191) (-3/17,-4/23) -> (12/41,5/17) Hyperbolic Matrix(3393,584,4096,705) (-5/29,-11/64) -> (53/64,29/35) Hyperbolic Matrix(3391,582,4096,703) (-11/64,-6/35) -> (24/29,53/64) Hyperbolic Matrix(865,136,1024,161) (-3/19,-5/32) -> (27/32,11/13) Hyperbolic Matrix(863,134,1024,159) (-5/32,-2/13) -> (16/19,27/32) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(385,46,544,65) (-1/8,-1/9) -> (29/41,17/24) Hyperbolic Matrix(223,24,288,31) (-1/9,0/1) -> (24/31,7/9) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(351,-76,448,-97) (3/14,2/9) -> (18/23,11/14) Hyperbolic Matrix(289,-66,416,-95) (2/9,3/13) -> (9/13,16/23) Hyperbolic Matrix(33,-8,128,-31) (3/13,1/4) -> (1/4,5/19) Parabolic Matrix(257,-68,480,-127) (5/19,4/15) -> (8/15,7/13) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(415,-116,576,-161) (5/18,2/7) -> (18/25,13/18) Hyperbolic Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(607,-216,992,-353) (6/17,5/14) -> (11/18,30/49) Hyperbolic Matrix(97,-36,256,-95) (4/11,3/8) -> (3/8,8/21) Parabolic Matrix(225,-86,416,-159) (8/21,5/13) -> (7/13,6/11) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(511,-212,928,-385) (12/29,5/12) -> (11/20,16/29) Hyperbolic Matrix(545,-228,992,-415) (5/12,13/31) -> (17/31,11/20) Hyperbolic Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(543,-244,928,-417) (4/9,9/20) -> (7/12,24/41) Hyperbolic Matrix(577,-260,992,-447) (9/20,5/11) -> (25/43,7/12) Hyperbolic Matrix(223,-102,352,-161) (5/11,1/2) -> (19/30,7/11) Hyperbolic Matrix(417,-220,544,-287) (1/2,8/15) -> (36/47,23/30) Hyperbolic Matrix(1183,-648,1696,-929) (6/11,17/31) -> (23/33,30/43) Hyperbolic Matrix(767,-424,928,-513) (16/29,5/9) -> (19/23,24/29) Hyperbolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(2303,-1338,2976,-1729) (18/31,25/43) -> (41/53,24/31) Hyperbolic Matrix(2879,-1686,3936,-2305) (24/41,17/29) -> (49/67,30/41) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(161,-100,256,-159) (8/13,5/8) -> (5/8,12/19) Parabolic Matrix(737,-466,960,-607) (12/19,19/30) -> (23/30,10/13) Hyperbolic Matrix(1311,-916,1696,-1185) (30/43,7/10) -> (17/22,58/75) Hyperbolic Matrix(1089,-796,1312,-959) (19/26,49/67) -> (29/35,5/6) Hyperbolic Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(223,200,320,287) -> Matrix(1,2,-2,-3) Matrix(159,140,544,479) -> Matrix(15,2,-8,-1) Matrix(65,56,224,193) -> Matrix(9,-2,-4,1) Matrix(129,110,-224,-191) -> Matrix(5,-2,8,-3) Matrix(95,78,-352,-289) -> Matrix(7,-4,2,-1) Matrix(287,234,352,287) -> Matrix(5,-4,-6,5) Matrix(129,104,160,129) -> Matrix(1,-2,0,1) Matrix(63,50,160,127) -> Matrix(1,-2,0,1) Matrix(97,76,-448,-351) -> Matrix(1,0,0,1) Matrix(127,98,-416,-321) -> Matrix(3,4,2,3) Matrix(95,72,-128,-97) -> Matrix(1,0,4,1) Matrix(223,164,-480,-353) -> Matrix(1,0,-2,1) Matrix(63,46,-352,-257) -> Matrix(7,-2,4,-1) Matrix(127,92,352,255) -> Matrix(1,0,-2,1) Matrix(161,116,-576,-415) -> Matrix(3,-2,2,-1) Matrix(31,22,224,159) -> Matrix(9,-4,-2,1) Matrix(65,46,544,385) -> Matrix(15,-8,2,-1) Matrix(897,634,1088,769) -> Matrix(7,-4,-12,7) Matrix(321,226,-544,-383) -> Matrix(3,-2,8,-5) Matrix(95,66,-416,-289) -> Matrix(3,-2,-4,3) Matrix(287,198,416,287) -> Matrix(9,-8,-10,9) Matrix(65,44,96,65) -> Matrix(1,-2,0,1) Matrix(225,146,544,353) -> Matrix(1,-2,0,1) Matrix(385,248,-992,-639) -> Matrix(1,2,0,1) Matrix(97,62,352,225) -> Matrix(1,-2,0,1) Matrix(159,100,-256,-161) -> Matrix(1,0,4,1) Matrix(191,118,-416,-257) -> Matrix(1,0,-2,1) Matrix(417,256,992,609) -> Matrix(1,-2,0,1) Matrix(353,216,-992,-607) -> Matrix(1,0,-2,1) Matrix(33,20,160,97) -> Matrix(1,0,-2,1) Matrix(257,152,-864,-511) -> Matrix(7,-2,4,-1) Matrix(191,112,544,319) -> Matrix(1,0,-2,1) Matrix(417,244,-928,-543) -> Matrix(5,-2,8,-3) Matrix(447,260,-992,-577) -> Matrix(3,-2,8,-5) Matrix(383,222,992,575) -> Matrix(1,0,-2,1) Matrix(97,56,-608,-351) -> Matrix(7,-4,2,-1) Matrix(127,72,224,127) -> Matrix(7,-6,-8,7) Matrix(161,90,288,161) -> Matrix(7,-8,-6,7) Matrix(385,212,-928,-511) -> Matrix(5,-8,2,-3) Matrix(415,228,-992,-545) -> Matrix(3,-8,2,-5) Matrix(129,70,-352,-191) -> Matrix(1,-2,0,1) Matrix(127,60,-544,-257) -> Matrix(1,-2,0,1) Matrix(513,232,-1696,-767) -> Matrix(7,-2,4,-1) Matrix(161,72,-928,-415) -> Matrix(7,-4,2,-1) Matrix(127,56,288,127) -> Matrix(7,-6,-8,7) Matrix(97,42,224,97) -> Matrix(7,-8,-6,7) Matrix(33,14,-224,-95) -> Matrix(5,-8,2,-3) Matrix(609,256,992,417) -> Matrix(1,-2,0,1) Matrix(673,282,-2976,-1247) -> Matrix(1,-2,0,1) Matrix(1057,438,-3936,-1631) -> Matrix(1,2,0,1) Matrix(353,146,544,225) -> Matrix(1,-2,0,1) Matrix(161,66,-544,-223) -> Matrix(3,-8,2,-5) Matrix(127,50,160,63) -> Matrix(1,-2,0,1) Matrix(2399,930,3712,1439) -> Matrix(1,-2,-2,5) Matrix(2561,992,3968,1537) -> Matrix(1,-4,0,1) Matrix(575,222,992,383) -> Matrix(1,0,-2,1) Matrix(95,36,-256,-97) -> Matrix(1,-4,0,1) Matrix(223,82,-960,-353) -> Matrix(1,0,0,1) Matrix(255,92,352,127) -> Matrix(1,0,-2,1) Matrix(2431,862,3968,1407) -> Matrix(1,0,-4,1) Matrix(2273,804,3712,1313) -> Matrix(1,-2,-2,5) Matrix(319,112,544,191) -> Matrix(1,0,-2,1) Matrix(31,10,96,31) -> Matrix(1,0,-2,1) Matrix(129,40,416,129) -> Matrix(9,-10,-8,9) Matrix(33,10,320,97) -> Matrix(1,-2,2,-3) Matrix(385,116,-1696,-511) -> Matrix(1,-2,0,1) Matrix(2881,856,4096,1217) -> Matrix(5,-8,-8,13) Matrix(2879,854,4096,1215) -> Matrix(5,-8,-8,13) Matrix(961,282,1312,385) -> Matrix(1,-2,0,1) Matrix(479,140,544,159) -> Matrix(1,-2,8,-15) Matrix(193,56,224,65) -> Matrix(1,-2,-4,9) Matrix(737,208,1024,289) -> Matrix(1,-2,-2,5) Matrix(735,206,1024,287) -> Matrix(1,-2,-2,5) Matrix(225,62,352,97) -> Matrix(1,-2,0,1) Matrix(223,60,-1312,-353) -> Matrix(1,-2,0,1) Matrix(927,248,1312,351) -> Matrix(1,0,-2,1) Matrix(31,8,-128,-33) -> Matrix(1,-4,0,1) Matrix(3137,736,4096,961) -> Matrix(1,2,2,5) Matrix(3135,734,4096,959) -> Matrix(1,2,2,5) Matrix(12673,2872,16384,3713) -> Matrix(5,2,2,1) Matrix(12671,2870,16384,3711) -> Matrix(5,2,2,1) Matrix(257,58,288,65) -> Matrix(1,0,2,1) Matrix(801,176,1024,225) -> Matrix(1,0,0,1) Matrix(799,174,1024,223) -> Matrix(1,0,0,1) Matrix(97,20,160,33) -> Matrix(1,0,-2,1) Matrix(31,6,160,31) -> Matrix(1,0,-2,1) Matrix(65,12,352,65) -> Matrix(5,-6,-4,5) Matrix(319,56,1088,191) -> Matrix(7,-12,-4,7) Matrix(3393,584,4096,705) -> Matrix(1,-2,-2,5) Matrix(3391,582,4096,703) -> Matrix(1,-2,-2,5) Matrix(865,136,1024,161) -> Matrix(1,-2,-2,5) Matrix(863,134,1024,159) -> Matrix(1,-2,-2,5) Matrix(159,22,224,31) -> Matrix(1,-4,-2,9) Matrix(385,46,544,65) -> Matrix(1,8,-2,-15) Matrix(223,24,288,31) -> Matrix(1,2,0,1) Matrix(95,-14,224,-33) -> Matrix(3,8,-2,-5) Matrix(257,-46,352,-63) -> Matrix(1,2,-4,-7) Matrix(351,-76,448,-97) -> Matrix(1,0,0,1) Matrix(289,-66,416,-95) -> Matrix(3,-2,-4,3) Matrix(33,-8,128,-31) -> Matrix(1,-4,0,1) Matrix(257,-68,480,-127) -> Matrix(1,2,0,1) Matrix(289,-78,352,-95) -> Matrix(1,4,-2,-7) Matrix(415,-116,576,-161) -> Matrix(1,2,-2,-3) Matrix(223,-66,544,-161) -> Matrix(5,8,-2,-3) Matrix(321,-98,416,-127) -> Matrix(3,4,2,3) Matrix(607,-216,992,-353) -> Matrix(1,0,-2,1) Matrix(97,-36,256,-95) -> Matrix(1,-4,0,1) Matrix(225,-86,416,-159) -> Matrix(1,2,0,1) Matrix(639,-248,992,-385) -> Matrix(1,2,0,1) Matrix(607,-248,864,-353) -> Matrix(1,4,-2,-7) Matrix(511,-212,928,-385) -> Matrix(3,8,-2,-5) Matrix(545,-228,992,-415) -> Matrix(5,8,-2,-3) Matrix(511,-216,608,-257) -> Matrix(1,2,-4,-7) Matrix(543,-244,928,-417) -> Matrix(3,2,-8,-5) Matrix(577,-260,992,-447) -> Matrix(5,2,-8,-3) Matrix(223,-102,352,-161) -> Matrix(1,0,2,1) Matrix(417,-220,544,-287) -> Matrix(1,0,2,1) Matrix(1183,-648,1696,-929) -> Matrix(1,4,-2,-7) Matrix(767,-424,928,-513) -> Matrix(1,2,-4,-7) Matrix(191,-110,224,-129) -> Matrix(3,2,-8,-5) Matrix(2303,-1338,2976,-1729) -> Matrix(1,0,2,1) Matrix(2879,-1686,3936,-2305) -> Matrix(1,0,-2,1) Matrix(383,-226,544,-321) -> Matrix(5,2,-8,-3) Matrix(161,-100,256,-159) -> Matrix(1,0,4,1) Matrix(737,-466,960,-607) -> Matrix(1,0,0,1) Matrix(1311,-916,1696,-1185) -> Matrix(1,0,2,1) Matrix(1089,-796,1312,-959) -> Matrix(1,0,2,1) Matrix(97,-72,128,-95) -> Matrix(1,0,4,1) 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): 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 1/8 1/7 1/6 1/5 1/4 3/10 1/3 3/8 5/12 1/2 9/16 7/12 5/8 9/14 31/48 11/16 45/64 3/4 25/32 13/16 27/32 7/8 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/8 1/0 1/7 -4/1 1/6 -3/1 -1/1 1/5 0/1 1/4 1/0 3/11 -2/1 5/18 -3/1 -1/1 2/7 1/0 7/24 -2/1 5/17 -2/1 3/10 -5/3 -1/1 1/3 0/1 3/8 1/0 5/13 -2/1 12/31 1/0 7/18 -3/1 -1/1 2/5 1/0 9/22 -3/1 -1/1 7/17 -2/1 5/12 -2/1 13/31 -2/1 8/19 -3/2 11/26 -5/3 -1/1 3/7 -4/3 1/2 -1/1 1/1 5/9 -4/3 9/16 -1/1 4/7 -3/4 11/19 -2/3 7/12 -1/2 17/29 0/1 10/17 -1/2 3/5 0/1 5/8 0/1 7/11 0/1 9/14 -1/1 1/1 20/31 1/0 31/48 -1/1 11/17 0/1 2/3 1/0 11/16 -1/1 9/13 -4/5 7/10 -1/1 -3/5 26/37 -1/2 45/64 -1/1 -3/5 19/27 -2/3 12/17 -1/2 17/24 -1/2 5/7 0/1 3/4 0/1 7/9 0/1 25/32 -1/1 1/1 18/23 1/0 11/14 -1/1 1/1 4/5 1/0 13/16 -1/1 9/11 -2/3 5/6 -1/1 -1/3 16/19 -1/2 27/32 -1/1 -1/3 11/13 0/1 6/7 -1/4 7/8 0/1 1/1 0/1 1/0 -1/1 1/1 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(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(65,-9,224,-31) (1/8,1/7) -> (2/7,7/24) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(79,-14,96,-17) (1/6,1/5) -> (9/11,5/6) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(113,-31,288,-79) (3/11,5/18) -> (7/18,2/5) Hyperbolic Matrix(401,-112,512,-143) (5/18,2/7) -> (18/23,11/14) Hyperbolic Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(417,-161,992,-383) (5/13,12/31) -> (13/31,8/19) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(241,-100,576,-239) (7/17,5/12) -> (5/12,13/31) Parabolic Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(145,-81,256,-143) (5/9,9/16) -> (9/16,4/7) Parabolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(337,-196,576,-335) (11/19,7/12) -> (7/12,17/29) Parabolic Matrix(353,-207,544,-319) (17/29,10/17) -> (11/17,2/3) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(81,-50,128,-79) (3/5,5/8) -> (5/8,7/11) Parabolic Matrix(177,-113,224,-143) (7/11,9/14) -> (11/14,4/5) Hyperbolic Matrix(1489,-961,2304,-1487) (20/31,31/48) -> (31/48,11/17) Parabolic Matrix(177,-121,256,-175) (2/3,11/16) -> (11/16,9/13) Parabolic Matrix(2881,-2025,4096,-2879) (26/37,45/64) -> (45/64,19/27) Parabolic Matrix(143,-101,160,-113) (12/17,17/24) -> (7/8,1/1) Hyperbolic Matrix(193,-137,224,-159) (17/24,5/7) -> (6/7,7/8) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(801,-625,1024,-799) (7/9,25/32) -> (25/32,18/23) Parabolic Matrix(209,-169,256,-207) (4/5,13/16) -> (13/16,9/11) Parabolic Matrix(865,-729,1024,-863) (16/19,27/32) -> (27/32,11/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(0,-1,1,0) Matrix(47,-5,160,-17) -> Matrix(2,-3,-1,2) Matrix(65,-9,224,-31) -> Matrix(2,9,-1,-4) Matrix(95,-14,224,-33) -> Matrix(3,8,-2,-5) Matrix(79,-14,96,-17) -> Matrix(1,2,-2,-3) Matrix(17,-4,64,-15) -> Matrix(1,-2,0,1) Matrix(113,-31,288,-79) -> Matrix(2,5,-1,-2) Matrix(401,-112,512,-143) -> Matrix(1,2,0,1) Matrix(223,-66,544,-161) -> Matrix(5,8,-2,-3) Matrix(111,-34,160,-49) -> Matrix(3,4,-4,-5) Matrix(49,-18,128,-47) -> Matrix(1,-2,0,1) Matrix(417,-161,992,-383) -> Matrix(2,1,-1,0) Matrix(639,-248,992,-385) -> Matrix(1,2,0,1) Matrix(607,-248,864,-353) -> Matrix(1,4,-2,-7) Matrix(241,-100,576,-239) -> Matrix(3,8,-2,-5) Matrix(511,-216,608,-257) -> Matrix(1,2,-4,-7) Matrix(17,-8,32,-15) -> Matrix(1,0,0,1) Matrix(145,-81,256,-143) -> Matrix(6,7,-7,-8) Matrix(191,-110,224,-129) -> Matrix(3,2,-8,-5) Matrix(337,-196,576,-335) -> Matrix(3,2,-8,-5) Matrix(353,-207,544,-319) -> Matrix(2,1,-1,0) Matrix(383,-226,544,-321) -> Matrix(5,2,-8,-3) Matrix(81,-50,128,-79) -> Matrix(1,0,2,1) Matrix(177,-113,224,-143) -> Matrix(0,-1,1,0) Matrix(1489,-961,2304,-1487) -> Matrix(0,-1,1,2) Matrix(177,-121,256,-175) -> Matrix(4,5,-5,-6) Matrix(2881,-2025,4096,-2879) -> Matrix(8,5,-13,-8) Matrix(143,-101,160,-113) -> Matrix(2,1,3,2) Matrix(193,-137,224,-159) -> Matrix(2,1,-9,-4) Matrix(49,-36,64,-47) -> Matrix(1,0,2,1) Matrix(801,-625,1024,-799) -> Matrix(0,-1,1,0) Matrix(209,-169,256,-207) -> Matrix(2,3,-3,-4) Matrix(865,-729,1024,-863) -> Matrix(2,1,-5,-2) 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): 16 ----------------------------------------------------------------------- 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 16 1/8 1/0 6 2 1/7 -4/1 1 16 1/6 (-3/1,-1/1) 0 8 3/16 -1/1 3 1 1/5 0/1 1 16 1/4 1/0 2 4 3/11 -2/1 1 16 5/18 (-3/1,-1/1) 0 8 9/32 (-3/1,-1/1).(-2/1,1/0) 0 1 2/7 1/0 1 16 7/24 -2/1 6 2 5/17 -2/1 1 16 3/10 (-5/3,-1/1) 0 8 5/16 -1/1 5 1 1/3 0/1 1 16 3/8 1/0 2 2 5/13 -2/1 1 16 12/31 1/0 1 16 31/80 -3/1 1 1 7/18 (-3/1,-1/1) 0 8 2/5 1/0 1 16 13/32 (-3/1,-1/1).(-2/1,1/0) 0 1 9/22 (-3/1,-1/1) 0 8 7/17 -2/1 1 16 5/12 -2/1 2 4 13/31 -2/1 1 16 8/19 -3/2 1 16 27/64 (-2/1,-3/2).(-5/3,-1/1) 0 1 11/26 (-5/3,-1/1) 0 8 3/7 -4/3 1 16 7/16 -1/1 7 1 1/2 (-1/1,1/1) 0 8 1/0 (-1/1,1/1).(0/1,1/0) 0 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(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(65,-9,224,-31) (1/8,1/7) -> (2/7,7/24) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(17,-3,96,-17) (1/6,3/16) -> (1/6,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(113,-31,288,-79) (3/11,5/18) -> (7/18,2/5) Hyperbolic Matrix(161,-45,576,-161) (5/18,9/32) -> (5/18,9/32) Reflection Matrix(127,-36,448,-127) (9/32,2/7) -> (9/32,2/7) Reflection Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(417,-161,992,-383) (5/13,12/31) -> (13/31,8/19) Hyperbolic Matrix(1921,-744,4960,-1921) (12/31,31/80) -> (12/31,31/80) Reflection Matrix(559,-217,1440,-559) (31/80,7/18) -> (31/80,7/18) Reflection Matrix(129,-52,320,-129) (2/5,13/32) -> (2/5,13/32) Reflection Matrix(287,-117,704,-287) (13/32,9/22) -> (13/32,9/22) Reflection Matrix(241,-100,576,-239) (7/17,5/12) -> (5/12,13/31) Parabolic Matrix(1025,-432,2432,-1025) (8/19,27/64) -> (8/19,27/64) Reflection Matrix(703,-297,1664,-703) (27/64,11/26) -> (27/64,11/26) Reflection Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(15,-7,32,-15) (7/16,1/2) -> (7/16,1/2) 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(47,-5,160,-17) -> Matrix(2,-3,-1,2) Matrix(65,-9,224,-31) -> Matrix(2,9,-1,-4) -3/1 Matrix(95,-14,224,-33) -> Matrix(3,8,-2,-5) -2/1 Matrix(17,-3,96,-17) -> Matrix(2,3,-1,-2) (1/6,3/16) -> (-3/1,-1/1) Matrix(31,-6,160,-31) -> Matrix(-1,0,2,1) (3/16,1/5) -> (-1/1,0/1) Matrix(17,-4,64,-15) -> Matrix(1,-2,0,1) 1/0 Matrix(113,-31,288,-79) -> Matrix(2,5,-1,-2) (-3/1,-1/1).(-2/1,1/0) Matrix(161,-45,576,-161) -> Matrix(2,3,-1,-2) (5/18,9/32) -> (-3/1,-1/1) Matrix(127,-36,448,-127) -> Matrix(1,4,0,-1) (9/32,2/7) -> (-2/1,1/0) Matrix(223,-66,544,-161) -> Matrix(5,8,-2,-3) -2/1 Matrix(49,-15,160,-49) -> Matrix(4,5,-3,-4) (3/10,5/16) -> (-5/3,-1/1) Matrix(31,-10,96,-31) -> Matrix(-1,0,2,1) (5/16,1/3) -> (-1/1,0/1) Matrix(49,-18,128,-47) -> Matrix(1,-2,0,1) 1/0 Matrix(417,-161,992,-383) -> Matrix(2,1,-1,0) -1/1 Matrix(1921,-744,4960,-1921) -> Matrix(1,6,0,-1) (12/31,31/80) -> (-3/1,1/0) Matrix(559,-217,1440,-559) -> Matrix(2,3,-1,-2) (31/80,7/18) -> (-3/1,-1/1) Matrix(129,-52,320,-129) -> Matrix(1,4,0,-1) (2/5,13/32) -> (-2/1,1/0) Matrix(287,-117,704,-287) -> Matrix(2,3,-1,-2) (13/32,9/22) -> (-3/1,-1/1) Matrix(241,-100,576,-239) -> Matrix(3,8,-2,-5) -2/1 Matrix(1025,-432,2432,-1025) -> Matrix(7,12,-4,-7) (8/19,27/64) -> (-2/1,-3/2) Matrix(703,-297,1664,-703) -> Matrix(4,5,-3,-4) (27/64,11/26) -> (-5/3,-1/1) Matrix(97,-42,224,-97) -> Matrix(7,8,-6,-7) (3/7,7/16) -> (-4/3,-1/1) Matrix(15,-7,32,-15) -> Matrix(0,1,1,0) (7/16,1/2) -> (-1/1,1/1) Matrix(-1,1,0,1) -> Matrix(0,1,1,0) (1/2,1/0) -> (-1/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.