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): 1152 Minimal number of generators: 193 Number of equivalence classes of cusps: 72 Genus: 61 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/7 -5/7 -2/3 -4/7 -7/15 -13/28 -19/42 -37/84 -3/7 -5/12 -23/63 -5/14 -1/3 -13/42 -25/84 -2/7 -5/18 -4/15 -11/42 -2/9 -3/14 -11/63 -1/6 -3/19 -3/20 -1/7 -2/15 -1/9 -1/10 0/1 1/9 1/8 1/7 2/13 1/6 3/17 2/11 3/16 4/21 1/5 4/19 3/14 2/9 3/13 4/17 5/21 1/4 3/11 5/18 2/7 1/3 5/14 8/21 41/105 2/5 5/12 3/7 13/28 7/15 1/2 4/7 64/105 13/21 40/63 2/3 5/7 16/21 17/21 52/63 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -8/9 -1/2 1/0 -7/8 1/1 1/0 -6/7 0/1 -11/13 1/2 1/1 -5/6 1/2 1/0 -19/23 5/6 1/1 -14/17 4/3 3/2 -9/11 2/1 1/0 -13/16 6/1 1/0 -17/21 1/0 -4/5 -2/1 1/0 -15/19 -3/2 -1/1 -11/14 -1/1 -18/23 -1/1 -3/4 -7/9 -1/2 -17/22 -1/2 0/1 -10/13 -1/3 -1/4 -13/17 -1/8 0/1 -16/21 0/1 -3/4 0/1 1/2 -14/19 1/1 9/8 -11/15 3/2 -41/56 2/1 -30/41 3/2 2/1 -19/26 7/4 2/1 -8/11 2/1 1/0 -13/18 7/2 1/0 -5/7 1/0 -17/24 -9/2 1/0 -29/41 -4/1 1/0 -12/17 -4/1 -7/2 -43/61 -31/10 -3/1 -74/105 -3/1 -31/44 -3/1 -23/8 -19/27 -5/2 -26/37 -9/4 -2/1 -7/10 -2/1 1/0 -23/33 -3/2 -16/23 -1/1 1/0 -25/36 -3/2 1/0 -9/13 -3/2 -1/1 -11/16 0/1 1/0 -2/3 -1/2 1/0 -11/17 0/1 1/0 -9/14 0/1 -16/25 1/2 2/3 -7/11 0/1 1/0 -12/19 3/4 1/1 -17/27 3/2 -22/35 2/1 -5/8 3/1 1/0 -13/21 1/0 -8/13 -7/1 1/0 -11/18 -7/2 1/0 -25/41 -4/1 1/0 -64/105 -4/1 -39/64 -4/1 -7/2 -14/23 -3/1 1/0 -3/5 -2/1 1/0 -16/27 -7/4 -3/2 -13/22 -2/1 -3/2 -10/17 -8/5 -3/2 -17/29 -3/2 -7/5 -41/70 -4/3 -24/41 -3/2 -4/3 -7/12 -3/2 -5/4 -11/19 -3/2 -1/1 -4/7 -1/1 -13/23 -1/1 -5/6 -22/39 -5/6 -3/4 -9/16 -4/5 -3/4 -23/41 -3/4 -2/3 -14/25 -7/10 -2/3 -19/34 -5/8 -3/5 -5/9 -1/2 -11/20 -1/2 0/1 -6/11 0/1 1/0 -7/13 -1/1 -1/2 -15/28 0/1 -8/15 -1/2 1/0 -1/2 -1/1 1/0 -7/15 -5/6 -13/28 -4/5 -6/13 -7/9 -3/4 -5/11 -3/4 -2/3 -19/42 -2/3 -14/31 -2/3 -5/8 -9/20 -2/3 -1/2 -4/9 -3/4 -1/2 -15/34 -3/4 -5/7 -26/59 -15/22 -2/3 -37/84 -2/3 -11/25 -2/3 -5/8 -18/41 -2/3 -1/2 -7/16 -2/3 -5/8 -3/7 -1/2 -11/26 -2/5 -3/8 -8/19 -3/8 -1/3 -13/31 -1/4 0/1 -44/105 0/1 -31/74 0/1 1/0 -18/43 -1/2 0/1 -5/12 -1/2 -1/4 -17/41 -1/4 0/1 -29/70 0/1 -12/29 -1/4 -1/5 -7/17 0/1 1/0 -16/39 -1/2 1/0 -25/61 -1/1 -1/2 -9/22 -1/2 0/1 -11/27 -1/2 -2/5 0/1 1/0 -9/23 -1/1 -1/2 -25/64 -2/3 -1/2 -41/105 -1/2 -16/41 -1/2 0/1 -7/18 -1/2 1/0 -12/31 -2/1 1/0 -5/13 -3/2 -1/1 -8/21 -1/1 -3/8 -1/1 -3/4 -10/27 -1/2 1/0 -7/19 -1/1 -5/6 -11/30 -5/6 -3/4 -15/41 -4/5 -3/4 -19/52 -10/13 -3/4 -23/63 -3/4 -4/11 -3/4 -2/3 -9/25 -3/4 -2/3 -5/14 -2/3 -6/17 -4/7 -1/2 -7/20 -2/3 -1/2 -1/3 -1/2 -7/22 -1/2 -2/5 -6/19 -3/8 -1/3 -5/16 -2/5 -3/8 -9/29 -5/14 -1/3 -13/42 -1/3 -4/13 -1/3 -1/4 -7/23 -1/3 -3/10 -3/10 -1/4 0/1 -14/47 -1/8 0/1 -25/84 0/1 -11/37 0/1 1/0 -8/27 -1/2 -1/4 -13/44 -1/3 -1/4 -5/17 -1/4 0/1 -2/7 0/1 -7/25 0/1 1/0 -12/43 0/1 1/2 -5/18 1/2 1/0 -8/29 -3/1 1/0 -3/11 0/1 1/0 -7/26 -4/1 1/0 -11/41 -2/1 1/0 -15/56 -2/1 -4/15 -3/2 1/0 -5/19 -7/6 -1/1 -11/42 -1/1 -6/23 -1/1 -11/12 -1/4 -2/3 -1/2 -5/21 -1/2 -4/17 -1/2 -8/17 -3/13 -1/2 -3/7 -8/35 -2/5 -13/57 -1/2 -5/22 -1/2 -2/5 -2/9 -1/2 -3/8 -9/41 -2/5 -3/8 -7/32 -3/8 -4/11 -5/23 -1/2 -1/3 -3/14 -1/3 -4/19 -1/3 -1/4 -1/5 -1/4 0/1 -4/21 0/1 -3/16 0/1 1/4 -2/11 0/1 1/0 -3/17 0/1 1/0 -7/40 1/1 1/0 -11/63 1/0 -4/23 -1/1 1/0 -1/6 -1/2 1/0 -3/19 -1/1 -5/6 -2/13 -1/1 -3/4 -3/20 -2/3 -1/2 -1/7 -1/2 -3/22 -1/2 -2/5 -2/15 -1/2 -3/8 -1/8 -1/3 -1/4 -1/9 -1/2 -1/10 -1/4 0/1 0/1 -1/2 0/1 1/9 -1/2 1/8 -1/1 1/0 1/7 -1/2 2/13 -3/8 -1/3 1/6 -1/2 -1/4 4/23 -1/3 -1/4 3/17 -1/4 0/1 2/11 -1/4 0/1 3/16 -1/8 0/1 4/21 0/1 1/5 0/1 1/0 4/19 -1/1 1/0 3/14 -1/1 5/23 -1/1 -1/2 2/9 -3/4 -1/2 5/22 -2/3 -1/2 3/13 -3/5 -1/2 4/17 -8/15 -1/2 5/21 -1/2 1/4 -1/2 -2/5 5/19 -1/3 -7/22 4/15 -3/10 -1/4 15/56 -2/7 11/41 -2/7 -1/4 7/26 -4/15 -1/4 3/11 -1/4 0/1 5/18 -1/4 -1/6 2/7 0/1 7/24 -1/2 1/0 12/41 -1/2 0/1 5/17 0/1 1/0 18/61 1/1 1/0 31/105 1/0 13/44 -1/1 1/0 8/27 -1/2 1/0 11/37 -1/4 0/1 3/10 0/1 1/0 10/33 -3/2 1/0 7/23 -3/2 -1/1 11/36 -3/4 -1/2 4/13 -1/1 1/0 5/16 -3/4 -2/3 1/3 -1/2 6/17 -1/2 -4/9 5/14 -2/5 9/25 -2/5 -3/8 4/11 -2/5 -3/8 7/19 -5/14 -1/3 10/27 -1/2 -1/4 13/35 -1/2 3/8 -3/8 -1/3 8/21 -1/3 5/13 -1/3 -3/10 7/18 -1/2 -1/4 16/41 -1/2 0/1 41/105 -1/2 25/64 -1/2 -2/5 9/23 -1/2 -1/3 2/5 -1/4 0/1 11/27 -1/2 9/22 -1/2 0/1 7/17 -1/4 0/1 12/29 1/1 1/0 29/70 0/1 17/41 0/1 1/0 5/12 -1/2 1/0 8/19 -1/1 -3/4 3/7 -1/2 10/23 -3/7 -5/12 17/39 -1/2 7/16 -5/12 -2/5 18/41 -1/2 -2/5 11/25 -5/12 -2/5 15/34 -5/13 -3/8 4/9 -1/2 -3/8 9/20 -1/2 -2/5 5/11 -2/5 -3/8 6/13 -3/8 -7/19 13/28 -4/11 7/15 -5/14 1/2 -1/3 -1/4 8/15 -1/2 -1/4 15/28 0/1 7/13 -1/2 -1/3 6/11 -1/4 0/1 23/42 0/1 17/31 0/1 1/0 11/20 -1/2 0/1 5/9 -1/2 19/34 -3/7 -5/12 33/59 -13/32 -2/5 47/84 -2/5 14/25 -2/5 -7/18 23/41 -2/5 -3/8 9/16 -3/8 -4/11 4/7 -1/3 15/26 -5/16 -4/13 11/19 -1/3 -3/10 18/31 -6/19 -5/16 61/105 -5/16 43/74 -5/16 -14/45 25/43 -5/16 -4/13 7/12 -5/16 -3/10 24/41 -4/13 -3/10 41/70 -4/13 17/29 -7/23 -3/10 10/17 -3/10 -8/27 23/39 -3/10 36/61 -5/17 -7/24 13/22 -3/10 -2/7 16/27 -3/10 -7/24 3/5 -2/7 -1/4 14/23 -3/11 -1/4 39/64 -7/26 -4/15 64/105 -4/15 25/41 -4/15 -1/4 11/18 -7/26 -1/4 19/31 -4/15 -1/4 8/13 -7/27 -1/4 13/21 -1/4 5/8 -1/4 -3/13 17/27 -3/14 12/19 -1/5 -3/16 19/30 -1/4 -1/6 26/41 -1/6 0/1 33/52 -1/8 0/1 40/63 0/1 7/11 -1/4 0/1 16/25 -2/11 -1/6 9/14 0/1 11/17 -1/4 0/1 13/20 -1/2 0/1 2/3 -1/2 -1/4 15/22 -1/2 0/1 13/19 -1/2 -1/3 11/16 -1/4 0/1 20/29 -3/8 -1/3 29/42 -1/3 9/13 -1/3 -3/10 16/23 -1/3 -1/4 7/10 -2/7 -1/4 33/47 -7/24 -2/7 59/84 -2/7 26/37 -2/7 -9/32 19/27 -5/18 31/44 -23/84 -3/11 12/17 -7/26 -4/15 5/7 -1/4 18/25 -9/38 -4/17 31/43 -1/4 -4/17 13/18 -1/4 -7/30 21/29 -3/13 -5/22 8/11 -1/4 -2/9 19/26 -2/9 -7/32 30/41 -2/9 -3/14 41/56 -2/9 11/15 -3/14 14/19 -9/44 -1/5 31/42 -1/5 17/23 -1/5 -9/46 3/4 -1/6 0/1 16/21 0/1 13/17 0/1 1/4 10/13 -1/1 1/0 27/35 -1/2 44/57 -1/2 -1/4 17/22 -1/2 0/1 7/9 -1/2 32/41 -1/2 -2/5 25/32 -2/5 -3/8 18/23 -3/8 -1/3 11/14 -1/3 15/19 -1/3 -3/10 4/5 -2/7 -1/4 17/21 -1/4 13/16 -1/4 -6/25 9/11 -1/4 -2/9 14/17 -3/14 -4/19 33/40 -13/64 -1/5 52/63 -1/5 19/23 -1/5 -5/26 5/6 -1/4 -1/6 16/19 -1/4 -1/5 11/13 -1/5 -1/6 17/20 -1/6 0/1 6/7 0/1 19/22 -1/2 0/1 13/15 -1/2 7/8 -1/4 -1/5 8/9 -1/2 -1/4 9/10 -2/7 -1/4 1/1 -1/4 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(83,74,-378,-337) (-1/1,-8/9) -> (-2/9,-9/41) Hyperbolic Matrix(167,148,378,335) (-8/9,-7/8) -> (15/34,4/9) Hyperbolic Matrix(211,184,-336,-293) (-7/8,-6/7) -> (-22/35,-5/8) Hyperbolic Matrix(125,106,-546,-463) (-6/7,-11/13) -> (-3/13,-8/35) Hyperbolic Matrix(379,320,-546,-461) (-11/13,-5/6) -> (-25/36,-9/13) Hyperbolic Matrix(295,244,966,799) (-5/6,-19/23) -> (7/23,11/36) Hyperbolic Matrix(1007,830,-1428,-1177) (-19/23,-14/17) -> (-12/17,-43/61) Hyperbolic Matrix(295,242,462,379) (-14/17,-9/11) -> (7/11,16/25) Hyperbolic Matrix(125,102,462,377) (-9/11,-13/16) -> (7/26,3/11) Hyperbolic Matrix(545,442,672,545) (-13/16,-17/21) -> (17/21,13/16) Hyperbolic Matrix(169,136,210,169) (-17/21,-4/5) -> (4/5,17/21) Hyperbolic Matrix(83,66,210,167) (-4/5,-15/19) -> (9/23,2/5) Hyperbolic Matrix(127,100,588,463) (-15/19,-11/14) -> (3/14,5/23) Hyperbolic Matrix(125,98,588,461) (-11/14,-18/23) -> (4/19,3/14) Hyperbolic Matrix(673,526,-966,-755) (-18/23,-7/9) -> (-23/33,-16/23) Hyperbolic Matrix(209,162,378,293) (-7/9,-17/22) -> (11/20,5/9) Hyperbolic Matrix(83,64,-546,-421) (-17/22,-10/13) -> (-2/13,-3/20) Hyperbolic Matrix(295,226,714,547) (-10/13,-13/17) -> (7/17,12/29) Hyperbolic Matrix(545,416,714,545) (-13/17,-16/21) -> (16/21,13/17) Hyperbolic Matrix(127,96,168,127) (-16/21,-3/4) -> (3/4,16/21) Hyperbolic Matrix(43,32,-168,-125) (-3/4,-14/19) -> (-6/23,-1/4) Hyperbolic Matrix(503,370,798,587) (-14/19,-11/15) -> (17/27,12/19) Hyperbolic Matrix(587,430,1260,923) (-11/15,-41/56) -> (13/28,7/15) Hyperbolic Matrix(3025,2214,5166,3781) (-41/56,-30/41) -> (24/41,41/70) Hyperbolic Matrix(755,552,1722,1259) (-30/41,-19/26) -> (7/16,18/41) Hyperbolic Matrix(85,62,462,337) (-19/26,-8/11) -> (2/11,3/16) Hyperbolic Matrix(293,212,-756,-547) (-8/11,-13/18) -> (-7/18,-12/31) Hyperbolic Matrix(209,150,-294,-211) (-13/18,-5/7) -> (-5/7,-17/24) Parabolic Matrix(1051,744,1722,1219) (-17/24,-29/41) -> (25/41,11/18) Hyperbolic Matrix(965,682,1722,1217) (-29/41,-12/17) -> (14/25,23/41) Hyperbolic Matrix(5167,3642,6258,4411) (-43/61,-74/105) -> (52/63,19/23) Hyperbolic Matrix(5753,4054,6972,4913) (-74/105,-31/44) -> (33/40,52/63) Hyperbolic Matrix(335,236,714,503) (-31/44,-19/27) -> (7/15,1/2) Hyperbolic Matrix(461,324,1134,797) (-19/27,-26/37) -> (2/5,11/27) Hyperbolic Matrix(251,176,-840,-589) (-26/37,-7/10) -> (-3/10,-14/47) Hyperbolic Matrix(43,30,-420,-293) (-7/10,-23/33) -> (-1/9,-1/10) Hyperbolic Matrix(167,116,966,671) (-16/23,-25/36) -> (1/6,4/23) Hyperbolic Matrix(209,144,-672,-463) (-9/13,-11/16) -> (-5/16,-9/29) Hyperbolic Matrix(379,260,-672,-461) (-11/16,-2/3) -> (-22/39,-9/16) Hyperbolic Matrix(293,190,-714,-463) (-2/3,-11/17) -> (-7/17,-16/39) Hyperbolic Matrix(211,136,588,379) (-11/17,-9/14) -> (5/14,9/25) Hyperbolic Matrix(209,134,588,377) (-9/14,-16/25) -> (6/17,5/14) Hyperbolic Matrix(379,242,462,295) (-16/25,-7/11) -> (9/11,14/17) Hyperbolic Matrix(335,212,-798,-505) (-7/11,-12/19) -> (-8/19,-13/31) Hyperbolic Matrix(587,370,798,503) (-12/19,-17/27) -> (11/15,14/19) Hyperbolic Matrix(671,422,-2940,-1849) (-17/27,-22/35) -> (-8/35,-13/57) Hyperbolic Matrix(209,130,336,209) (-5/8,-13/21) -> (13/21,5/8) Hyperbolic Matrix(337,208,546,337) (-13/21,-8/13) -> (8/13,13/21) Hyperbolic Matrix(209,128,-756,-463) (-8/13,-11/18) -> (-5/18,-8/29) Hyperbolic Matrix(1091,666,1512,923) (-11/18,-25/41) -> (31/43,13/18) Hyperbolic Matrix(5249,3200,8610,5249) (-25/41,-64/105) -> (64/105,25/41) Hyperbolic Matrix(8191,4992,13440,8191) (-64/105,-39/64) -> (39/64,64/105) Hyperbolic Matrix(2603,1586,4410,2687) (-39/64,-14/23) -> (36/61,13/22) Hyperbolic Matrix(43,26,210,127) (-14/23,-3/5) -> (1/5,4/19) Hyperbolic Matrix(337,200,1134,673) (-3/5,-16/27) -> (8/27,11/37) Hyperbolic Matrix(125,74,-924,-547) (-16/27,-13/22) -> (-3/22,-2/15) Hyperbolic Matrix(251,148,-714,-421) (-13/22,-10/17) -> (-6/17,-7/20) Hyperbolic Matrix(167,98,714,419) (-10/17,-17/29) -> (3/13,4/17) Hyperbolic Matrix(925,542,1722,1009) (-17/29,-41/70) -> (15/28,7/13) Hyperbolic Matrix(3781,2214,5166,3025) (-41/70,-24/41) -> (30/41,41/56) Hyperbolic Matrix(1091,638,1722,1007) (-24/41,-7/12) -> (19/30,26/41) Hyperbolic Matrix(293,170,-798,-463) (-7/12,-11/19) -> (-7/19,-11/30) Hyperbolic Matrix(167,96,-294,-169) (-11/19,-4/7) -> (-4/7,-13/23) Parabolic Matrix(1343,758,-3276,-1849) (-13/23,-22/39) -> (-16/39,-25/61) Hyperbolic Matrix(463,260,1722,967) (-9/16,-23/41) -> (11/41,7/26) Hyperbolic Matrix(1513,848,2100,1177) (-23/41,-14/25) -> (18/25,31/43) Hyperbolic Matrix(1259,704,-2856,-1597) (-14/25,-19/34) -> (-15/34,-26/59) Hyperbolic Matrix(43,24,378,211) (-19/34,-5/9) -> (1/9,1/8) Hyperbolic Matrix(293,162,378,209) (-5/9,-11/20) -> (17/22,7/9) Hyperbolic Matrix(379,208,-840,-461) (-11/20,-6/11) -> (-14/31,-9/20) Hyperbolic Matrix(335,182,462,251) (-6/11,-7/13) -> (21/29,8/11) Hyperbolic Matrix(1009,542,1722,925) (-7/13,-15/28) -> (41/70,17/29) Hyperbolic Matrix(337,180,1260,673) (-15/28,-8/15) -> (4/15,15/56) Hyperbolic Matrix(211,112,714,379) (-8/15,-1/2) -> (13/44,8/27) Hyperbolic Matrix(503,236,714,335) (-1/2,-7/15) -> (19/27,31/44) Hyperbolic Matrix(923,430,1260,587) (-7/15,-13/28) -> (41/56,11/15) Hyperbolic Matrix(713,330,1722,797) (-13/28,-6/13) -> (12/29,29/70) Hyperbolic Matrix(335,154,546,251) (-6/13,-5/11) -> (19/31,8/13) Hyperbolic Matrix(967,438,1764,799) (-5/11,-19/42) -> (23/42,17/31) Hyperbolic Matrix(965,436,1764,797) (-19/42,-14/31) -> (6/11,23/42) Hyperbolic Matrix(85,38,378,169) (-9/20,-4/9) -> (2/9,5/22) Hyperbolic Matrix(335,148,378,167) (-4/9,-15/34) -> (7/8,8/9) Hyperbolic Matrix(3949,1740,7056,3109) (-26/59,-37/84) -> (47/84,14/25) Hyperbolic Matrix(3947,1738,7056,3107) (-37/84,-11/25) -> (33/59,47/84) Hyperbolic Matrix(505,222,1722,757) (-11/25,-18/41) -> (12/41,5/17) Hyperbolic Matrix(1259,552,1722,755) (-18/41,-7/16) -> (19/26,30/41) Hyperbolic Matrix(125,54,-294,-127) (-7/16,-3/7) -> (-3/7,-11/26) Parabolic Matrix(251,106,-798,-337) (-11/26,-8/19) -> (-6/19,-5/16) Hyperbolic Matrix(1975,828,3108,1303) (-13/31,-44/105) -> (40/63,7/11) Hyperbolic Matrix(6425,2692,10122,4241) (-44/105,-31/74) -> (33/52,40/63) Hyperbolic Matrix(3443,1442,4410,1847) (-31/74,-18/43) -> (32/41,25/32) Hyperbolic Matrix(589,246,1008,421) (-18/43,-5/12) -> (7/12,24/41) Hyperbolic Matrix(587,244,1008,419) (-5/12,-17/41) -> (25/43,7/12) Hyperbolic Matrix(1385,574,5166,2141) (-17/41,-29/70) -> (15/56,11/41) Hyperbolic Matrix(797,330,1722,713) (-29/70,-12/29) -> (6/13,13/28) Hyperbolic Matrix(547,226,714,295) (-12/29,-7/17) -> (13/17,10/13) Hyperbolic Matrix(1723,706,4410,1807) (-25/61,-9/22) -> (25/64,9/23) Hyperbolic Matrix(421,172,-1848,-755) (-9/22,-11/27) -> (-13/57,-5/22) Hyperbolic Matrix(797,324,1134,461) (-11/27,-2/5) -> (26/37,19/27) Hyperbolic Matrix(167,66,210,83) (-2/5,-9/23) -> (15/19,4/5) Hyperbolic Matrix(1177,460,1722,673) (-9/23,-25/64) -> (15/22,13/19) Hyperbolic Matrix(5249,2050,13440,5249) (-25/64,-41/105) -> (41/105,25/64) Hyperbolic Matrix(3361,1312,8610,3361) (-41/105,-16/41) -> (16/41,41/105) Hyperbolic Matrix(503,196,1722,671) (-16/41,-7/18) -> (7/24,12/41) Hyperbolic Matrix(295,114,546,211) (-12/31,-5/13) -> (7/13,6/11) Hyperbolic Matrix(209,80,546,209) (-5/13,-8/21) -> (8/21,5/13) Hyperbolic Matrix(127,48,336,127) (-8/21,-3/8) -> (3/8,8/21) Hyperbolic Matrix(43,16,-336,-125) (-3/8,-10/27) -> (-2/15,-1/8) Hyperbolic Matrix(211,78,798,295) (-10/27,-7/19) -> (5/19,4/15) Hyperbolic Matrix(715,262,1722,631) (-11/30,-15/41) -> (17/41,5/12) Hyperbolic Matrix(421,154,462,169) (-15/41,-19/52) -> (9/10,1/1) Hyperbolic Matrix(5881,2148,10122,3697) (-19/52,-23/63) -> (61/105,43/74) Hyperbolic Matrix(1805,658,3108,1133) (-23/63,-4/11) -> (18/31,61/105) Hyperbolic Matrix(83,30,462,167) (-4/11,-9/25) -> (3/17,2/11) Hyperbolic Matrix(379,136,588,211) (-9/25,-5/14) -> (9/14,11/17) Hyperbolic Matrix(377,134,588,209) (-5/14,-6/17) -> (16/25,9/14) Hyperbolic Matrix(41,14,-126,-43) (-7/20,-1/3) -> (-1/3,-7/22) Parabolic Matrix(1049,332,1722,545) (-7/22,-6/19) -> (14/23,39/64) Hyperbolic Matrix(1219,378,1764,547) (-9/29,-13/42) -> (29/42,9/13) Hyperbolic Matrix(1217,376,1764,545) (-13/42,-4/13) -> (20/29,29/42) Hyperbolic Matrix(85,26,-546,-167) (-4/13,-7/23) -> (-3/19,-2/13) Hyperbolic Matrix(211,64,-966,-293) (-7/23,-3/10) -> (-7/32,-5/23) Hyperbolic Matrix(4957,1476,7056,2101) (-14/47,-25/84) -> (59/84,26/37) Hyperbolic Matrix(4955,1474,7056,2099) (-25/84,-11/37) -> (33/47,59/84) Hyperbolic Matrix(673,200,1134,337) (-11/37,-8/27) -> (16/27,3/5) Hyperbolic Matrix(379,112,714,211) (-8/27,-13/44) -> (1/2,8/15) Hyperbolic Matrix(251,74,-1428,-421) (-13/44,-5/17) -> (-3/17,-7/40) Hyperbolic Matrix(83,24,-294,-85) (-5/17,-2/7) -> (-2/7,-7/25) Parabolic Matrix(923,258,2100,587) (-7/25,-12/43) -> (18/41,11/25) Hyperbolic Matrix(589,164,1512,421) (-12/43,-5/18) -> (7/18,16/41) Hyperbolic Matrix(211,58,462,127) (-8/29,-3/11) -> (5/11,6/13) Hyperbolic Matrix(377,102,462,125) (-3/11,-7/26) -> (13/16,9/11) Hyperbolic Matrix(967,260,1722,463) (-7/26,-11/41) -> (23/41,9/16) Hyperbolic Matrix(2141,574,5166,1385) (-11/41,-15/56) -> (29/70,17/41) Hyperbolic Matrix(673,180,1260,337) (-15/56,-4/15) -> (8/15,15/28) Hyperbolic Matrix(295,78,798,211) (-4/15,-5/19) -> (7/19,10/27) Hyperbolic Matrix(1303,342,1764,463) (-5/19,-11/42) -> (31/42,17/23) Hyperbolic Matrix(1301,340,1764,461) (-11/42,-6/23) -> (14/19,31/42) Hyperbolic Matrix(41,10,168,41) (-1/4,-5/21) -> (5/21,1/4) Hyperbolic Matrix(169,40,714,169) (-5/21,-4/17) -> (4/17,5/21) Hyperbolic Matrix(419,98,714,167) (-4/17,-3/13) -> (17/29,10/17) Hyperbolic Matrix(169,38,378,85) (-5/22,-2/9) -> (4/9,9/20) Hyperbolic Matrix(2563,562,4410,967) (-9/41,-7/32) -> (43/74,25/43) Hyperbolic Matrix(463,100,588,127) (-5/23,-3/14) -> (11/14,15/19) Hyperbolic Matrix(461,98,588,125) (-3/14,-4/19) -> (18/23,11/14) Hyperbolic Matrix(127,26,210,43) (-4/19,-1/5) -> (3/5,14/23) Hyperbolic Matrix(41,8,210,41) (-1/5,-4/21) -> (4/21,1/5) Hyperbolic Matrix(127,24,672,127) (-4/21,-3/16) -> (3/16,4/21) Hyperbolic Matrix(337,62,462,85) (-3/16,-2/11) -> (8/11,19/26) Hyperbolic Matrix(167,30,462,83) (-2/11,-3/17) -> (9/25,4/11) Hyperbolic Matrix(2059,360,6972,1219) (-7/40,-11/63) -> (31/105,13/44) Hyperbolic Matrix(1847,322,6258,1091) (-11/63,-4/23) -> (18/61,31/105) Hyperbolic Matrix(211,36,252,43) (-4/23,-1/6) -> (5/6,16/19) Hyperbolic Matrix(209,34,252,41) (-1/6,-3/19) -> (19/23,5/6) Hyperbolic Matrix(41,6,-294,-43) (-3/20,-1/7) -> (-1/7,-3/22) Parabolic Matrix(211,24,378,43) (-1/8,-1/9) -> (5/9,19/34) Hyperbolic Matrix(293,26,462,41) (-1/10,0/1) -> (26/41,33/52) Hyperbolic Matrix(295,-32,378,-41) (0/1,1/9) -> (7/9,32/41) Hyperbolic Matrix(125,-16,336,-43) (1/8,1/7) -> (13/35,3/8) Hyperbolic Matrix(421,-64,546,-83) (1/7,2/13) -> (10/13,27/35) Hyperbolic Matrix(167,-26,546,-85) (2/13,1/6) -> (11/36,4/13) Hyperbolic Matrix(421,-74,1428,-251) (4/23,3/17) -> (5/17,18/61) Hyperbolic Matrix(293,-64,966,-211) (5/23,2/9) -> (10/33,7/23) Hyperbolic Matrix(463,-106,546,-125) (5/22,3/13) -> (11/13,17/20) Hyperbolic Matrix(125,-32,168,-43) (1/4,5/19) -> (17/23,3/4) Hyperbolic Matrix(463,-128,756,-209) (3/11,5/18) -> (11/18,19/31) Hyperbolic Matrix(85,-24,294,-83) (5/18,2/7) -> (2/7,7/24) Parabolic Matrix(589,-176,840,-251) (11/37,3/10) -> (7/10,33/47) Hyperbolic Matrix(377,-114,420,-127) (3/10,10/33) -> (8/9,9/10) Hyperbolic Matrix(463,-144,672,-209) (4/13,5/16) -> (11/16,20/29) Hyperbolic Matrix(293,-92,672,-211) (5/16,1/3) -> (17/39,7/16) Hyperbolic Matrix(421,-148,714,-251) (1/3,6/17) -> (10/17,23/39) Hyperbolic Matrix(463,-170,798,-293) (4/11,7/19) -> (11/19,18/31) Hyperbolic Matrix(2269,-842,2940,-1091) (10/27,13/35) -> (27/35,44/57) Hyperbolic Matrix(547,-212,756,-293) (5/13,7/18) -> (13/18,21/29) Hyperbolic Matrix(799,-326,924,-377) (11/27,9/22) -> (19/22,13/15) Hyperbolic Matrix(463,-190,714,-293) (9/22,7/17) -> (11/17,13/20) Hyperbolic Matrix(505,-212,798,-335) (5/12,8/19) -> (12/19,19/30) Hyperbolic Matrix(127,-54,294,-125) (8/19,3/7) -> (3/7,10/23) Parabolic Matrix(1933,-842,3276,-1427) (10/23,17/39) -> (23/39,36/61) Hyperbolic Matrix(1597,-704,2856,-1259) (11/25,15/34) -> (19/34,33/59) Hyperbolic Matrix(461,-208,840,-379) (9/20,5/11) -> (17/31,11/20) Hyperbolic Matrix(169,-96,294,-167) (9/16,4/7) -> (4/7,15/26) Parabolic Matrix(547,-316,798,-461) (15/26,11/19) -> (13/19,11/16) Hyperbolic Matrix(1427,-844,1848,-1093) (13/22,16/27) -> (44/57,17/22) Hyperbolic Matrix(293,-184,336,-211) (5/8,17/27) -> (13/15,7/8) Hyperbolic Matrix(85,-56,126,-83) (13/20,2/3) -> (2/3,15/22) Parabolic Matrix(461,-320,546,-379) (9/13,16/23) -> (16/19,11/13) Hyperbolic Matrix(755,-526,966,-673) (16/23,7/10) -> (25/32,18/23) Hyperbolic Matrix(1177,-830,1428,-1007) (31/44,12/17) -> (14/17,33/40) Hyperbolic Matrix(211,-150,294,-209) (12/17,5/7) -> (5/7,18/25) Parabolic Matrix(253,-216,294,-251) (17/20,6/7) -> (6/7,19/22) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(83,74,-378,-337) -> Matrix(3,2,-8,-5) Matrix(167,148,378,335) -> Matrix(3,2,-8,-5) Matrix(211,184,-336,-293) -> Matrix(1,2,0,1) Matrix(125,106,-546,-463) -> Matrix(5,-2,-12,5) Matrix(379,320,-546,-461) -> Matrix(1,-2,0,1) Matrix(295,244,966,799) -> Matrix(3,-2,-4,3) Matrix(1007,830,-1428,-1177) -> Matrix(13,-16,-4,5) Matrix(295,242,462,379) -> Matrix(1,-2,-4,9) Matrix(125,102,462,377) -> Matrix(1,-2,-4,9) Matrix(545,442,672,545) -> Matrix(1,-12,-4,49) Matrix(169,136,210,169) -> Matrix(1,4,-4,-15) Matrix(83,66,210,167) -> Matrix(1,2,-4,-7) Matrix(127,100,588,463) -> Matrix(3,4,-4,-5) Matrix(125,98,588,461) -> Matrix(5,4,-4,-3) Matrix(673,526,-966,-755) -> Matrix(5,4,-4,-3) Matrix(209,162,378,293) -> Matrix(1,0,0,1) Matrix(83,64,-546,-421) -> Matrix(5,2,-8,-3) Matrix(295,226,714,547) -> Matrix(1,0,4,1) Matrix(545,416,714,545) -> Matrix(1,0,12,1) Matrix(127,96,168,127) -> Matrix(1,0,-8,1) Matrix(43,32,-168,-125) -> Matrix(3,-2,-4,3) Matrix(503,370,798,587) -> Matrix(5,-6,-24,29) Matrix(587,430,1260,923) -> Matrix(13,-22,-36,61) Matrix(3025,2214,5166,3781) -> Matrix(11,-18,-36,59) Matrix(755,552,1722,1259) -> Matrix(3,-4,-8,11) Matrix(85,62,462,337) -> Matrix(1,-2,-4,9) Matrix(293,212,-756,-547) -> Matrix(1,-4,0,1) Matrix(209,150,-294,-211) -> Matrix(1,-8,0,1) Matrix(1051,744,1722,1219) -> Matrix(1,8,-4,-31) Matrix(965,682,1722,1217) -> Matrix(3,14,-8,-37) Matrix(5167,3642,6258,4411) -> Matrix(15,46,-76,-233) Matrix(5753,4054,6972,4913) -> Matrix(21,62,-104,-307) Matrix(335,236,714,503) -> Matrix(7,20,-20,-57) Matrix(461,324,1134,797) -> Matrix(1,2,0,1) Matrix(251,176,-840,-589) -> Matrix(1,2,-4,-7) Matrix(43,30,-420,-293) -> Matrix(1,2,-4,-7) Matrix(167,116,966,671) -> Matrix(1,2,-4,-7) Matrix(209,144,-672,-463) -> Matrix(3,2,-8,-5) Matrix(379,260,-672,-461) -> Matrix(3,4,-4,-5) Matrix(293,190,-714,-463) -> Matrix(1,0,0,1) Matrix(211,136,588,379) -> Matrix(3,2,-8,-5) Matrix(209,134,588,377) -> Matrix(5,-2,-12,5) Matrix(379,242,462,295) -> Matrix(1,-2,-4,9) Matrix(335,212,-798,-505) -> Matrix(1,0,-4,1) Matrix(587,370,798,503) -> Matrix(5,-6,-24,29) Matrix(671,422,-2940,-1849) -> Matrix(3,-4,-8,11) Matrix(209,130,336,209) -> Matrix(1,-6,-4,25) Matrix(337,208,546,337) -> Matrix(1,14,-4,-55) Matrix(209,128,-756,-463) -> Matrix(1,4,0,1) Matrix(1091,666,1512,923) -> Matrix(1,0,-4,1) Matrix(5249,3200,8610,5249) -> Matrix(1,8,-4,-31) Matrix(8191,4992,13440,8191) -> Matrix(15,56,-56,-209) Matrix(2603,1586,4410,2687) -> Matrix(7,26,-24,-89) Matrix(43,26,210,127) -> Matrix(1,2,0,1) Matrix(337,200,1134,673) -> Matrix(1,2,-4,-7) Matrix(125,74,-924,-547) -> Matrix(5,8,-12,-19) Matrix(251,148,-714,-421) -> Matrix(3,4,-4,-5) Matrix(167,98,714,419) -> Matrix(11,16,-20,-29) Matrix(925,542,1722,1009) -> Matrix(3,4,-4,-5) Matrix(3781,2214,5166,3025) -> Matrix(13,18,-60,-83) Matrix(1091,638,1722,1007) -> Matrix(3,4,-16,-21) Matrix(293,170,-798,-463) -> Matrix(7,8,-8,-9) Matrix(167,96,-294,-169) -> Matrix(7,8,-8,-9) Matrix(1343,758,-3276,-1849) -> Matrix(5,4,-4,-3) Matrix(463,260,1722,967) -> Matrix(11,8,-40,-29) Matrix(1513,848,2100,1177) -> Matrix(13,10,-56,-43) Matrix(1259,704,-2856,-1597) -> Matrix(25,16,-36,-23) Matrix(43,24,378,211) -> Matrix(3,2,-8,-5) Matrix(293,162,378,209) -> Matrix(1,0,0,1) Matrix(379,208,-840,-461) -> Matrix(5,2,-8,-3) Matrix(335,182,462,251) -> Matrix(1,-2,-4,9) Matrix(1009,542,1722,925) -> Matrix(11,4,-36,-13) Matrix(337,180,1260,673) -> Matrix(1,2,-4,-7) Matrix(211,112,714,379) -> Matrix(1,0,0,1) Matrix(503,236,714,335) -> Matrix(23,20,-84,-73) Matrix(923,430,1260,587) -> Matrix(27,22,-124,-101) Matrix(713,330,1722,797) -> Matrix(5,4,-4,-3) Matrix(335,154,546,251) -> Matrix(19,14,-72,-53) Matrix(967,438,1764,799) -> Matrix(3,2,4,3) Matrix(965,436,1764,797) -> Matrix(3,2,-20,-13) Matrix(85,38,378,169) -> Matrix(1,0,0,1) Matrix(335,148,378,167) -> Matrix(3,2,-8,-5) Matrix(3949,1740,7056,3109) -> Matrix(65,44,-164,-111) Matrix(3947,1738,7056,3107) -> Matrix(55,36,-136,-89) Matrix(505,222,1722,757) -> Matrix(3,2,-8,-5) Matrix(1259,552,1722,755) -> Matrix(5,4,-24,-19) Matrix(125,54,-294,-127) -> Matrix(7,4,-16,-9) Matrix(251,106,-798,-337) -> Matrix(1,0,0,1) Matrix(1975,828,3108,1303) -> Matrix(1,0,0,1) Matrix(6425,2692,10122,4241) -> Matrix(1,0,-8,1) Matrix(3443,1442,4410,1847) -> Matrix(3,2,-8,-5) Matrix(589,246,1008,421) -> Matrix(11,4,-36,-13) Matrix(587,244,1008,419) -> Matrix(11,4,-36,-13) Matrix(1385,574,5166,2141) -> Matrix(9,2,-32,-7) Matrix(797,330,1722,713) -> Matrix(13,4,-36,-11) Matrix(547,226,714,295) -> Matrix(1,0,4,1) Matrix(1723,706,4410,1807) -> Matrix(3,2,-8,-5) Matrix(421,172,-1848,-755) -> Matrix(3,2,-8,-5) Matrix(797,324,1134,461) -> Matrix(9,2,-32,-7) Matrix(167,66,210,83) -> Matrix(1,2,-4,-7) Matrix(1177,460,1722,673) -> Matrix(3,2,-8,-5) Matrix(5249,2050,13440,5249) -> Matrix(7,4,-16,-9) Matrix(3361,1312,8610,3361) -> Matrix(1,0,0,1) Matrix(503,196,1722,671) -> Matrix(1,0,0,1) Matrix(295,114,546,211) -> Matrix(1,2,-4,-7) Matrix(209,80,546,209) -> Matrix(5,6,-16,-19) Matrix(127,48,336,127) -> Matrix(7,6,-20,-17) Matrix(43,16,-336,-125) -> Matrix(3,2,-8,-5) Matrix(211,78,798,295) -> Matrix(1,2,-4,-7) Matrix(715,262,1722,631) -> Matrix(5,4,-4,-3) Matrix(421,154,462,169) -> Matrix(5,4,-24,-19) Matrix(5881,2148,10122,3697) -> Matrix(121,92,-388,-295) Matrix(1805,658,3108,1133) -> Matrix(39,28,-124,-89) Matrix(83,30,462,167) -> Matrix(3,2,-8,-5) Matrix(379,136,588,211) -> Matrix(3,2,-8,-5) Matrix(377,134,588,209) -> Matrix(3,2,-20,-13) Matrix(41,14,-126,-43) -> Matrix(7,4,-16,-9) Matrix(1049,332,1722,545) -> Matrix(27,10,-100,-37) Matrix(1219,378,1764,547) -> Matrix(23,8,-72,-25) Matrix(1217,376,1764,545) -> Matrix(13,4,-36,-11) Matrix(85,26,-546,-167) -> Matrix(5,2,-8,-3) Matrix(211,64,-966,-293) -> Matrix(13,4,-36,-11) Matrix(4957,1476,7056,2101) -> Matrix(25,2,-88,-7) Matrix(4955,1474,7056,2099) -> Matrix(7,-2,-24,7) Matrix(673,200,1134,337) -> Matrix(1,2,-4,-7) Matrix(379,112,714,211) -> Matrix(1,0,0,1) Matrix(251,74,-1428,-421) -> Matrix(1,0,4,1) Matrix(83,24,-294,-85) -> Matrix(1,0,4,1) Matrix(923,258,2100,587) -> Matrix(5,-2,-12,5) Matrix(589,164,1512,421) -> Matrix(1,0,-4,1) Matrix(211,58,462,127) -> Matrix(3,2,-8,-5) Matrix(377,102,462,125) -> Matrix(1,-2,-4,9) Matrix(967,260,1722,463) -> Matrix(3,8,-8,-21) Matrix(2141,574,5166,1385) -> Matrix(1,2,0,1) Matrix(673,180,1260,337) -> Matrix(1,2,-4,-7) Matrix(295,78,798,211) -> Matrix(1,2,-4,-7) Matrix(1303,342,1764,463) -> Matrix(15,16,-76,-81) Matrix(1301,340,1764,461) -> Matrix(21,20,-104,-99) Matrix(41,10,168,41) -> Matrix(7,4,-16,-9) Matrix(169,40,714,169) -> Matrix(33,16,-64,-31) Matrix(419,98,714,167) -> Matrix(35,16,-116,-53) Matrix(169,38,378,85) -> Matrix(1,0,0,1) Matrix(2563,562,4410,967) -> Matrix(57,22,-184,-71) Matrix(463,100,588,127) -> Matrix(11,4,-36,-13) Matrix(461,98,588,125) -> Matrix(13,4,-36,-11) Matrix(127,26,210,43) -> Matrix(9,2,-32,-7) Matrix(41,8,210,41) -> Matrix(1,0,4,1) Matrix(127,24,672,127) -> Matrix(1,0,-12,1) Matrix(337,62,462,85) -> Matrix(1,-2,-4,9) Matrix(167,30,462,83) -> Matrix(3,2,-8,-5) Matrix(2059,360,6972,1219) -> Matrix(1,-2,0,1) Matrix(1847,322,6258,1091) -> Matrix(1,2,0,1) Matrix(211,36,252,43) -> Matrix(1,0,-4,1) Matrix(209,34,252,41) -> Matrix(1,0,-4,1) Matrix(41,6,-294,-43) -> Matrix(7,4,-16,-9) Matrix(211,24,378,43) -> Matrix(3,2,-8,-5) Matrix(293,26,462,41) -> Matrix(1,0,-4,1) Matrix(295,-32,378,-41) -> Matrix(3,2,-8,-5) Matrix(125,-16,336,-43) -> Matrix(3,2,-8,-5) Matrix(421,-64,546,-83) -> Matrix(5,2,-8,-3) Matrix(167,-26,546,-85) -> Matrix(5,2,-8,-3) Matrix(421,-74,1428,-251) -> Matrix(1,0,4,1) Matrix(293,-64,966,-211) -> Matrix(5,4,-4,-3) Matrix(463,-106,546,-125) -> Matrix(3,2,-20,-13) Matrix(125,-32,168,-43) -> Matrix(5,2,-28,-11) Matrix(463,-128,756,-209) -> Matrix(17,4,-64,-15) Matrix(85,-24,294,-83) -> Matrix(1,0,4,1) Matrix(589,-176,840,-251) -> Matrix(1,2,-4,-7) Matrix(377,-114,420,-127) -> Matrix(1,2,-4,-7) Matrix(463,-144,672,-209) -> Matrix(3,2,-8,-5) Matrix(293,-92,672,-211) -> Matrix(7,4,-16,-9) Matrix(421,-148,714,-251) -> Matrix(11,4,-36,-13) Matrix(463,-170,798,-293) -> Matrix(23,8,-72,-25) Matrix(2269,-842,2940,-1091) -> Matrix(1,0,0,1) Matrix(547,-212,756,-293) -> Matrix(15,4,-64,-17) Matrix(799,-326,924,-377) -> Matrix(1,0,0,1) Matrix(463,-190,714,-293) -> Matrix(1,0,0,1) Matrix(505,-212,798,-335) -> Matrix(1,0,-4,1) Matrix(127,-54,294,-125) -> Matrix(7,4,-16,-9) Matrix(1933,-842,3276,-1427) -> Matrix(11,4,-36,-13) Matrix(1597,-704,2856,-1259) -> Matrix(41,16,-100,-39) Matrix(461,-208,840,-379) -> Matrix(5,2,-8,-3) Matrix(169,-96,294,-167) -> Matrix(23,8,-72,-25) Matrix(547,-316,798,-461) -> Matrix(13,4,-36,-11) Matrix(1427,-844,1848,-1093) -> Matrix(7,2,-4,-1) Matrix(293,-184,336,-211) -> Matrix(9,2,-32,-7) Matrix(85,-56,126,-83) -> Matrix(1,0,0,1) Matrix(461,-320,546,-379) -> Matrix(7,2,-32,-9) Matrix(755,-526,966,-673) -> Matrix(13,4,-36,-11) Matrix(1177,-830,1428,-1007) -> Matrix(59,16,-284,-77) Matrix(211,-150,294,-209) -> Matrix(31,8,-128,-33) Matrix(253,-216,294,-251) -> Matrix(1,0,4,1) 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: 192 ----------------------------------------------------------------------- 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): 576 Minimal number of generators: 97 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: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/7 -11/15 -5/7 -2/3 -4/7 -5/9 -19/42 -3/7 -1/3 -2/7 -11/42 -2/9 -1/6 -1/7 0/1 1/8 1/7 1/6 3/17 4/21 1/5 2/9 3/13 5/21 1/4 5/19 3/11 2/7 31/105 3/10 1/3 8/21 41/105 2/5 3/7 1/2 5/9 4/7 61/105 13/21 2/3 5/7 11/15 16/21 17/21 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -7/8 1/1 1/0 -6/7 0/1 -11/13 1/2 1/1 -5/6 1/2 1/0 -9/11 2/1 1/0 -4/5 -2/1 1/0 -7/9 -1/2 -17/22 -1/2 0/1 -10/13 -1/3 -1/4 -13/17 -1/8 0/1 -16/21 0/1 -3/4 0/1 1/2 -14/19 1/1 9/8 -11/15 3/2 -19/26 7/4 2/1 -8/11 2/1 1/0 -5/7 1/0 -7/10 -2/1 1/0 -9/13 -3/2 -1/1 -2/3 -1/2 1/0 -7/11 0/1 1/0 -12/19 3/4 1/1 -17/27 3/2 -22/35 2/1 -5/8 3/1 1/0 -3/5 -2/1 1/0 -7/12 -3/2 -5/4 -11/19 -3/2 -1/1 -4/7 -1/1 -5/9 -1/2 -11/20 -1/2 0/1 -6/11 0/1 1/0 -1/2 -1/1 1/0 -5/11 -3/4 -2/3 -19/42 -2/3 -14/31 -2/3 -5/8 -9/20 -2/3 -1/2 -4/9 -3/4 -1/2 -3/7 -1/2 -5/12 -1/2 -1/4 -12/29 -1/4 -1/5 -7/17 0/1 1/0 -9/22 -1/2 0/1 -2/5 0/1 1/0 -9/23 -1/1 -1/2 -16/41 -1/2 0/1 -7/18 -1/2 1/0 -5/13 -3/2 -1/1 -8/21 -1/1 -3/8 -1/1 -3/4 -10/27 -1/2 1/0 -7/19 -1/1 -5/6 -11/30 -5/6 -3/4 -15/41 -4/5 -3/4 -4/11 -3/4 -2/3 -1/3 -1/2 -3/10 -1/4 0/1 -5/17 -1/4 0/1 -2/7 0/1 -3/11 0/1 1/0 -4/15 -3/2 1/0 -5/19 -7/6 -1/1 -11/42 -1/1 -6/23 -1/1 -11/12 -1/4 -2/3 -1/2 -3/13 -1/2 -3/7 -8/35 -2/5 -5/22 -1/2 -2/5 -2/9 -1/2 -3/8 -1/5 -1/4 0/1 -4/21 0/1 -3/16 0/1 1/4 -2/11 0/1 1/0 -1/6 -1/2 1/0 -2/13 -1/1 -3/4 -1/7 -1/2 -1/8 -1/3 -1/4 0/1 -1/2 0/1 1/8 -1/1 1/0 1/7 -1/2 2/13 -3/8 -1/3 1/6 -1/2 -1/4 4/23 -1/3 -1/4 3/17 -1/4 0/1 2/11 -1/4 0/1 3/16 -1/8 0/1 4/21 0/1 1/5 0/1 1/0 2/9 -3/4 -1/2 5/22 -2/3 -1/2 3/13 -3/5 -1/2 4/17 -8/15 -1/2 5/21 -1/2 1/4 -1/2 -2/5 5/19 -1/3 -7/22 4/15 -3/10 -1/4 3/11 -1/4 0/1 5/18 -1/4 -1/6 2/7 0/1 7/24 -1/2 1/0 12/41 -1/2 0/1 5/17 0/1 1/0 18/61 1/1 1/0 31/105 1/0 13/44 -1/1 1/0 8/27 -1/2 1/0 3/10 0/1 1/0 1/3 -1/2 4/11 -2/5 -3/8 7/19 -5/14 -1/3 10/27 -1/2 -1/4 3/8 -3/8 -1/3 8/21 -1/3 5/13 -1/3 -3/10 7/18 -1/2 -1/4 16/41 -1/2 0/1 41/105 -1/2 25/64 -1/2 -2/5 9/23 -1/2 -1/3 2/5 -1/4 0/1 11/27 -1/2 9/22 -1/2 0/1 7/17 -1/4 0/1 12/29 1/1 1/0 17/41 0/1 1/0 5/12 -1/2 1/0 3/7 -1/2 4/9 -1/2 -3/8 9/20 -1/2 -2/5 5/11 -2/5 -3/8 1/2 -1/3 -1/4 8/15 -1/2 -1/4 7/13 -1/2 -1/3 6/11 -1/4 0/1 23/42 0/1 17/31 0/1 1/0 11/20 -1/2 0/1 5/9 -1/2 9/16 -3/8 -4/11 4/7 -1/3 15/26 -5/16 -4/13 11/19 -1/3 -3/10 18/31 -6/19 -5/16 61/105 -5/16 43/74 -5/16 -14/45 25/43 -5/16 -4/13 7/12 -5/16 -3/10 3/5 -2/7 -1/4 8/13 -7/27 -1/4 13/21 -1/4 5/8 -1/4 -3/13 17/27 -3/14 12/19 -1/5 -3/16 7/11 -1/4 0/1 9/14 0/1 11/17 -1/4 0/1 13/20 -1/2 0/1 2/3 -1/2 -1/4 9/13 -1/3 -3/10 16/23 -1/3 -1/4 7/10 -2/7 -1/4 5/7 -1/4 8/11 -1/4 -2/9 19/26 -2/9 -7/32 30/41 -2/9 -3/14 41/56 -2/9 11/15 -3/14 14/19 -9/44 -1/5 31/42 -1/5 17/23 -1/5 -9/46 3/4 -1/6 0/1 16/21 0/1 13/17 0/1 1/4 10/13 -1/1 1/0 17/22 -1/2 0/1 7/9 -1/2 11/14 -1/3 15/19 -1/3 -3/10 4/5 -2/7 -1/4 17/21 -1/4 13/16 -1/4 -6/25 9/11 -1/4 -2/9 5/6 -1/4 -1/6 16/19 -1/4 -1/5 11/13 -1/5 -1/6 17/20 -1/6 0/1 6/7 0/1 19/22 -1/2 0/1 13/15 -1/2 7/8 -1/4 -1/5 8/9 -1/2 -1/4 1/1 -1/4 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(148,131,357,316) (-1/1,-7/8) -> (12/29,17/41) Hyperbolic Matrix(211,184,-336,-293) (-7/8,-6/7) -> (-22/35,-5/8) Hyperbolic Matrix(125,106,-546,-463) (-6/7,-11/13) -> (-3/13,-8/35) Hyperbolic Matrix(146,123,273,230) (-11/13,-5/6) -> (8/15,7/13) Hyperbolic Matrix(62,51,-231,-190) (-5/6,-9/11) -> (-3/11,-4/15) Hyperbolic Matrix(188,153,231,188) (-9/11,-4/5) -> (13/16,9/11) Hyperbolic Matrix(106,83,189,148) (-4/5,-7/9) -> (5/9,9/16) Hyperbolic Matrix(209,162,378,293) (-7/9,-17/22) -> (11/20,5/9) Hyperbolic Matrix(22,17,-189,-146) (-17/22,-10/13) -> (-1/8,0/1) Hyperbolic Matrix(295,226,714,547) (-10/13,-13/17) -> (7/17,12/29) Hyperbolic Matrix(545,416,714,545) (-13/17,-16/21) -> (16/21,13/17) Hyperbolic Matrix(127,96,168,127) (-16/21,-3/4) -> (3/4,16/21) Hyperbolic Matrix(43,32,-168,-125) (-3/4,-14/19) -> (-6/23,-1/4) Hyperbolic Matrix(503,370,798,587) (-14/19,-11/15) -> (17/27,12/19) Hyperbolic Matrix(316,231,777,568) (-11/15,-19/26) -> (2/5,11/27) Hyperbolic Matrix(85,62,462,337) (-19/26,-8/11) -> (2/11,3/16) Hyperbolic Matrix(104,75,-147,-106) (-8/11,-5/7) -> (-5/7,-7/10) Parabolic Matrix(148,103,273,190) (-7/10,-9/13) -> (7/13,6/11) Hyperbolic Matrix(106,73,-273,-188) (-9/13,-2/3) -> (-7/18,-5/13) Hyperbolic Matrix(64,41,231,148) (-2/3,-7/11) -> (3/11,5/18) Hyperbolic Matrix(106,67,-231,-146) (-7/11,-12/19) -> (-1/2,-5/11) Hyperbolic Matrix(587,370,798,503) (-12/19,-17/27) -> (11/15,14/19) Hyperbolic Matrix(1492,939,2037,1282) (-17/27,-22/35) -> (41/56,11/15) Hyperbolic Matrix(64,39,105,64) (-5/8,-3/5) -> (3/5,8/13) Hyperbolic Matrix(22,13,-105,-62) (-3/5,-7/12) -> (-2/9,-1/5) Hyperbolic Matrix(293,170,-798,-463) (-7/12,-11/19) -> (-7/19,-11/30) Hyperbolic Matrix(314,181,399,230) (-11/19,-4/7) -> (11/14,15/19) Hyperbolic Matrix(148,83,189,106) (-4/7,-5/9) -> (7/9,11/14) Hyperbolic Matrix(293,162,378,209) (-5/9,-11/20) -> (17/22,7/9) Hyperbolic Matrix(379,208,-840,-461) (-11/20,-6/11) -> (-14/31,-9/20) Hyperbolic Matrix(190,103,273,148) (-6/11,-1/2) -> (16/23,7/10) Hyperbolic Matrix(967,438,1764,799) (-5/11,-19/42) -> (23/42,17/31) Hyperbolic Matrix(965,436,1764,797) (-19/42,-14/31) -> (6/11,23/42) Hyperbolic Matrix(85,38,378,169) (-9/20,-4/9) -> (2/9,5/22) Hyperbolic Matrix(62,27,-147,-64) (-4/9,-3/7) -> (-3/7,-5/12) Parabolic Matrix(316,131,357,148) (-5/12,-12/29) -> (7/8,8/9) Hyperbolic Matrix(547,226,714,295) (-12/29,-7/17) -> (13/17,10/13) Hyperbolic Matrix(314,129,1071,440) (-7/17,-9/22) -> (12/41,5/17) Hyperbolic Matrix(568,231,777,316) (-9/22,-2/5) -> (19/26,30/41) Hyperbolic Matrix(167,66,210,83) (-2/5,-9/23) -> (15/19,4/5) Hyperbolic Matrix(944,369,2415,944) (-9/23,-16/41) -> (25/64,9/23) Hyperbolic Matrix(503,196,1722,671) (-16/41,-7/18) -> (7/24,12/41) Hyperbolic Matrix(209,80,546,209) (-5/13,-8/21) -> (8/21,5/13) Hyperbolic Matrix(127,48,336,127) (-8/21,-3/8) -> (3/8,8/21) Hyperbolic Matrix(62,23,-399,-148) (-3/8,-10/27) -> (-1/6,-2/13) Hyperbolic Matrix(211,78,798,295) (-10/27,-7/19) -> (5/19,4/15) Hyperbolic Matrix(715,262,1722,631) (-11/30,-15/41) -> (17/41,5/12) Hyperbolic Matrix(2038,745,3507,1282) (-15/41,-4/11) -> (43/74,25/43) Hyperbolic Matrix(20,7,-63,-22) (-4/11,-1/3) -> (-1/3,-3/10) Parabolic Matrix(64,19,357,106) (-3/10,-5/17) -> (3/17,2/11) Hyperbolic Matrix(230,67,357,104) (-5/17,-2/7) -> (9/14,11/17) Hyperbolic Matrix(148,41,231,64) (-2/7,-3/11) -> (7/11,9/14) Hyperbolic Matrix(295,78,798,211) (-4/15,-5/19) -> (7/19,10/27) Hyperbolic Matrix(1303,342,1764,463) (-5/19,-11/42) -> (31/42,17/23) Hyperbolic Matrix(1301,340,1764,461) (-11/42,-6/23) -> (14/19,31/42) Hyperbolic Matrix(64,15,273,64) (-1/4,-3/13) -> (3/13,4/17) Hyperbolic Matrix(1952,445,2667,608) (-8/35,-5/22) -> (30/41,41/56) Hyperbolic Matrix(169,38,378,85) (-5/22,-2/9) -> (4/9,9/20) Hyperbolic Matrix(41,8,210,41) (-1/5,-4/21) -> (4/21,1/5) Hyperbolic Matrix(127,24,672,127) (-4/21,-3/16) -> (3/16,4/21) Hyperbolic Matrix(337,62,462,85) (-3/16,-2/11) -> (8/11,19/26) Hyperbolic Matrix(106,19,357,64) (-2/11,-1/6) -> (8/27,3/10) Hyperbolic Matrix(20,3,-147,-22) (-2/13,-1/7) -> (-1/7,-1/8) Parabolic Matrix(146,-17,189,-22) (0/1,1/8) -> (10/13,17/22) Hyperbolic Matrix(22,-3,147,-20) (1/8,1/7) -> (1/7,2/13) Parabolic Matrix(148,-23,399,-62) (2/13,1/6) -> (10/27,3/8) Hyperbolic Matrix(190,-33,357,-62) (1/6,4/23) -> (1/2,8/15) Hyperbolic Matrix(421,-74,1428,-251) (4/23,3/17) -> (5/17,18/61) Hyperbolic Matrix(62,-13,105,-22) (1/5,2/9) -> (7/12,3/5) Hyperbolic Matrix(463,-106,546,-125) (5/22,3/13) -> (11/13,17/20) Hyperbolic Matrix(106,-25,441,-104) (4/17,5/21) -> (5/21,1/4) Parabolic Matrix(125,-32,168,-43) (1/4,5/19) -> (17/23,3/4) Hyperbolic Matrix(190,-51,231,-62) (4/15,3/11) -> (9/11,5/6) Hyperbolic Matrix(85,-24,294,-83) (5/18,2/7) -> (2/7,7/24) Parabolic Matrix(3256,-961,11025,-3254) (18/61,31/105) -> (31/105,13/44) Parabolic Matrix(652,-193,777,-230) (13/44,8/27) -> (5/6,16/19) Hyperbolic Matrix(22,-7,63,-20) (3/10,1/3) -> (1/3,4/11) Parabolic Matrix(463,-170,798,-293) (4/11,7/19) -> (11/19,18/31) Hyperbolic Matrix(188,-73,273,-106) (5/13,7/18) -> (2/3,9/13) Hyperbolic Matrix(316,-123,483,-188) (7/18,16/41) -> (13/20,2/3) Hyperbolic Matrix(4306,-1681,11025,-4304) (16/41,41/105) -> (41/105,25/64) Parabolic Matrix(400,-157,693,-272) (9/23,2/5) -> (15/26,11/19) Hyperbolic Matrix(799,-326,924,-377) (11/27,9/22) -> (19/22,13/15) Hyperbolic Matrix(463,-190,714,-293) (9/22,7/17) -> (11/17,13/20) Hyperbolic Matrix(64,-27,147,-62) (5/12,3/7) -> (3/7,4/9) Parabolic Matrix(461,-208,840,-379) (9/20,5/11) -> (17/31,11/20) Hyperbolic Matrix(146,-67,231,-106) (5/11,1/2) -> (12/19,7/11) Hyperbolic Matrix(169,-96,294,-167) (9/16,4/7) -> (4/7,15/26) Parabolic Matrix(6406,-3721,11025,-6404) (18/31,61/105) -> (61/105,43/74) Parabolic Matrix(356,-207,399,-232) (25/43,7/12) -> (8/9,1/1) Hyperbolic Matrix(274,-169,441,-272) (8/13,13/21) -> (13/21,5/8) Parabolic Matrix(293,-184,336,-211) (5/8,17/27) -> (13/15,7/8) Hyperbolic Matrix(461,-320,546,-379) (9/13,16/23) -> (16/19,11/13) Hyperbolic Matrix(106,-75,147,-104) (7/10,5/7) -> (5/7,8/11) Parabolic Matrix(358,-289,441,-356) (4/5,17/21) -> (17/21,13/16) Parabolic Matrix(253,-216,294,-251) (17/20,6/7) -> (6/7,19/22) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(148,131,357,316) -> Matrix(1,0,0,1) Matrix(211,184,-336,-293) -> Matrix(1,2,0,1) Matrix(125,106,-546,-463) -> Matrix(5,-2,-12,5) Matrix(146,123,273,230) -> Matrix(1,0,-4,1) Matrix(62,51,-231,-190) -> Matrix(1,-2,0,1) Matrix(188,153,231,188) -> Matrix(1,-4,-4,17) Matrix(106,83,189,148) -> Matrix(3,2,-8,-5) Matrix(209,162,378,293) -> Matrix(1,0,0,1) Matrix(22,17,-189,-146) -> Matrix(1,0,0,1) Matrix(295,226,714,547) -> Matrix(1,0,4,1) Matrix(545,416,714,545) -> Matrix(1,0,12,1) Matrix(127,96,168,127) -> Matrix(1,0,-8,1) Matrix(43,32,-168,-125) -> Matrix(3,-2,-4,3) Matrix(503,370,798,587) -> Matrix(5,-6,-24,29) Matrix(316,231,777,568) -> Matrix(1,-2,0,1) Matrix(85,62,462,337) -> Matrix(1,-2,-4,9) Matrix(104,75,-147,-106) -> Matrix(1,-4,0,1) Matrix(148,103,273,190) -> Matrix(1,2,-4,-7) Matrix(106,73,-273,-188) -> Matrix(1,0,0,1) Matrix(64,41,231,148) -> Matrix(1,0,-4,1) Matrix(106,67,-231,-146) -> Matrix(3,-2,-4,3) Matrix(587,370,798,503) -> Matrix(5,-6,-24,29) Matrix(1492,939,2037,1282) -> Matrix(7,-12,-32,55) Matrix(64,39,105,64) -> Matrix(1,4,-4,-15) Matrix(22,13,-105,-62) -> Matrix(1,2,-4,-7) Matrix(293,170,-798,-463) -> Matrix(7,8,-8,-9) Matrix(314,181,399,230) -> Matrix(5,6,-16,-19) Matrix(148,83,189,106) -> Matrix(3,2,-8,-5) Matrix(293,162,378,209) -> Matrix(1,0,0,1) Matrix(379,208,-840,-461) -> Matrix(5,2,-8,-3) Matrix(190,103,273,148) -> Matrix(1,2,-4,-7) Matrix(967,438,1764,799) -> Matrix(3,2,4,3) Matrix(965,436,1764,797) -> Matrix(3,2,-20,-13) Matrix(85,38,378,169) -> Matrix(1,0,0,1) Matrix(62,27,-147,-64) -> Matrix(3,2,-8,-5) Matrix(316,131,357,148) -> Matrix(1,0,0,1) Matrix(547,226,714,295) -> Matrix(1,0,4,1) Matrix(314,129,1071,440) -> Matrix(1,0,0,1) Matrix(568,231,777,316) -> Matrix(7,2,-32,-9) Matrix(167,66,210,83) -> Matrix(1,2,-4,-7) Matrix(944,369,2415,944) -> Matrix(3,2,-8,-5) Matrix(503,196,1722,671) -> Matrix(1,0,0,1) Matrix(209,80,546,209) -> Matrix(5,6,-16,-19) Matrix(127,48,336,127) -> Matrix(7,6,-20,-17) Matrix(62,23,-399,-148) -> Matrix(1,0,0,1) Matrix(211,78,798,295) -> Matrix(1,2,-4,-7) Matrix(715,262,1722,631) -> Matrix(5,4,-4,-3) Matrix(2038,745,3507,1282) -> Matrix(41,32,-132,-103) Matrix(20,7,-63,-22) -> Matrix(3,2,-8,-5) Matrix(64,19,357,106) -> Matrix(1,0,0,1) Matrix(230,67,357,104) -> Matrix(1,0,0,1) Matrix(148,41,231,64) -> Matrix(1,0,-4,1) Matrix(295,78,798,211) -> Matrix(1,2,-4,-7) Matrix(1303,342,1764,463) -> Matrix(15,16,-76,-81) Matrix(1301,340,1764,461) -> Matrix(21,20,-104,-99) Matrix(64,15,273,64) -> Matrix(13,6,-24,-11) Matrix(1952,445,2667,608) -> Matrix(19,8,-88,-37) Matrix(169,38,378,85) -> Matrix(1,0,0,1) Matrix(41,8,210,41) -> Matrix(1,0,4,1) Matrix(127,24,672,127) -> Matrix(1,0,-12,1) Matrix(337,62,462,85) -> Matrix(1,-2,-4,9) Matrix(106,19,357,64) -> Matrix(1,0,0,1) Matrix(20,3,-147,-22) -> Matrix(3,2,-8,-5) Matrix(146,-17,189,-22) -> Matrix(1,0,0,1) Matrix(22,-3,147,-20) -> Matrix(3,2,-8,-5) Matrix(148,-23,399,-62) -> Matrix(1,0,0,1) Matrix(190,-33,357,-62) -> Matrix(1,0,0,1) Matrix(421,-74,1428,-251) -> Matrix(1,0,4,1) Matrix(62,-13,105,-22) -> Matrix(1,2,-4,-7) Matrix(463,-106,546,-125) -> Matrix(3,2,-20,-13) Matrix(106,-25,441,-104) -> Matrix(19,10,-40,-21) Matrix(125,-32,168,-43) -> Matrix(5,2,-28,-11) Matrix(190,-51,231,-62) -> Matrix(7,2,-32,-9) Matrix(85,-24,294,-83) -> Matrix(1,0,4,1) Matrix(3256,-961,11025,-3254) -> Matrix(1,-2,0,1) Matrix(652,-193,777,-230) -> Matrix(1,0,-4,1) Matrix(22,-7,63,-20) -> Matrix(3,2,-8,-5) Matrix(463,-170,798,-293) -> Matrix(23,8,-72,-25) Matrix(188,-73,273,-106) -> Matrix(1,0,0,1) Matrix(316,-123,483,-188) -> Matrix(1,0,0,1) Matrix(4306,-1681,11025,-4304) -> Matrix(3,2,-8,-5) Matrix(400,-157,693,-272) -> Matrix(11,4,-36,-13) Matrix(799,-326,924,-377) -> Matrix(1,0,0,1) Matrix(463,-190,714,-293) -> Matrix(1,0,0,1) Matrix(64,-27,147,-62) -> Matrix(3,2,-8,-5) Matrix(461,-208,840,-379) -> Matrix(5,2,-8,-3) Matrix(146,-67,231,-106) -> Matrix(5,2,-28,-11) Matrix(169,-96,294,-167) -> Matrix(23,8,-72,-25) Matrix(6406,-3721,11025,-6404) -> Matrix(319,100,-1024,-321) Matrix(356,-207,399,-232) -> Matrix(13,4,-36,-11) Matrix(274,-169,441,-272) -> Matrix(39,10,-160,-41) Matrix(293,-184,336,-211) -> Matrix(9,2,-32,-7) Matrix(461,-320,546,-379) -> Matrix(7,2,-32,-9) Matrix(106,-75,147,-104) -> Matrix(15,4,-64,-17) Matrix(358,-289,441,-356) -> Matrix(31,8,-128,-33) Matrix(253,-216,294,-251) -> Matrix(1,0,4,1) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 4 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 (-1/2,0/1) 0 21 1/8 (-1/1,1/0) 0 21 1/7 -1/2 2 3 2/13 (-3/8,-1/3) 0 21 1/6 0 7 2/11 (-1/4,0/1) 0 21 3/16 (-1/8,0/1) 0 21 4/21 0/1 8 1 1/5 (0/1,1/0) 0 21 2/9 0 7 5/22 (-2/3,-1/2) 0 21 3/13 (-3/5,-1/2) 0 21 5/21 -1/2 10 1 1/4 (-1/2,-2/5) 0 21 5/19 (-1/3,-7/22) 0 21 4/15 0 7 3/11 (-1/4,0/1) 0 21 2/7 0/1 2 3 5/17 (0/1,1/0) 0 21 31/105 1/0 2 1 13/44 (-1/1,1/0) 0 21 8/27 0 7 3/10 (0/1,1/0) 0 21 1/3 -1/2 2 7 4/11 (-2/5,-3/8) 0 21 7/19 (-5/14,-1/3) 0 21 10/27 0 7 3/8 (-3/8,-1/3) 0 21 8/21 -1/3 6 1 5/13 (-1/3,-3/10) 0 21 7/18 0 7 16/41 (-1/2,0/1) 0 21 41/105 -1/2 2 1 9/23 (-1/2,-1/3) 0 21 2/5 (-1/4,0/1) 0 21 9/22 (-1/2,0/1) 0 21 7/17 (-1/4,0/1) 0 21 12/29 (1/1,1/0) 0 21 5/12 0 7 3/7 -1/2 2 3 4/9 0 7 9/20 (-1/2,-2/5) 0 21 5/11 (-2/5,-3/8) 0 21 1/2 (-1/3,-1/4) 0 21 6/11 (-1/4,0/1) 0 21 23/42 0/1 4 1 17/31 (0/1,1/0) 0 21 11/20 (-1/2,0/1) 0 21 5/9 -1/2 2 7 4/7 -1/3 4 3 11/19 (-1/3,-3/10) 0 21 18/31 (-6/19,-5/16) 0 21 61/105 -5/16 4 1 25/43 (-5/16,-4/13) 0 21 7/12 0 7 3/5 (-2/7,-1/4) 0 21 13/21 -1/4 10 1 5/8 (-1/4,-3/13) 0 21 17/27 -3/14 2 7 12/19 (-1/5,-3/16) 0 21 7/11 (-1/4,0/1) 0 21 9/14 0/1 2 3 11/17 (-1/4,0/1) 0 21 13/20 (-1/2,0/1) 0 21 2/3 0 7 9/13 (-1/3,-3/10) 0 21 16/23 (-1/3,-1/4) 0 21 7/10 (-2/7,-1/4) 0 21 5/7 -1/4 4 3 8/11 (-1/4,-2/9) 0 21 19/26 (-2/9,-7/32) 0 21 30/41 (-2/9,-3/14) 0 21 11/15 -3/14 2 7 14/19 (-9/44,-1/5) 0 21 31/42 -1/5 18 1 17/23 (-1/5,-9/46) 0 21 3/4 (-1/6,0/1) 0 21 16/21 0/1 10 1 13/17 (0/1,1/4) 0 21 10/13 (-1/1,1/0) 0 21 17/22 (-1/2,0/1) 0 21 7/9 -1/2 2 7 11/14 -1/3 4 3 15/19 (-1/3,-3/10) 0 21 4/5 (-2/7,-1/4) 0 21 17/21 -1/4 8 1 9/11 (-1/4,-2/9) 0 21 5/6 0 7 16/19 (-1/4,-1/5) 0 21 11/13 (-1/5,-1/6) 0 21 17/20 (-1/6,0/1) 0 21 6/7 0/1 2 3 19/22 (-1/2,0/1) 0 21 13/15 -1/2 2 7 7/8 (-1/4,-1/5) 0 21 8/9 0 7 1/1 (-1/4,0/1) 0 21 1/0 0/1 2 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(146,-17,189,-22) (0/1,1/8) -> (10/13,17/22) Hyperbolic Matrix(22,-3,147,-20) (1/8,1/7) -> (1/7,2/13) Parabolic Matrix(148,-23,399,-62) (2/13,1/6) -> (10/27,3/8) Hyperbolic Matrix(106,-19,357,-64) (1/6,2/11) -> (8/27,3/10) Glide Reflection Matrix(337,-62,462,-85) (2/11,3/16) -> (8/11,19/26) Glide Reflection Matrix(127,-24,672,-127) (3/16,4/21) -> (3/16,4/21) Reflection Matrix(41,-8,210,-41) (4/21,1/5) -> (4/21,1/5) Reflection Matrix(62,-13,105,-22) (1/5,2/9) -> (7/12,3/5) Hyperbolic Matrix(169,-38,378,-85) (2/9,5/22) -> (4/9,9/20) Glide Reflection Matrix(463,-106,546,-125) (5/22,3/13) -> (11/13,17/20) Hyperbolic Matrix(64,-15,273,-64) (3/13,5/21) -> (3/13,5/21) Reflection Matrix(41,-10,168,-41) (5/21,1/4) -> (5/21,1/4) Reflection Matrix(125,-32,168,-43) (1/4,5/19) -> (17/23,3/4) Hyperbolic Matrix(295,-78,798,-211) (5/19,4/15) -> (7/19,10/27) Glide Reflection Matrix(190,-51,231,-62) (4/15,3/11) -> (9/11,5/6) Hyperbolic Matrix(148,-41,231,-64) (3/11,2/7) -> (7/11,9/14) Glide Reflection Matrix(230,-67,357,-104) (2/7,5/17) -> (9/14,11/17) Glide Reflection Matrix(526,-155,1785,-526) (5/17,31/105) -> (5/17,31/105) Reflection Matrix(2729,-806,9240,-2729) (31/105,13/44) -> (31/105,13/44) Reflection Matrix(652,-193,777,-230) (13/44,8/27) -> (5/6,16/19) Hyperbolic Matrix(22,-7,63,-20) (3/10,1/3) -> (1/3,4/11) Parabolic Matrix(463,-170,798,-293) (4/11,7/19) -> (11/19,18/31) Hyperbolic Matrix(127,-48,336,-127) (3/8,8/21) -> (3/8,8/21) Reflection Matrix(209,-80,546,-209) (8/21,5/13) -> (8/21,5/13) Reflection Matrix(188,-73,273,-106) (5/13,7/18) -> (2/3,9/13) Hyperbolic Matrix(316,-123,483,-188) (7/18,16/41) -> (13/20,2/3) Hyperbolic Matrix(3361,-1312,8610,-3361) (16/41,41/105) -> (16/41,41/105) Reflection Matrix(944,-369,2415,-944) (41/105,9/23) -> (41/105,9/23) Reflection Matrix(167,-66,210,-83) (9/23,2/5) -> (15/19,4/5) Glide Reflection Matrix(568,-231,777,-316) (2/5,9/22) -> (19/26,30/41) Glide Reflection Matrix(463,-190,714,-293) (9/22,7/17) -> (11/17,13/20) Hyperbolic Matrix(547,-226,714,-295) (7/17,12/29) -> (13/17,10/13) Glide Reflection Matrix(316,-131,357,-148) (12/29,5/12) -> (7/8,8/9) Glide Reflection Matrix(64,-27,147,-62) (5/12,3/7) -> (3/7,4/9) Parabolic Matrix(461,-208,840,-379) (9/20,5/11) -> (17/31,11/20) Hyperbolic Matrix(146,-67,231,-106) (5/11,1/2) -> (12/19,7/11) Hyperbolic Matrix(190,-103,273,-148) (1/2,6/11) -> (16/23,7/10) Glide Reflection Matrix(505,-276,924,-505) (6/11,23/42) -> (6/11,23/42) Reflection Matrix(1427,-782,2604,-1427) (23/42,17/31) -> (23/42,17/31) Reflection Matrix(293,-162,378,-209) (11/20,5/9) -> (17/22,7/9) Glide Reflection Matrix(148,-83,189,-106) (5/9,4/7) -> (7/9,11/14) Glide Reflection Matrix(314,-181,399,-230) (4/7,11/19) -> (11/14,15/19) Glide Reflection Matrix(3781,-2196,6510,-3781) (18/31,61/105) -> (18/31,61/105) Reflection Matrix(2624,-1525,4515,-2624) (61/105,25/43) -> (61/105,25/43) Reflection Matrix(356,-207,399,-232) (25/43,7/12) -> (8/9,1/1) Hyperbolic Matrix(64,-39,105,-64) (3/5,13/21) -> (3/5,13/21) Reflection Matrix(209,-130,336,-209) (13/21,5/8) -> (13/21,5/8) Reflection Matrix(293,-184,336,-211) (5/8,17/27) -> (13/15,7/8) Hyperbolic Matrix(587,-370,798,-503) (17/27,12/19) -> (11/15,14/19) Glide Reflection Matrix(461,-320,546,-379) (9/13,16/23) -> (16/19,11/13) Hyperbolic Matrix(106,-75,147,-104) (7/10,5/7) -> (5/7,8/11) Parabolic Matrix(818,-599,945,-692) (30/41,11/15) -> (19/22,13/15) Glide Reflection Matrix(1177,-868,1596,-1177) (14/19,31/42) -> (14/19,31/42) Reflection Matrix(1427,-1054,1932,-1427) (31/42,17/23) -> (31/42,17/23) Reflection Matrix(127,-96,168,-127) (3/4,16/21) -> (3/4,16/21) Reflection Matrix(545,-416,714,-545) (16/21,13/17) -> (16/21,13/17) Reflection Matrix(169,-136,210,-169) (4/5,17/21) -> (4/5,17/21) Reflection Matrix(188,-153,231,-188) (17/21,9/11) -> (17/21,9/11) Reflection Matrix(253,-216,294,-251) (17/20,6/7) -> (6/7,19/22) Parabolic Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(146,-17,189,-22) -> Matrix(1,0,0,1) Matrix(22,-3,147,-20) -> Matrix(3,2,-8,-5) -1/2 Matrix(148,-23,399,-62) -> Matrix(1,0,0,1) Matrix(106,-19,357,-64) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(337,-62,462,-85) -> Matrix(9,2,-40,-9) *** -> (-1/4,-1/5) Matrix(127,-24,672,-127) -> Matrix(-1,0,16,1) (3/16,4/21) -> (-1/8,0/1) Matrix(41,-8,210,-41) -> Matrix(1,0,0,-1) (4/21,1/5) -> (0/1,1/0) Matrix(62,-13,105,-22) -> Matrix(1,2,-4,-7) Matrix(169,-38,378,-85) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(463,-106,546,-125) -> Matrix(3,2,-20,-13) Matrix(64,-15,273,-64) -> Matrix(11,6,-20,-11) (3/13,5/21) -> (-3/5,-1/2) Matrix(41,-10,168,-41) -> Matrix(9,4,-20,-9) (5/21,1/4) -> (-1/2,-2/5) Matrix(125,-32,168,-43) -> Matrix(5,2,-28,-11) Matrix(295,-78,798,-211) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(190,-51,231,-62) -> Matrix(7,2,-32,-9) -1/4 Matrix(148,-41,231,-64) -> Matrix(-1,0,8,1) *** -> (-1/4,0/1) Matrix(230,-67,357,-104) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(526,-155,1785,-526) -> Matrix(1,0,0,-1) (5/17,31/105) -> (0/1,1/0) Matrix(2729,-806,9240,-2729) -> Matrix(1,2,0,-1) (31/105,13/44) -> (-1/1,1/0) Matrix(652,-193,777,-230) -> Matrix(1,0,-4,1) 0/1 Matrix(22,-7,63,-20) -> Matrix(3,2,-8,-5) -1/2 Matrix(463,-170,798,-293) -> Matrix(23,8,-72,-25) -1/3 Matrix(127,-48,336,-127) -> Matrix(17,6,-48,-17) (3/8,8/21) -> (-3/8,-1/3) Matrix(209,-80,546,-209) -> Matrix(19,6,-60,-19) (8/21,5/13) -> (-1/3,-3/10) Matrix(188,-73,273,-106) -> Matrix(1,0,0,1) Matrix(316,-123,483,-188) -> Matrix(1,0,0,1) Matrix(3361,-1312,8610,-3361) -> Matrix(-1,0,4,1) (16/41,41/105) -> (-1/2,0/1) Matrix(944,-369,2415,-944) -> Matrix(5,2,-12,-5) (41/105,9/23) -> (-1/2,-1/3) Matrix(167,-66,210,-83) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(568,-231,777,-316) -> Matrix(1,2,-4,-9) Matrix(463,-190,714,-293) -> Matrix(1,0,0,1) Matrix(547,-226,714,-295) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(316,-131,357,-148) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(64,-27,147,-62) -> Matrix(3,2,-8,-5) -1/2 Matrix(461,-208,840,-379) -> Matrix(5,2,-8,-3) -1/2 Matrix(146,-67,231,-106) -> Matrix(5,2,-28,-11) Matrix(190,-103,273,-148) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(505,-276,924,-505) -> Matrix(-1,0,8,1) (6/11,23/42) -> (-1/4,0/1) Matrix(1427,-782,2604,-1427) -> Matrix(1,0,0,-1) (23/42,17/31) -> (0/1,1/0) Matrix(293,-162,378,-209) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(148,-83,189,-106) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(314,-181,399,-230) -> Matrix(19,6,-60,-19) *** -> (-1/3,-3/10) Matrix(3781,-2196,6510,-3781) -> Matrix(191,60,-608,-191) (18/31,61/105) -> (-6/19,-5/16) Matrix(2624,-1525,4515,-2624) -> Matrix(129,40,-416,-129) (61/105,25/43) -> (-5/16,-4/13) Matrix(356,-207,399,-232) -> Matrix(13,4,-36,-11) -1/3 Matrix(64,-39,105,-64) -> Matrix(15,4,-56,-15) (3/5,13/21) -> (-2/7,-1/4) Matrix(209,-130,336,-209) -> Matrix(25,6,-104,-25) (13/21,5/8) -> (-1/4,-3/13) Matrix(293,-184,336,-211) -> Matrix(9,2,-32,-7) -1/4 Matrix(587,-370,798,-503) -> Matrix(29,6,-140,-29) *** -> (-3/14,-1/5) Matrix(461,-320,546,-379) -> Matrix(7,2,-32,-9) -1/4 Matrix(106,-75,147,-104) -> Matrix(15,4,-64,-17) -1/4 Matrix(818,-599,945,-692) -> Matrix(9,2,-4,-1) Matrix(1177,-868,1596,-1177) -> Matrix(89,18,-440,-89) (14/19,31/42) -> (-9/44,-1/5) Matrix(1427,-1054,1932,-1427) -> Matrix(91,18,-460,-91) (31/42,17/23) -> (-1/5,-9/46) Matrix(127,-96,168,-127) -> Matrix(-1,0,12,1) (3/4,16/21) -> (-1/6,0/1) Matrix(545,-416,714,-545) -> Matrix(1,0,8,-1) (16/21,13/17) -> (0/1,1/4) Matrix(169,-136,210,-169) -> Matrix(15,4,-56,-15) (4/5,17/21) -> (-2/7,-1/4) Matrix(188,-153,231,-188) -> Matrix(17,4,-72,-17) (17/21,9/11) -> (-1/4,-2/9) Matrix(253,-216,294,-251) -> Matrix(1,0,4,1) 0/1 Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.