# Stored data for newform 77.4.a.d, downloaded from the LMFDB on 20 April 2024. {"label": "77.4.a.d", "space_label": "77.4.a", "level": 77, "weight": 4, "hecke_orbit": 4, "hecke_orbit_code": 13510798949220429, "dim": 4, "is_polredabs": true, "nf_label": "4.4.522072.1", "trace_hash": 2111832923050804655, "analytic_rank": 0, "is_twist_minimal": true, "hecke_cutters": [[2, [112, -40, -27, 2, 1]]], "qexp_display": "q+\\beta _{2}q^{2}+(4+\\beta _{1})q^{3}+(6-\\beta _{2}-2\\beta _{3})q^{4}+\\cdots", "char_order": 1, "char_parity": 1, "char_degree": 1, "char_conductor": 1, "char_orbit_label": "a", "char_is_real": true, "Nk2": 1232, "fricke_eigenval": 1, "atkin_lehner_string": "++", "analytic_conductor": 4.543147070442028, "hecke_ring_index": 2, "level_radical": 77, "field_poly_is_cyclotomic": false, "field_poly_root_of_unity": 0, "field_poly_is_real_cyclotomic": false, "hecke_ring_index_proved": true, "self_twist_type": 0, "is_self_dual": true, "is_self_twist": false, "is_cm": false, "is_rm": false, "trace_zratio": 0, "analytic_rank_proved": true, "char_values": [77, 1, [45, 57], [1, 1]], "hecke_ring_generator_nbound": 3, "inner_twist_count": 1, "level_primes": [7, 11], "conrey_indexes": [1], "field_disc": 522072, "field_disc_factorization": [[2, 3], [3, 3], [2417, 1]], "field_poly": [1, 5, -12, -1, 1], "hecke_ring_index_factorization": [[2, 1]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.030", "prec": 17}, {"__RealLiteral__": 0, "data": "3.866", "prec": 17}, {"__RealLiteral__": 0, "data": "0.258", "prec": 17}, {"__RealLiteral__": 0, "data": "40.564", "prec": 20}, {"__RealLiteral__": 0, "data": "7.842", "prec": 17}, {"__RealLiteral__": 0, "data": "632.212", "prec": 24}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 4, "sato_tate_group": "3.2.3.c1", "atkin_lehner_eigenvals": [[7, 1], [11, 1]], "trace_display": [-2, 14, 10, -28], "traces": [0, 4, -2, 14, 26, 10, 14, -28, -18, 76, -2, -44, 70, 58, 14, 284, 2, 4, -62, 258, 182, -98, 22, 8, -498, 80, -482, 428, -182, -396, -628, -56, 134, -154, 472, -70, -418, 84, -942, -412, -1026, 52, -98, 408, -286, 826, 368, 8, 982, 196, -1642, -388, 2030, 624, 92, -110, 126, 48, 864, -238, 1420, -162, 688, -532, -902, -32, -154, 1340, -1384, 2416, 14, 1788, -2622, 1456, 996, -806, 3042, 308, -2632, -1324, 2342, 1444, 1984, 450, -490, -1736, -4380, 588, 198, -3072, -218, -406, 544, -1264, -1696, 24, 862, -652, -98, -836, 874, 14, 4592, 3056, -66, -1988, -888, 348, -5300, -3632, 22, -1572, -14, -636, -2280, 4856, -1848, -4958, 7682, -28, -4116, 484, -102, -1852, 3128, -2244, 434, -304, -2850, 144, -4616, 4098, -770, -1806, 2636, 872, 3960, -3556, -4184, 1462, -1274, -5552, 2076, -638, 1670, 2988, 9556, 686, 5004, -5740, -3686, 3188, -1806, 1276, -154, -860, 12568, 134, 4184, 1020, -370, -56, 12370, 5500, -3424, -3124, 378, -9540, 3486, 4428, 9952, -174, 4164, -4398, -4092, -560, -22, 2176, 4908, 3432, -322, -7826, 3374, 3852, -7728, -1812, -988, -44, 1048, -2996, -4248, 8564, -5338, -5024, -7048, 4112, 1274, -2048, 682, 2688, -4026, -1160, -7390, 2772, -12032, -6824, -8296, 7112, 974, -2838, 4396, 1308, 2928, 4680, -10812, 1296, -8388, 392, 10372, 3308, -2002, -8156, 972, 124, -938, 920, -4704, 8610, 15336, 1614, -14008, 1078, -2784, -2544, -17330, -9796, -902, -4856, -3304, -4464, 652, 8452, -242, 17690, -18726, 490, 16928, 17196, 7824, -8160, -3324, 4126, 2926, -88, -5776, -2440, -12494, 7336, -12888, -588, 21320, -3804, -27438, -1432, 5478, -1044, 6594, -16308, -9620, 2302, 17768, 10280, -8392, 2884, -4348, -880, 6440, -14084, 11158, -17792, 7182, 2424, 3772, 8818, -3012, 22008, 5302, -364, 17978, -7264, -10524, 14812, -2524, 15918, 686, 1964, 12276, -4708, 1208, -6448, 8738, -2856, 16712, 23572, -990, -312, 10072, 12650, 2002, 7528, -2156, 7524, -4536, 3692, 14282, -5782, 6664, 14416, 4188, 4356, -12362, -5784, -2576, -8520, -31138, 14198, 2836, -21748, 17952, -56, 6908, 2608, 9930, -22356, -4572, -19168, -6874, -16, -14970, 11028, -26464, 616, -18078, -1372, -17580, 640, -31794, 4408, -1260, -2838, 11494, -41728, -1474, 11880, 18652, -7488, -5532, 2716, 7872, 4276, -25338, 6752, -5558, 1694, -14210, -11072, 16464, -5612, 19840, -1892, 1260, -4368, 16612, 12280, -5192, -16680, 13824, -11364, -644, 6340, 30936, 16552, -20140, -15112, 7086, 770, -24380, 10608, -8192, -13380, -57064, -10984, -882, 3408, -30764, 2120, 4598, 13918, 10296, -336, -4382, 3740, 55456, 16552, -14318, 25186, -6048, -924, 12768, 3756, 36400, -18860, 12208, 1666, 4256, -15144, -14146, 1760, 10362, 30830, -9940, -27128, -14268, -20080, -1032, -9700, 41760, 1134, -26028, 4532, -20760, -8964, -8660, 22180, -4816, -32280, -20596, -11592, 62648, 6016, 11286, 3724, 26944, -21248, 22044, -13260, 25492, -49508, 6314, 16244, -5566, -572, 10824, -18704, 18114, 224, -37992, -4008, 16026, 29408, 23272, -6438, 1078, -4580, 16440, -19696, 6132, 17010, 45950, -9380, 20684, -21812, 26850, -4488, -14768, 29166, 9688, 12312, -26112, -460, 26092, 23916, 9856, -16912, 3146, 34268, 60794, -7048, -17862, 34616, -98, -17228, -36656, -7872, -55980, -9086, 584, -12516, 12648, -19804, 180, -14208, 14350, -27872, 18354, 43568, -4048, 59538, -704, -5754, 41216, -10192, 25078, -12192, -64748, 860, 33192, -88, -6972, -71652, -39912, 23292, 6168, 21506, 20754, 5642, 8168, -4820, -10802, -3820, 7452, -17110, -21294, -22004, -9492, -21960, 33900, 4488, -11750, -2156, -50456, -30872, -15160, -20164, 8312, -29924, 18424, 57992, -29636, 68166, 18062, -38556, -21240, 9268, -26312, -17064, -25018, 12112, -36152, 38208, -16394, 4268, -19068, 29134, 3068, 34908, 35890, -10108, 13116, -28968, -104760, 37472, -22330, -20552, -13888, -19928, 21982, 19536, -34894, -9772, -1644, -3150, -56348, -6864, 52020, -34400, 25194, -54890, 3430, 5292, 21404, -29140, -3588, -10288, -1012, 12152, -71408, 19128, -18088, 73132, -40422, 63704, 30660, 47132, 232, 1210, -51224, -11272, 3186, -4116, 34416, 30296, -45544, -892, -83542, -12952, -1386, 9360, 4828, -10030, 12548, 12880, -55284, 21504, 35608, -23656, 35744, -528, 16738, -1824, 1526, 13156, 24456, -47640, -14180, 38072, 3852, 2842, -9504, 62652, 52254, 2328, 20424, 36590, -3808, 65136, 47856, -49168, -37806, 2618, -65902, 8848, 102548, 1756, 30788, -1872, -31552, 25624, 11872, -39224, -15620, 30322, -18752, -83392, 39930, -168, -50964, 37584, -65820, 34904, 116768, 1782, -6034, -22864, 111308, 388, 9114, 15858, -68748, 4564, 38688, 81312, -7568, 78104, 39762, -21916, 686, 8796, 2820, 9780, 21616, -11806, -48354, 5852, -24212, -26264, 44580, 44236, -30426, -58404, -6118, 36440, -61888, 16548, 9922, 4336, 2388, -98, -27988, -39124, 76128, -82420, 52908, -8200, -32144, 352, 38928, 28536, -7232, -50832, 28142, -21392, -115678, 62516, -93262, -34524, 1694, 29256, 462, 112108, 108520, -35472, -108048, 22686, -548, 13916, -4880, -14740, 20896, 10352, 14844, 67224, 6216, -10352, 57108, -89404, 9604, -53214, 15224, -2436, -27912, -2552, -38264, -7504, 49356, -71152, 37100, 23276, -29756, -26576, -108984, 9108, -58952, 25424, 107908, -152, -14608, -47068, -119434, -75988, -154, 4604, 54860, 4338, -16716, -18160, -102312, 11004, 46728, -24792, 168664, -19668, 66752, -44280, 98, -43468, -43824, 31186, 1196, 50032, -32080, 4452, 28842, -103056, 31714, 49992, 112944, 63182, 15960, 9044, 27022, -134016, -102712, -16016, -122272, -33992, 49324, -5044, -131070, -39504, 67966, -53702, 12936, 40192, -10956, 75088, -15584, 84768, -17184, 34706, -90640, -49700, -11276, 46284, -34920, 8866, -53774, 49484, -21488, 10258, 36984, -113284, -63986, 196, 105224, 17280, -33462, -119032, 36494, 20416, 28812, -22484, 40408, -45108, 59604, 95694, -28024, -3388, -11232, -4384, 23144, -19008, -80352, -31318, 714, 8520, -26748, -6724, 28952, 12606, 90504, 12964, 44256, 8364, 129052, -116400, -36920, -2702, -21896, 14564, 95928, -71200, 95268, 89972, 11400, 15708, -102296, -77472, -49448, 69804, -25762, -18592, -3038, 17992, -84784, -87800, -116768, -11812, 66348, 2128, 2892, -15884, 27788, 45780, 84076, -2952, 19950, 6976, -77764, 9840, 83350, 6624, -21824, -1008, -75264, -21572, 165688, 59292, 78498, 159734, 32312, -46008, 102600, -4950, 98580, 34512, -39582, -28686, -30208, 20032, -14328, 44032, -15330, -57360, 5390, -23340, 153412, 88664, -29520, 68748, -5896, 12642, 23004, -20472, 167250, 19096, 65262, 6200, -18452, 6604, -17060, -6954, 71452, -45784, 10762, -6104, 48180, 85752, 59792, -84884, -144834, -3052, -27720, 40504, -3024, -45592, -82416, -6468, 7916, 24892, -18676, -52248, 59964, 95388, 110336, 60268, 29288, -50996, -2178, -119856, -128956, -87358, -105566, -10234, -16072, 60940, -17382, 91800, -92128, 33792, 8918, -139856, -86648, -129672, 64368, -25916, 4176, 38864, 169572, -21936, 2398, -51788, -68896, -40408, -14532, 68076, 31224, -2170, 63164, -143112, 23172], "weight_parity": 1, "has_non_self_twist": 0, "level_is_prime": false, "level_is_prime_power": false, "level_is_square": false, "level_is_squarefree": true, "inner_twists": [[1, 1, 1, 1, 1, 1, 1]], "char_orbit_index": 1, "prim_orbit_index": 1, "minimal_twist": "77.4.a.d", "char_is_minimal": true, "conrey_index": 1, "id": null, "projective_image": null, "projective_image_type": null, "artin_degree": null, "artin_image": null, "artin_field_label": null, "projective_field_label": null, "projective_field": null, "artin_field": null, "embedded_related_objects": null, "qexp": [[0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [4, 1, 0, 0], [6, 0, -1, -2], [3, 2, -1, -1], [5, -2, 5, 3], [-7, 0, 0, 0], [-4, -4, 5, 0], [21, 4, 0, 6], [1, -6, 9, 7], [-11, 0, 0, 0], [13, 2, -11, -13], [8, -3, -10, -10], [0, 0, -7, 0], [68, 2, -8, 4], [2, 8, -5, -6], [5, -5, 13, 9], [-10, 4, 7, 18], [55, -4, -15, -7], [37, 10, -27, -21], [-28, -7, 0, 0], [0, 0, -11, 0], [10, 18, -2, -6], [-119, -14, 25, -9], [5, -4, -26, 2], [-105, -14, 45, 1], [122, 12, 18, 54], [-42, 0, 7, 14], [-90, 14, 4, -8], [-122, 4, 66, 26], [-15, -9, 7, -25], [32, 4, -7, 28], [-44, -11, 0, 0], [112, 28, -40, -32], [-21, -14, 7, 7], [-140, -4, -67, -32], [20, -16, 14, -32], [-195, -6, 87, 11], [-144, -2, -80, -98], [-231, -14, 65, 7], [27, -23, 51, 23], [-35, 14, -35, -21], [66, 2, -74, 0], [-66, 0, 11, 22], [227, 42, -1, 35], [92, -48, 48, 52], [-25, -39, -15, 1], [221, -6, -43, 3], [49, 0, 0, 0], [-394, 12, 21, 42], [-48, 6, 92, 54], [491, 54, -87, -51], [150, -8, -4, 10], [42, 84, -46, 54], [-33, -22, 11, 11], [28, 28, -35, 0], [-4, 64, -96, -104], [166, -44, -56, 26], [4, 3, 124, -24], [270, 28, -198, -126], [-14, 29, 24, 120], [178, -32, 44, -66], [-147, -28, 0, -42], [-234, -16, -1, 102], [-54, 54, -146, -142], [-55, 22, -55, -33], [364, -62, 120, 124], [-300, -80, 172, 60], [588, -4, -28, 72], [-7, 42, -63, -49], [480, -26, 92, 96], [-718, -88, -37, -54], [449, -47, 217, 61], [276, -32, 86, -108], [-232, 63, -124, -92], [693, 66, -201, -125], [77, 0, 0, 0], [-640, -192, 228, 56], [-320, -6, 28, -114], [509, -38, -115, 3], [557, 140, 252, 234], [484, 92, -116, -148], [87, -76, 25, -43], [-91, -14, 77, 91], [-342, -2, 186, 82], [-1026, -4, 142, 154], [84, -122, -4, 56], [44, 44, -55, 0], [-770, -48, 44, -104], [21, -14, 165, 163], [-56, 21, 70, 70], [92, 56, -144, -140], [-388, -104, -40, -158], [-410, 80, -52, -86], [-4, 196, -216, -220], [305, 130, 49, 143], [-174, 154, -176, 32], [0, 0, 49, 0], [-231, -44, 0, -66], [104, 92, -321, 20]]}