# Stored data for newform 47.4.a.b, downloaded from the LMFDB on 19 February 2026. {"Nk2": 752, "analytic_conductor": 2.773089770269809, "analytic_rank": 0, "analytic_rank_proved": true, "atkin_lehner_eigenvals": [[47, 1]], "atkin_lehner_string": "+", "char_conductor": 1, "char_degree": 1, "char_is_minimal": true, "char_is_real": true, "char_orbit_index": 1, "char_orbit_label": "a", "char_order": 1, "char_parity": 1, "char_values": [47, 1, [5], [1]], "cm_discs": [], "conrey_index": 1, "dim": 8, "field_disc": 860381024818537416, "field_disc_factorization": [[2, 3], [3, 1], [331, 1], [108305768481689, 1]], "field_poly": [10316, 5418, -5393, -1549, 844, 124, -50, -3, 1], "field_poly_is_cyclotomic": false, "field_poly_is_real_cyclotomic": false, "field_poly_root_of_unity": 0, "fricke_eigenval": 1, "has_non_self_twist": 0, "hecke_cutters": [[2, [10316, 5418, -5393, -1549, 844, 124, -50, -3, 1]]], "hecke_orbit": 2, "hecke_orbit_code": 4503599694479407, "hecke_ring_generator_nbound": 5, "hecke_ring_index": 1, "hecke_ring_index_factorization": [], "hecke_ring_index_proved": true, "inner_twist_count": 1, "inner_twists": [[1, 1, 1, 1, 1, 1, 1]], "is_cm": false, "is_largest": true, "is_maximal": true, "is_polredabs": true, "is_rm": false, "is_self_dual": true, "is_self_twist": false, "is_twist_minimal": true, "label": "47.4.a.b", "level": 47, "level_is_powerful": false, "level_is_prime": true, "level_is_prime_power": true, "level_is_prime_square": false, "level_is_square": false, "level_is_squarefree": true, "level_primes": [47], "level_radical": 47, "minimal_twist": "47.4.a.b", "prim_orbit_index": 1, "qexp_display": "q+\\beta _{1}q^{2}+(1+\\beta _{5})q^{3}+(5+\\beta _{2}-\\beta _{3}+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 8, "rm_discs": [], "sato_tate_group": "3.2.3.c1", "self_twist_discs": [], "self_twist_type": 0, "space_label": "47.4.a", "trace_display": [3, 7, 14, 39], "trace_hash": 146837199919982895, "trace_moments": [{"__RealLiteral__": 0, "data": "0.136", "prec": 17}, {"__RealLiteral__": 0, "data": "7.383", "prec": 17}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "162.907", "prec": 24}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "5833.903", "prec": 27}], "trace_zratio": 0, "traces": [0, 8, 3, 7, 45, 14, 16, 39, 57, 93, 60, 68, -19, 112, -277, -126, -87, -7, -474, 74, -432, -160, 20, 64, -325, 440, -620, 256, -84, 20, -526, 176, 214, 456, 398, -116, -55, 811, -10, 320, 446, 466, 293, 1164, 1150, 774, 410, -376, -84, 665, 273, -1495, 832, 747, -1721, -1548, -1536, 740, 1492, -1661, -1406, 1051, -1864, -984, -1087, -2572, -24, 174, -793, -2830, 3106, -631, -1143, 1982, 566, -4595, 1552, -1092, 1932, -2387, -1800, 2144, 1456, -3064, -1443, 2802, 2798, -1984, 3298, 1167, 1358, 2052, 3204, -800, -141, -740, 3636, 1901, 1771, -3816, 6035, -615, 5397, 1759, 582, 3726, 3579, -2694, 5431, 902, -1648, -3445, -5209, 2526, -5054, -4436, 1850, 4804, 1587, -6118, -8254, 3572, -6009, 1470, -8692, 516, -1211, -1056, 917, 1906, -7576, -903, -1148, -182, 308, -4086, -1073, 3114, 10140, 620, -3842, -329, -12208, 236, -9734, 2932, -1134, 1090, -551, 5101, 2104, 4104, 6042, 5433, -21220, 1600, 10210, -2293, 11796, -5628, -14718, 3702, -8816, -106, 360, 9792, -10196, -3866, 3633, -1940, 7806, 9372, 8304, 1383, -3424, 2633, 16206, 8564, 915, -2488, 5594, 6876, -5104, 8024, -2618, 8614, 26196, -4560, -2115, -1287, -19084, -4780, 1832, 12562, 9405, 432, 8462, -3908, 3700, -7508, 14807, -8794, -778, 9912, -16180, 5044, -9196, -6198, -3686, -13316, -6460, -394, 15678, -25465, -21926, -1776, -15111, 10344, 7658, 7768, -34380, -11160, 15279, 15060, -8704, -12781, -5464, -822, 29496, -8162, 10048, -9844, -2194, 9672, -29188, -658, 48, -19257, -10763, 11, 4780, -1001, 14255, 16964, -4034, -24196, 4278, 18972, -17010, 14804, 20456, -19687, 13105, -10440, -6362, -9004, -11095, 9948, 16732, 11058, -39496, -32244, -2259, -1569, 24934, 3932, 23044, -13436, 11896, -12524, 52296, 2107, -906, -5300, -18014, 24572, -12532, 10473, 9422, 16400, 37720, -3696, -752, 11, 4147, -30852, 8988, 1100, 22292, 7385, 39664, 15080, 29038, 1400, 33105, -17172, 19661, 8164, -11398, 7048, -34693, -19542, 3508, -5115, -11760, -1460, 35429, -3949, -31378, -20815, -22268, -264, 4180, 18086, -28393, -13498, -3145, 5270, -18287, 11308, 8884, 7896, -5946, -28520, 10630, -14432, 30972, 38500, -34872, -1833, -40020, 4727, -52852, -17569, 1566, 17496, -6297, 26655, 2785, 1886, 31826, -28712, -80490, 18942, 15164, -2340, 17135, 969, -42168, -16556, -63539, -196, 27498, 22171, -24407, -3578, 8046, -31449, 24222, -4386, 22506, 17320, -24126, -8845, 10250, -25040, 1133, 1372, -8602, 29946, -17438, 11961, -5386, 28978, 18284, 3604, -2679, -37092, -31801, 10735, -484, -49410, 53960, 16615, 19665, 24240, -9444, 29802, 25270, -5502, 37512, 16262, -14858, 4164, -18922, 13942, -15812, 7855, -34340, 34434, 27815, -19673, -36410, 29640, -36393, -8996, -1096, 21232, -48032, 23664, 65632, 38820, -6631, -29235, 9788, 35960, 38340, 26306, 36164, 25398, 50256, 24084, -72906, -4371, 37542, 22803, 9805, 5461, 22956, 12488, -28808, -3597, -27181, 6850, 31984, 23400, -32694, -1260, -6138, 4244, -45740, -63800, -30816, 9214, -27704, -7788, 68916, -1327, -34379, -14906, 16678, -53136, 26626, -43924, 14584, -608, -7514, 47895, -59632, -10411, -17236, -6748, -17348, -23198, 10458, 22176, 16206, 12052, 12742, -27452, -2820, 12188, -20760, 3780, 10613, -51170, 9355, -20132, -4956, 27111, 26142, 448, 1042, -25084, 37261, -49596, -23239, -35760, -10454, 15918, 1034, -46231, -35500, -6680, 10912, -21916, -48382, 49688, -54620, -19032, 32856, 17944, -40880, -28432, 55251, 55578, -16836, -37643, -2052, 12992, -6082, -28466, -44039, -2922, 40042, 52778, 11630, -3196, -30305, 22232, -29992, -49097, 79952, -61924, 10872, -21212, 1503, 34056, 13070, -15632, 966, -9856, -41700, -22080, 84121, -96668, 59242, 8718, -16726, 33644, 33092, -579, 54708, 2586, -4910, 14532, 9848, 14660, -14416, 44752, 13876, -67140, -3740, 16624, 85934, -4692, 36458, -13022, 36784, 25844, 74092, 29172, 3788, 11794, 893, -756, -429, 1882, -27975, 54236, 97288, -102323, 70772, -3368, 41014, 27904, 13730, 35906, 9400, 39682, 50940, -23444, 48435, -29276, -18572, 41092, -24464, -11324, -29233, -7920, 2786, 19022, 22242, 19904, -65196, -46246, 13139, 2456, -60, 3320, 24529, -35463, -63994, 6406, 31584, -50762, -13119, -72192, 11360, 22856, 5366, -5264, -69206, 25949, 31772, -7648, -74680, 7897, -19869, -49816, -52484, -34788, -18306, -59575, -2968, 8000, 34464, 40360, 1626, 3659, -53656, -93932, -27483, 33488, -10918, 25108, 24325, 36464, -103888, 41555, -59056, 41278, -60258, 18099, 72640, -46892, -41466, 17397, 28018, 18020, -101212, -46792, -14568, -18009, 29982, -49988, -92618, 89604, 13019, 46209, -14740, -31043, 79, -81556, -36688, 72532, 22611, 16616, -18866, -27736, 43800, 69908, -83088, 62018, -86665, -9948, 43219, 72882, -55302, 9931, 41950, -24506, -61224, -66457, 60832, -9576, 57717, 36622, 8700, 8772, -64624, -74402, 610, 34620, 4668, 352, 52734, 84702, -12112, 58990, -33368, -40602, -179500, -38192, -47032, 5922, 24619, 30026, 34621, 37317, 146830, -121209, 97474, 40424, -17544, -96776, -41830, 18133, 34830, 17751, 2400, 53444, -105817, 62059, 31862, 110420, 124504, -32840, -77258, 37497, 108196, 67902, 5085, 35065, 84748, 126426, 7010, 53804, 46838, -41659, -80946, 67200, 43420, 59130, -100732, 5698, -20820, 67948, 7760, 34878, -171924, 97886, 4089, 112647, -36364, 2768, 115044, -5196, -41637, -40456, -76544, 27792, 90356, -100798, -38948, 106852, -58976, -29220, 38499, 55179, 234120, -8718, 48110, -3549, -71144, -98344, -16426, 3121, 88244, -32544, 107172, -99152, -112472, -3776, 21081, -63892, 19999, 80754, -24082, -92680, 88910, 41416, -169502, 36652, -167142, -125466, -120622, 4000, -44126, 329, -26694, 10208, 15282, 45736, -25702, -131992, 68108, -57686, 44359, -156702, -128352, -105245, 38266, 33625, 12428, -51896, 73013, 29568, 9048, 32780, 161540, 18400, -85086, -66624, 73659, 34288, 10216, 53545, -122224, 41510, -69984, 45997, 25800, -79232, -174708, 76448, 33600, -147176, -54268, 56452, 36870, 9980, 60694, -90560, -141126, 62690, 22278, 28707, -11699, -30992, 30450, -6502, -58464, 45847, -122924, 39976, -964, -65150, -21388, -10246, -95116, -59310, 141407, -52205, 160336, 236, 160716, 131267, 162742, -162536, -266948, -60644, 48070, -69840, 162632, -48328, -2120, 29128, 103784, 47924, -69176, -64142, 78999, -9449, -145514, 61190, 28852, -88264, -9472, -68854, -169842, -13440, 190214, -3478, 32081, 102216, 179148, -95264, 101424, -111792, -93965, -81570, -159444, -96896, 9280, -110192, 183580, -71792, 45288, 20865, 216284, 33011, -39702, 504, -133206, 90510, 31768, 40695, 322692, 51360, 15056, 20317, 63080, 49544, -177506, 133505, 112510, 17537, -128446, -96073, 247128, -59242, -39676, 101390, 200082, -91344, 41444, 40530, -128158, 131050, 20304, 11427, -81683, 82612, 49163, -205256, 180144, 115564, -120552, 41032, 130890, 33040, 91793, 101406, -196323, -29400, 79105, -114016, 117231, 229774, -19510, 123808, 18168, -29744, -142895, 79124, 164306, 93733, 219693, -292530, -49366, -87238, 138889, -3198, -136076, -17032, 8845, -155944, 161738, 117412, -289926, 44336, -110095, 61092, -184246, -48836, 21140, 7520, 97992, 48672, 204104, -101789, -21120, -139428, 5163, 12992, 73304, -44202, 79484, -204597, 98804], "weight": 4, "weight_parity": 1, "id": null, "artin_degree": null, "artin_field": null, "artin_field_label": null, "artin_image": null, "embedded_related_objects": null, "nf_label": null, "projective_field": null, "projective_field_label": null, "projective_image": null, "projective_image_type": null, "qexp": [[0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0], [5, 0, 1, -1, 0, 0, 0, 0], [2, 1, -1, 1, 1, 0, 0, -1], [2, 1, -1, 1, -2, -3, -1, 2], [5, -1, -1, 1, -1, -1, 3, 0], [7, 3, 1, 0, 2, -1, -1, 1], [14, -7, -1, -3, 3, 3, -3, -2], [6, -7, 3, -1, -3, 0, 2, -5], [10, -1, 5, 3, -3, 0, -2, 3], [-1, -1, -1, -4, 5, 5, 0, 1], [16, -5, -1, -3, 1, 0, 0, 3], [-37, 9, -10, 0, 5, 3, 8, -4], [-14, -4, 2, 6, -10, 0, 2, 4], [-11, 6, 6, 3, -2, -2, -4, 5], [-1, -2, 0, 0, -2, -5, -10, -2], [-61, 8, 0, 14, -15, -9, -2, 2], [8, 3, -5, -11, 13, 6, 4, -1], [-60, 7, -7, -3, 11, 6, 4, -19], [-22, 11, -5, 1, 9, 4, 11, -4], [4, 11, 7, -7, 11, 0, -20, 9], [0, 2, 0, -8, -6, -12, 2, 2], [-42, 2, 11, 3, -7, -1, 8, 8], [53, -2, 14, 6, -6, -16, -2, 0], [-80, 31, -7, 9, -7, -4, 2, 19], [34, -39, 17, -17, 1, 26, -3, -16], [-9, -38, -7, -4, -16, -5, 13, -9], [-8, 17, 7, 13, -13, -8, 16, -5], [-48, -10, -28, -4, 22, 12, -10, 4], [32, -6, -26, 10, -8, -14, -4, 10], [34, -35, 17, -17, -5, 11, -18, 8], [68, 4, -8, 4, 18, -6, -16, 4], [58, -33, 17, -3, 6, 31, -13, -22], [-8, 48, -8, 16, 22, 10, -6, 14], [10, -48, -2, -7, 20, 33, 13, -21], [107, 22, -8, 28, -6, -23, 4, 12], [-20, 39, 7, 19, -43, -42, 24, 31], [28, 2, 20, -20, 18, 44, 16, -24], [66, -43, -15, 17, -1, -18, 22, -29], [44, -6, -12, -16, -8, 8, -2, -22], [25, -20, 3, -3, -21, -32, 31, -4], [142, 9, -1, -15, 3, 0, -28, 7], [132, 1, 51, -35, -13, -18, -40, 39], [94, 7, -11, -17, 5, -34, -30, 15], [44, 48, -16, -6, 36, 44, 22, -12], [-47, 0, 0, 0, 0, 0, 0, 0], [-9, 23, -1, 15, -2, -50, -16, 20], [64, 32, 2, 14, -48, -25, 32, 30], [10, 57, 34, -34, 70, 48, -22, -44], [-195, 37, 1, 39, -67, -11, -37, 22], [124, -39, 13, -13, -17, 12, -6, 47], [65, 45, -1, -27, 31, -7, 35, -12], [-225, 94, -39, 65, -7, -78, -33, -10], [-174, 28, -40, 68, 4, 40, 46, -26], [-154, -42, -45, 44, 10, 26, -6, -35], [80, -72, 54, -62, 20, 68, 58, -28], [170, 5, -19, -45, 77, 32, 18, -67], [-217, -9, 27, -23, -5, 39, -11, -6], [-148, -100, -12, 6, -58, -46, 8, 38], [173, -67, 11, 29, -17, 17, -25, 12], [-208, -44, -50, 8, -22, 78, 38, 4], [-120, -25, 39, -43, 39, 2, -23, 6], [-118, 55, -56, 22, 12, -14, -40, 34], [-278, -18, -74, 10, 26, -8, -14, 32], [-20, -22, 84, -6, -60, 2, -18, 22], [28, -21, 45, -5, 31, -4, -26, -21], [-109, 31, 11, 50, -15, -47, 14, -23], [-398, 122, -16, 12, -16, -24, 16, 12], [382, -100, 116, -46, -100, -74, -36, 96], [-75, -122, 2, -50, 10, -61, 18, -28], [-133, -53, -58, -22, 44, -57, 41, -28], [244, -98, 56, -64, -18, 34, -56, -16], [40, 21, 31, -65, 34, 73, 7, -20], [-561, 100, -24, 60, -4, -25, 12, 64], [208, 71, -23, -19, 73, 102, -6, 17], [-48, -208, 34, 94, -56, -24, -12, -26], [178, 130, 0, 42, -44, -180, -8, 6], [-389, 138, -34, -22, -6, -53, 72, -38], [-202, -79, -29, 47, -29, 30, 72, -47], [315, -85, -17, -67, 97, 108, 3, -8], [160, 42, -74, 42, -4, 28, 50, -86], [-300, -70, -30, 66, 38, 72, -28, -18], [-194, 61, -70, -20, 75, 76, 7, -70], [310, 4, 6, -90, 54, 0, -50, -52], [310, 127, 57, 7, -43, 16, -46, 53], [-216, 88, -112, 64, 22, -146, -4, 76], [362, 257, 71, 21, -3, 30, -8, 109], [183, -3, 23, 21, -9, -79, -111, 74], [148, 75, 77, -13, -9, 118, -2, 29], [236, -64, -14, -18, -38, -18, 52, -38], [390, 26, 30, 82, -110, -102, 34, 54], [-196, 186, 40, 88, -134, 48, 12, -52], [0, -47, 0, 0, 0, 0, 0, 0], [-70, -188, -4, -44, -60, -120, -10, 26], [476, -93, 24, -111, 122, 159, -79, -81], [191, 81, 11, -11, -69, 3, 21, 50], [200, 216, -133, -73, 152, 107, 41, 38], [-546, 21, -7, -33, -49, 92, 102, -69], [649, 6, 185, -57, -98, -124, -66, -20]]}