# Stored data for newform 405.4.b.e, downloaded from the LMFDB on 09 May 2024.
{"label": "405.4.b.e", "space_label": "405.4.b", "level": 405, "weight": 4, "hecke_orbit": 5, "hecke_orbit_code": 18014467296067989, "dim": 16, "is_polredabs": true, "trace_hash": 1274292470740273881, "analytic_rank": 0, "is_twist_minimal": false, "hecke_cutters": [[2, [614656, 0, 4069504, 0, 6555196, 0, 2881141, 0, 571975, 0, 59128, 0, 3268, 0, 91, 0, 1]], [11, [137637673416, -26757336852, -809472312, 158752116, 4235328, -160551, -3975, 45, 1]]], "qexp_display": "q+\\beta _{1}q^{2}+(-3+\\beta _{2})q^{4}+\\beta _{9}q^{5}+(-\\beta _{1}+\\cdots)q^{7}+\\cdots", "char_order": 2, "char_parity": 1, "char_degree": 1, "char_conductor": 5, "char_orbit_label": "b", "char_is_real": true, "Nk2": 6480, "analytic_conductor": 23.89577355232495, "hecke_ring_index": 104162436, "level_radical": 15, "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": false, "is_self_twist": false, "is_cm": false, "is_rm": false, "trace_zratio": {"__RealLiteral__": 0, "data": "0.507", "prec": 17}, "analytic_rank_proved": true, "char_values": [405, 2, [326, 82], [2, 1]], "hecke_ring_generator_nbound": 7, "inner_twist_count": 2, "level_primes": [3, 5], "conrey_indexes": [244], "field_disc": 19990255128347287155396854827528013224083456, "field_disc_factorization": [[2, 28], [3, 8], [1097, 2], [1583, 2], [1940066729, 2]], "field_poly": [614656, 0, 4069504, 0, 6555196, 0, 2881141, 0, 571975, 0, 59128, 0, 3268, 0, 91, 0, 1], "hecke_ring_index_factorization": [[2, 2], [3, 12], [7, 2]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.090", "prec": 17}, {"__RealLiteral__": 0, "data": "12.929", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "941.566", "prec": 24}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "107262.648", "prec": 34}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 16, "sato_tate_group": "3.2.3.c2", "trace_display": [0, 0, -3, 0], "traces": [0, 16, 0, 0, -54, -3, 0, 0, 0, 0, -10, -90, 0, 0, 102, 0, 146, 0, 0, -4, 6, 0, 0, 0, 0, -71, -468, 0, 0, 516, 0, 38, 0, 0, -212, -267, 0, 0, 0, 0, -44, -576, 0, 0, 1644, 0, -290, 0, 0, 4, -558, 0, 0, 0, 0, 15, -2430, 0, 0, 2202, 0, 20, 0, 0, 322, -339, 0, 0, 0, 0, -636, -2952, 0, 0, 4080, 0, -396, 0, 0, 218, 1266, 0, 0, 0, 0, 704, -6108, 0, 0, 4074, 0, -942, 0, 0, 1078, 1692, 0, 0, 0, 0, 594, -7134, 0, 0, 6288, 0, -1256, 0, 0, -958, 5658, 0, 0, 0, 0, 2285, -12414, 0, 0, 5304, 0, -1346, 0, 0, 576, 4932, 0, 0, 0, 0, -2550, -11898, 0, 0, 4770, 0, 3740, 0, 0, -3526, 13272, 0, 0, 0, 0, -201, -15492, 0, 0, 4812, 0, 3098, 0, 0, -996, 7353, 0, 0, 0, 0, -1188, -13452, 0, 0, -114, 0, 5686, 0, 0, -1814, 19812, 0, 0, 0, 0, 2847, -25764, 0, 0, 1944, 0, 986, 0, 0, 2306, 12840, 0, 0, 0, 0, -5024, -18822, 0, 0, -2520, 0, -5292, 0, 0, 4568, 31380, 0, 0, 0, 0, -3633, -29424, 0, 0, 9708, 0, 2414, 0, 0, 5062, 17733, 0, 0, 0, 0, 756, -25632, 0, 0, 6978, 0, -8444, 0, 0, 7940, 38784, 0, 0, 0, 0, 2501, -45348, 0, 0, 9798, 0, -2392, 0, 0, -2718, 20256, 0, 0, 0, 0, 9740, -37572, 0, 0, -3222, 0, -20358, 0, 0, -5184, 44976, 0, 0, 0, 0, -5380, -41400, 0, 0, 17682, 0, 7448, 0, 0, -13880, 22311, 0, 0, 0, 0, 16782, -37812, 0, 0, 12888, 0, 20868, 0, 0, 4692, 47694, 0, 0, 0, 0, -6063, -60816, 0, 0, 32490, 0, 14550, 0, 0, -2732, 17379, 0, 0, 0, 0, 9442, -45906, 0, 0, -6324, 0, -6120, 0, 0, -3378, 36378, 0, 0, 0, 0, -21153, -56892, 0, 0, 34476, 0, -17206, 0, 0, -12458, 13629, 0, 0, 0, 0, -12708, -39450, 0, 0, 24336, 0, 11860, 0, 0, 16220, 42168, 0, 0, 0, 0, -2592, -57582, 0, 0, 53184, 0, -12372, 0, 0, 27000, -2832, 0, 0, 0, 0, 1728, -56976, 0, 0, 11148, 0, -9502, 0, 0, 2732, 32880, 0, 0, 0, 0, 17349, -45960, 0, 0, 39636, 0, -38768, 0, 0, -35456, -2097, 0, 0, 0, 0, -14266, -49458, 0, 0, 35052, 0, 35994, 0, 0, -19042, 47946, 0, 0, 0, 0, 31301, -42816, 0, 0, 56790, 0, 45866, 0, 0, 14720, -16488, 0, 0, 0, 0, -12894, -55176, 0, 0, -2268, 0, 40050, 0, 0, 12674, 27240, 0, 0, 0, 0, 8856, -53346, 0, 0, 63300, 0, -16278, 0, 0, -12044, -9981, 0, 0, 0, 0, -61308, -47892, 0, 0, 2034, 0, -50180, 0, 0, -28212, 35172, 0, 0, 0, 0, -5524, -37464, 0, 0, 68586, 0, 26784, 0, 0, 49086, -29325, 0, 0, 0, 0, 4628, -32670, 0, 0, -45192, 0, -22160, 0, 0, 41618, 25146, 0, 0, 0, 0, 6009, 23964, 0, 0, 73116, 0, -36288, 0, 0, 32848, -21387, 0, 0, 0, 0, 45156, -23994, 0, 0, -5424, 0, -81272, 0, 0, -51588, 28380, 0, 0, 0, 0, -9358, 7710, 0, 0, 30708, 0, 41018, 0, 0, -68940, -43812, 0, 0, 0, 0, 78234, -14568, 0, 0, -83340, 0, 84780, 0, 0, 8994, 1404, 0, 0, 0, 0, -18811, 86478, 0, 0, 20838, 0, 31178, 0, 0, 10122, -44469, 0, 0, 0, 0, 24678, 43716, 0, 0, -50028, 0, -23060, 0, 0, 2416, 37302, 0, 0, 0, 0, -59208, 59826, 0, 0, 51558, 0, -38194, 0, 0, -53568, -61356, 0, 0, 0, 0, -13382, 3138, 0, 0, -176466, 0, 27240, 0, 0, 57206, 22008, 0, 0, 0, 0, 6229, 143616, 0, 0, 44544, 0, -15736, 0, 0, 118492, -67842, 0, 0, 0, 0, -6508, 80784, 0, 0, -151242, 0, -34876, 0, 0, 30810, 41886, 0, 0, 0, 0, 45993, 165246, 0, 0, 57306, 0, -72646, 0, 0, -98710, -100824, 0, 0, 0, 0, -51744, -1722, 0, 0, -220080, 0, 47898, 0, 0, -90642, -43692, 0, 0, 0, 0, 80891, 148722, 0, 0, -16992, 0, 87266, 0, 0, 18340, -68775, 0, 0, 0, 0, -97092, 82818, 0, 0, -121668, 0, 133216, 0, 0, -25684, 36036, 0, 0, 0, 0, -1203, 217944, 0, 0, 52800, 0, -23946, 0, 0, 14346, -102993, 0, 0, 0, 0, -124524, 50988, 0, 0, -195888, 0, -24390, 0, 0, -88924, -46872, 0, 0, 0, 0, -29985, 171876, 0, 0, 9438, 0, 13070, 0, 0, 163428, -70341, 0, 0, 0, 0, 62768, 56460, 0, 0, -232536, 0, -52040, 0, 0, 99380, 18162, 0, 0, 0, 0, 13733, 216732, 0, 0, 62148, 0, -36936, 0, 0, 127868, -150507, 0, 0, 0, 0, 115534, 81258, 0, 0, -254232, 0, -201960, 0, 0, -89240, -80508, 0, 0, 0, 0, -40407, 317508, 0, 0, 62868, 0, 46796, 0, 0, -228096, -127014, 0, 0, 0, 0, 149418, 103524, 0, 0, -134892, 0, 99708, 0, 0, 25034, 22368, 0, 0, 0, 0, -70003, 189228, 0, 0, 39666, 0, 65884, 0, 0, -86472, -213276, 0, 0, 0, 0, -39080, 72840, 0, 0, -202806, 0, -15538, 0, 0, 38954, -61716, 0, 0, 0, 0, -116569, 292596, 0, 0, 56922, 0, -38046, 0, 0, -166360, -161952, 0, 0, 0, 0, 55848, 162786, 0, 0, -151176, 0, 44818, 0, 0, 171990, -58308, 0, 0, 0, 0, 108108, 230874, 0, 0, 126858, 0, 26560, 0, 0, 213320, -233712, 0, 0, 0, 0, -16806, 15234, 0, 0, -352752, 0, -141390, 0, 0, 85352, -140478, 0, 0, 0, 0, 95472, 329064, 0, 0, 187602, 0, -66988, 0, 0, -143312, -171372, 0, 0, 0, 0, -47520, 134364, 0, 0, -102564, 0, -21192, 0, 0, -220536, -36300, 0, 0, 0, 0, 107907, 321816, 0, 0, 201990, 0, 74682, 0, 0, 2142, -240507, 0, 0, 0, 0, -167010, -38784, 0, 0, -229380, 0, 154978, 0, 0, -111564, -81516, 0, 0, 0, 0, -74212, 84168, 0, 0, 30126, 0, -65272, 0, 0, 92664, -174912, 0, 0, 0, 0, -162884], "weight_parity": 1, "has_non_self_twist": 1, "level_is_prime": false, "level_is_prime_power": false, "level_is_square": false, "level_is_squarefree": false, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 5, 2, 1, 2, 0]], "char_orbit_index": 2, "prim_orbit_index": 2, "minimal_twist": "45.4.j.a", "char_is_minimal": true, "conrey_index": 244, "id": null, "nf_label": null, "fricke_eigenval": null, "atkin_lehner_string": null, "projective_image": null, "projective_image_type": null, "artin_degree": null, "artin_image": null, "artin_field_label": null, "projective_field_label": null, "atkin_lehner_eigenvals": null, "projective_field": null, "artin_field": null, "embedded_related_objects": null, "qexp": [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1], [5, 0, -2, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -3, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, -1, -2, -1, 1, 1, 0, 1, 0, 0, 0], [-1, 1, 0, 1, 1, -1, -1, -1, 0, -2, 0, 0, -1, 3, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 3, 0, 0, 0, 0, -3, -3, 0, 1, 0, 1, 1, 0], [0, 0, 0, -1, 0, 0, 0, 0, -1, -1, 2, 1, -2, -3, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 2, 0, -3, 0, 1, -1, 1, -3, 0, 0, -1, -1, -1, 2, 0], [-30, 0, 6, 0, -2, -2, -1, 0, 1, -1, -3, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 19, 0, -2, 0, 0, 0, 0, -6, -6, 1, 2, -1, 1, -2, 1], [28, 0, -7, 0, -1, -1, 1, -1, 0, 0, -2, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -2, -5, -2, -1, -4, 4, -2, 0, -2, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0, 1, 1, 1, -5, -1, 0, 3, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 5, 0, -3, 1, 2, 1, -9, 9, 0, 0, 0, 0, 0, 0], [-12, 4, 10, -1, 0, 2, 3, 2, -1, 0, 1, -2, -2, 3, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -23, 0, 6, 0, 0, 0, 0, -6, -6, -1, -7, 1, 1, 2, 0], [0, -4, 0, -2, 0, 0, 0, 0, 2, 2, 1, -2, -1, -9, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -7, 0, -2, 1, 5, 3, 0, -13, 1, 4, -1, 3, 2, -5, 1], [-38, 0, -2, 0, 4, -1, 2, -3, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 35, 0, 1, 0, 0, 0, 0, -9, -9, -1, -3, 1, 2, 2, 1], [94, 0, -29, 0, 4, 2, -1, -3, -1, 1, 3, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 6, 0, -3, 7, 5, 7, -18, 18, 2, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 3, 5, -3, 6, 8, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -3, 0, 2, 5, 2, 4, -11, 11, 0, 0, 0, 0, 0, 0], [-25, 2, 25, 2, -5, 1, -1, 1, -3, -1, -1, 9, -1, 9, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -46, 0, 9, 0, 0, 0, 0, -18, -18, 1, 12, -1, 8, -18, 0], [0, -1, 0, -3, 0, 0, 0, 0, 1, 1, 2, -5, -2, -12, 5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -27, 0, -7, -1, -7, -1, -2, -11, -4, 0, 13, -6, 0, -4, -1], [-136, 0, 20, 0, -1, 7, 4, 9, -7, 7, 3, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 92, 0, 0, 0, 0, 0, 0, -21, -21, -1, -5, 1, 5, -11, 6], [125, 0, -27, 0, -3, 1, -3, -1, 3, -3, -2, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, -3, 0, 6, 1, 2, -8, -9, 9, 5, 0, 5, 0, 0, 0], [0, 5, 0, 6, 0, 0, 0, 0, -1, -1, -4, 13, 4, 6, -31, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [20, 0, -9, 0, -1, -7, -1, 4, -20, 20, -4, 0, -4, 0, 0, 0], [-16, 5, 30, -5, -1, -6, -2, -1, 11, 2, -5, -8, -2, -6, -2, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -22, 0, -5, 0, 0, 0, 0, -12, -12, 1, 7, -1, -7, 18, 1], [0, -23, 0, 7, 0, 0, 0, 0, 3, 3, -9, 11, 9, -3, -1, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-29, -82, 15, 8, -6, -2, -1, -2, -21, 3, -2, -12, 4, 5, 16, -1], [-191, 0, -2, 0, 2, -9, -8, -3, -1, 1, -3, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 89, 0, 6, 0, 0, 0, 0, -3, -3, -5, 7, 5, -7, 7, -6], [235, 0, -58, 0, 1, -5, -4, 2, -1, 1, 4, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, -3, 0, 1, 5, -6, -3, -19, 19, -3, 0, -3, 0, 0, 0], [0, 1, 0, 8, 0, 0, 0, 0, -5, -5, -3, -31, 3, 15, 2, 6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, -8, 0, 3, -3, -9, 0, -3, 3, -6, 0, -6, 0, 0, 0], [95, 4, 20, 4, 5, 7, -2, -3, 4, 4, 3, -17, 18, 3, 19, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -46, 0, -16, 0, 0, 0, 0, -21, -21, 7, -35, -7, -10, 17, -1], [0, -22, 0, 13, 0, 0, 0, 0, -11, -11, -2, 3, 2, 15, -19, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [35, -123, -10, -3, 4, -10, -3, 0, 8, 4, -5, -14, 2, -12, -5, -6], [-367, 0, 35, 0, 1, -3, -4, -3, 9, -9, 9, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 208, 0, -16, 0, 0, 0, 0, -18, -18, 5, -6, -5, -11, 28, -7], [250, 0, -23, 0, -11, 5, 11, 15, -1, 1, -3, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-54, 0, 12, 0, 5, 4, -9, -9, 4, -4, 5, 0, 5, 0, 0, 0], [0, -20, 0, 1, 0, 0, 0, 0, 15, 15, -15, 11, 15, -21, 29, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [66, 0, -12, 0, -1, -15, -2, -13, -17, 17, 3, 0, 3, 0, 0, 0], [117, 26, 25, -14, 9, 2, 0, 2, -3, -1, 5, 29, 0, -24, -11, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -25, 0, 0, 0, 0, 15, 15, -7, 22, 7, -23, -34, -1], [0, 39, 0, -11, 0, 0, 0, 0, -7, -7, -9, -7, 9, 15, 29, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [38, -181, -20, 24, 11, 3, -5, -7, -17, -1, 5, 21, 15, -1, -9, 6]]}