# Stored data for newform 363.3.h.n, downloaded from the LMFDB on 10 July 2025. {"Nk2": 3267, "analytic_conductor": 9.891033596278778, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 33, "char_degree": 4, "char_is_minimal": false, "char_is_real": false, "char_orbit_index": 8, "char_orbit_label": "h", "char_order": 10, "char_parity": -1, "char_values": [363, 10, [122, 244], [5, 8]], "cm_discs": [], "conrey_index": 245, "dim": 16, "field_disc": 556599580127587562500000000, "field_disc_factorization": [[2, 8], [5, 12], [11, 2], [29, 4], [101, 4]], "field_poly": [1771561, 0, -536393, 0, 239275, 0, -96398, 0, 25179, 0, -2562, 0, 180, 0, -12, 0, 1], "field_poly_is_cyclotomic": false, "field_poly_is_real_cyclotomic": false, "field_poly_root_of_unity": 0, "has_non_self_twist": 1, "hecke_cutters": [[2, [1771561, 0, -875798, 0, 245930, 0, -48408, 0, 18829, 0, -1372, 0, 175, 0, -17, 0, 1]], [5, [7929856, 0, -28971008, 0, 42508800, 0, -12321408, 0, 1364209, 0, 35613, 0, 3020, 0, 38, 0, 1]], [7, [156816, -140184, 72504, -23148, 5629, -861, 151, -17, 1]]], "hecke_orbit": 14, "hecke_orbit_code": 58547276242485611, "hecke_ring_generator_nbound": 3, "hecke_ring_index": 1, "hecke_ring_index_factorization": [], "hecke_ring_index_proved": true, "inner_twist_count": 4, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 3, 2, -1, 2, 0], [1, 1, 11, 3, 1, 5, 0], [1, 1, 33, 8, -1, 10, 0]], "is_cm": false, "is_largest": false, "is_maximal": false, "is_polredabs": true, "is_rm": false, "is_self_dual": false, "is_self_twist": false, "is_twist_minimal": false, "label": "363.3.h.n", "level": 363, "level_is_powerful": false, "level_is_prime": false, "level_is_prime_power": false, "level_is_prime_square": false, "level_is_square": false, "level_is_squarefree": false, "level_primes": [3, 11], "level_radical": 33, "minimal_twist": "33.3.h.b", "prim_orbit_index": 8, "qexp_display": "q+\\beta _{4}q^{2}+(-\\beta _{2}+\\beta _{6}+\\beta _{13})q^{3}+(1+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 4, "rm_discs": [], "sato_tate_group": "2.2.3.c10", "self_twist_discs": [], "self_twist_type": 0, "space_label": "363.3.h", "trace_display": [0, 5, 0, 34], "trace_hash": 364581127300126549, "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "15.158", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "1654.562", "prec": 27}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "414205.233", "prec": 34}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.508", "prec": 17}, "traces": [0, 16, 0, 5, 18, 0, -32, 34, 0, 17, 12, 0, 106, 2, 0, -28, 102, 0, -42, -66, 0, 12, 0, 0, 74, -176, 0, -55, 146, 0, -110, -126, 0, 0, -132, 0, 226, -230, 0, -136, -226, 0, -72, 156, 0, -72, -308, 0, 255, 170, 0, 169, 224, 0, -1046, 0, 0, 259, 184, 0, -316, -104, 0, 108, -184, 0, 0, 368, 0, -22, -52, 0, -73, -354, 0, 54, 900, 0, -492, -566, 0, 377, 200, 0, 720, 162, 0, -132, 0, 0, 774, -226, 0, -370, 530, 0, 67, 252, 0, 0, -200, 0, -71, -460, 0, 114, -658, 0, 15, 1496, 0, 88, -472, 0, -171, 440, 0, 324, -772, 0, -848, 0, 0, 427, -518, 0, -456, -452, 0, 116, 1374, 0, 0, 372, 0, 176, -76, 0, -616, 492, 0, 636, 460, 0, 474, 12, 0, 511, 686, 0, 339, 164, 0, 190, 0, 0, -46, 8, 0, -160, 870, 0, -1023, -1060, 0, 0, 536, 0, 480, 162, 0, -242, 650, 0, 66, -1108, 0, 196, 4, 0, 408, -1266, 0, -930, 1782, 0, -728, 0, 0, 390, -582, 0, 306, 602, 0, 416, -1078, 0, 0, -3088, 0, 1267, 42, 0, 77, -78, 0, -594, 764, 0, -36, 62, 0, -174, 972, 0, -878, -1124, 0, -134, 0, 0, -1706, -238, 0, -237, -638, 0, 701, -996, 0, 0, 640, 0, -1130, -694, 0, -174, -1742, 0, -966, -2380, 0, -500, 738, 0, -2462, -860, 0, -450, 798, 0, 2472, 0, 0, -766, 2262, 0, 1421, -1378, 0, -82, 936, 0, 0, 1632, 0, -552, 1638, 0, 720, 1558, 0, 690, -3952, 0, 660, -330, 0, 1118, -936, 0, 134, -1184, 0, 2280, 0, 0, 119, -212, 0, 677, 60, 0, -417, -346, 0, 0, 596, 0, -1839, 178, 0, 1848, 2008, 0, -1094, 288, 0, 828, 3328, 0, -884, 1504, 0, 660, -3336, 0, 12, 0, 0, 707, 1914, 0, 1356, -1320, 0, 810, 3206, 0, 0, 1196, 0, -1250, 2336, 0, 84, -1528, 0, 1335, 1084, 0, -1234, 178, 0, 462, -3550, 0, -288, -738, 0, 1520, 0, 0, -3203, 396, 0, 636, -1370, 0, -802, -618, 0, 0, 1812, 0, -822, -1474, 0, 2191, 1440, 0, -756, -3136, 0, -1140, 1578, 0, -2946, 158, 0, -984, 4974, 0, 1654, 0, 0, -2043, -1860, 0, 1088, 638, 0, 687, -150, 0, 0, 240, 0, 1494, -762, 0, -1945, 908, 0, 1372, 1600, 0, -2084, 1276, 0, 2212, -3284, 0, -1936, -114, 0, 3704, 0, 0, -1308, 22, 0, 1608, -1762, 0, -576, 1364, 0, 0, -5828, 0, 150, -184, 0, -20, 808, 0, -1889, 600, 0, 1370, -522, 0, -1196, -48, 0, -984, 498, 0, 606, 0, 0, 1856, 2598, 0, -1961, 2846, 0, 2827, -3586, 0, 0, -2076, 0, -22, 5150, 0, 2378, 1190, 0, 1074, -6760, 0, 3920, -1372, 0, -18, 758, 0, -652, 2234, 0, -1716, 0, 0, -1622, -866, 0, -1797, 1640, 0, -1608, -132, 0, 0, -4832, 0, -1164, -1384, 0, -2528, 2746, 0, 684, -5420, 0, 2186, 2676, 0, -572, -1232, 0, 175, -1546, 0, 1914, 0, 0, 578, 174, 0, -102, -1680, 0, -1056, 5198, 0, 0, 1072, 0, 3891, 3274, 0, -1369, -1666, 0, 341, 6016, 0, -1828, -172, 0, 0, -678, 0, -168, -3992, 0, -348, 0, 0, -220, -2110, 0, 1814, 464, 0, 822, -4110, 0, 0, 8772, 0, 1406, -2758, 0, -312, -1382, 0, -1902, -2980, 0, -2760, -1288, 0, -1408, 6226, 0, -220, -544, 0, -858, 0, 0, 672, -2426, 0, 648, 1806, 0, 262, -1460, 0, 0, -1456, 0, -2714, -3894, 0, -876, 1644, 0, 1666, 12424, 0, 1284, -1126, 0, -552, 2582, 0, 2953, 1904, 0, -3064, 0, 0, 4584, -1306, 0, -3366, 1320, 0, -1332, 5858, 0, 0, -368, 0, -1464, -2904, 0, 2099, -8186, 0, 512, 4968, 0, 2628, -5398, 0, 5200, -1150, 0, 780, -1036, 0, -9042, 0, 0, 1848, 2966, 0, -2442, -5622, 0, -2183, -4874, 0, 0, 4428, 0, -1342, -2596, 0, -2762, -528, 0, 102, 2136, 0, -24, -3208, 0, 1590, 3658, 0, 6387, -2054, 0, -3332, 0, 0, 3960, 1690, 0, 2450, 1332, 0, -1474, -6214, 0, 0, 4300, 0, 2216, -718, 0, 3085, -4606, 0, -2146, 3484, 0, -5132, 4374, 0, 5029, 1190, 0, 838, 3380, 0, -5928, 0, 0, 216, -2592, 0, -1464, 2432, 0, 2583, 6142, 0, 0, 660, 0, 1781, 2572, 0, -2184, -508, 0, -3052, -4268, 0, -7100, -5932, 0, 1314, 286, 0, 1186, 522, 0, 2076, 0, 0, 3258, 1046, 0, -2626, 556, 0, 2100, 726, 0, 0, 4412, 0, 3552, 1264, 0, -2446, 3490, 0, 988, 2380, 0, 402, -3804, 0, -931, -1194, 0, 1452, 3442, 0, -6900, 0, 0, -1046, -3110, 0, -5031, 1374, 0, -1158, -3342, 0, 0, -1788, 0, 1808, -9732, 0, 2268, -4414, 0, -817, 3444, 0, 5024, 2508, 0, -1284, 4642, 0, 1984, -1574, 0, -3084, 0, 0, 783, -528, 0, -282, -742, 0, 266, 1254, 0, 0, -16300, 0, -2926, 7188, 0, -920, -6630, 0, 4868, 5708, 0, 3700, -10994, 0, -1914, -3500, 0, 526, -5436, 0, -500, 0, 0, 3716, 4814, 0, 126, -1214, 0, 1048, -2186, 0, 0, -7596, 0, 0, -4618, 0, -5209, 668, 0, -215, -5916, 0, -664, 4492, 0, 1809, 5390, 0, -6446, 8774, 0, -984, 0, 0, 1241, -322, 0, -5100, 9292, 0, 2736, -964, 0, 0, -3020, 0, 4680, 906, 0, -3740, 5456, 0, 1665, 1740, 0, -1824, -1852, 0, -2970, -3106, 0, -3744, -1032, 0, 5208, 0, 0, 222, -13268, 0, -500, 3478, 0, -6242, 4834, 0, 0, 4684, 0, 840, 1168, 0, 4470, 4608, 0, -84, 8148, 0, 2836, 3768, 0, -203, -3426, 0, -294, -7194, 0, 6852, 0, 0, -2188, 1980, 0, 2882, 690, 0, -1916, -1316, 0, 0, 13456, 0, -1560, 5550, 0, -247, 10276, 0, 3630, -1084, 0, -2554, 3554, 0, 4024, 4834, 0, 3324, 7732, 0, 5964, 0, 0, -48, -3600, 0, 3491, 4462, 0, 2970, 5598, 0, 0, -8444, 0, -2464, -964, 0, -1329, 262, 0, 900, 3712], "weight": 3, "weight_parity": -1, "id": null, "artin_degree": null, "artin_field": null, "artin_field_label": null, "artin_image": null, "atkin_lehner_eigenvals": null, "atkin_lehner_string": null, "embedded_related_objects": null, "fricke_eigenval": 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, -1, 1, 0, 0, -1, 3, 0, -2, 0, 1, 0, -1, 1, 1, -1], [0, 2, 0, 0, -2, 2, 0, 1, 1, 1, -1, 1, -1, -1, 0, 0], [-3, 2, -2, -3, 0, 3, 0, 3, 2, 1, -2, 0, -1, 0, 0, 2], [2, 0, 2, 0, 0, 0, 3, 0, 0, 1, 1, 0, -1, 1, 0, 0], [0, 2, 0, 0, 2, -1, 0, 2, 2, 1, 0, 1, 2, 2, -1, -1], [-1, 0, -3, -3, 0, 0, 1, -3, 0, 1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 3, 0, -1, -3, -4, 2, 0, 1, 0, 2, -2, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -2, 0, -7, -4, 4, 7, 0, -2, 4, -1, -3, -6, -1, 0, -3], [3, 0, 5, 5, 0, 0, -3, -3, 3, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 3, -5, 0, 3, 3, -2, 0, 5, 1, 1, 2, 2], [-5, -2, -5, 0, 0, -6, 9, 0, -2, -5, 2, 3, 1, 1, 3, 1], [11, -1, 5, 11, 0, -1, 0, 2, -4, 3, 4, 0, -3, 3, 1, -1], [0, 0, 0, 0, -1, -1, 0, -1, -1, 0, 1, -4, 0, 0, -1, -1], [-6, -2, -6, 0, 0, 9, 2, 0, 7, -1, -1, -3, -2, -10, -6, 1], [0, -1, 9, -5, 0, -1, -9, -1, -1, -1, 0, 0, -2, 2, 1, -1], [0, -1, 0, 0, 2, -1, 0, -3, -3, -5, 5, 0, 0, 0, 0, 0], [-3, -3, 0, -9, 0, 0, 9, -3, -3, 0, 3, 0, -3, 3, 3, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 5, -2, 0, -4, -4, -5, -1, -3, 1, 1, 6, 6], [6, 1, -1, -1, 8, 1, -6, 0, 4, 2, -7, 0, 0, 0, 0, 0], [0, 1, 15, 13, 0, 1, -15, 1, 1, 1, 0, 0, -1, 1, -1, 1], [0, -1, 0, 0, 0, 2, 0, 0, 0, -5, 3, -4, -7, -7, 2, 2], [-7, 10, -6, -7, -1, -8, 0, 0, 1, -9, -1, 9, 9, 8, 12, 9], [23, -5, 30, 23, 0, -5, 0, 0, -10, 5, 10, 0, -5, 5, 5, -5], [0, -2, 0, 0, 0, 4, 0, 0, 0, 2, 0, -2, -2, -2, -2, -2], [0, -3, 13, 5, 8, -20, -13, -3, -3, -12, 0, 9, 17, 11, 12, 8], [-9, 0, 6, 6, 0, 0, 9, -5, 5, 5, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, -19, 0, -4, -4, -13, 5, 5, 9, 9, 8, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 3, 0, 26, 0, 3, -26, 5, 1, 0, -3, 0, 1, -1, -3, 3], [0, -8, 0, 0, -3, -8, 0, -3, -3, -4, 4, 0, 0, 0, 0, 0], [0, -9, -23, -16, -10, 7, 23, -9, -9, -3, 0, -6, -7, -4, 3, -10], [-16, 0, -16, 0, 0, 0, -5, 0, 0, 9, 9, 0, -9, 9, 0, 0], [0, -4, 0, 0, -9, 23, 0, 5, 5, 18, -5, -20, -18, -18, -13, -13], [-15, -11, -10, -15, 12, -21, 0, -9, -11, -10, 11, 9, 10, 12, 9, 1], [-27, 3, -27, 0, 0, 3, 20, 0, 6, 0, -3, 0, 3, -3, -3, 3], [0, -19, 0, 0, 10, -8, 0, -19, -19, 2, 0, -21, -3, -3, -2, -2], [-21, -9, -24, -24, 12, -9, 21, 6, 6, -6, 0, 0, 0, 0, 0, 0], [9, 5, 0, -6, 0, 5, 6, 12, -2, 0, -5, 0, -2, 2, -5, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, -8, 0, -14, -8, -1, 14, -3, 1, 8, 2, -6, -7, -7, -6, -10], [-44, 0, -33, -33, 0, 0, 44, 0, 0, -11, -11, 0, 0, 0, 0, 0], [0, -14, 0, 0, 23, -25, 0, -14, -14, -2, 0, -12, 5, 5, 2, 2], [12, -3, 12, 0, 0, 6, 33, 0, -3, 12, 0, -9, -9, 6, 3, 0], [12, -5, -8, 12, 0, -5, 0, -1, -9, 4, 9, 0, -4, 4, 5, -5], [0, 18, 0, 0, -21, 11, 0, 4, 4, 7, -4, -2, -7, -7, -3, -3], [14, 9, 14, 0, 0, 6, -1, 0, 0, 7, -3, 3, 6, 1, -9, -9], [0, -7, -25, -2, 0, -7, 25, -7, -7, -7, 0, 0, 5, -5, 7, -7], [0, -3, 0, 0, -7, -3, 0, 0, 0, -9, 9, 0, 0, 0, 0, 0], [-65, -7, 0, 3, 0, 9, -3, 3, 2, 0, 1, -6, 2, -8, -12, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 38, 0, 0, 9, -26, 0, 14, 14, -9, -1, 37, 23, 23, 10, 10], [13, -15, -15, -15, 12, -15, -13, -6, -15, -16, 12, 0, 0, 0, 0, 0], [0, -2, -16, -18, 0, -2, 16, -2, -2, -2, 0, 0, -2, 2, 2, -2], [0, 28, 0, 0, 0, 3, 0, 0, 0, 10, 1, 8, 7, 7, -11, -11], [-15, 13, 23, -15, -3, 0, 0, 12, -2, 2, 2, 6, -2, 12, -6, 10], [-17, 2, -15, -17, 0, 2, 0, -9, 13, -11, -13, 0, 11, -11, -2, 2], [0, 1, 0, 0, 0, -35, 0, 0, 0, -30, 20, 25, 5, 5, 10, 10], [0, -9, 0, -33, -6, -3, 0, -9, -9, -9, 0, 0, -6, 0, 9, 3], [1, 0, 15, 15, 0, 0, -1, 10, -10, -14, -14, 0, 0, 0, 0, 0], [0, -10, 0, 0, -16, 7, 0, 1, 1, 16, -4, -14, -15, -15, -12, -12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [38, -12, 0, 21, 0, -12, -21, -9, -15, 0, 12, 0, -15, 15, 12, -12], [0, -2, 0, 0, -11, -2, 0, -12, -12, -5, 5, 0, 0, 0, 0, 0], [0, 18, 11, -11, -17, 29, -11, 18, 18, 12, 0, 6, -2, -2, -12, 10], [11, 5, 11, 0, 0, 5, -55, 0, 10, -3, -8, 0, 8, -8, -5, 5], [0, -16, 0, 0, 6, -8, 0, -9, -9, 1, 9, -5, -1, -1, -10, -10], [-32, 7, -80, -32, 10, 14, 0, 15, 16, -2, -16, 3, 2, 1, 6, 17], [-18, 3, -18, 0, 0, 3, -38, 0, 6, 2, -1, 0, 1, -1, -3, 3], [0, 27, 0, 0, -39, 53, 0, 27, 27, 14, 0, 13, -23, -23, -14, -14], [9, -15, 9, 9, 18, -15, -9, -3, -15, -3, 18, 0, 0, 0, 0, 0], [53, -11, 0, -6, 0, -11, 6, -17, -5, 0, 11, 0, -5, 5, 11, -11], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-51, -7, 0, -43, -1, 13, 43, 18, 2, 1, 4, -3, 1, -5, -27, -5], [-49, 0, -1, -1, 0, 0, 49, 9, -9, 8, 8, 0, 0, 0, 0, 0], [0, 6, 0, 0, -21, 17, 0, 6, 6, -4, 0, 10, -13, -13, 4, 4], [34, 8, 34, 0, 0, -30, -19, 0, 17, -21, 10, 30, 20, 9, 0, 17], [5, 24, -37, 5, 0, 24, 0, 14, 34, -10, -34, 0, 10, -10, -24, 24], [0, -5, 0, 0, -3, 2, 0, -3, -3, 5, 3, 5, -5, -5, -8, -8], [57, -30, 57, 0, 0, 0, 36, 0, -30, 18, 15, -15, -30, 18, 15, -15], [0, 7, -28, 7, 0, 7, 28, 7, 7, 7, 0, 0, -8, 8, -7, 7], [0, 22, 0, 0, -9, 22, 0, 11, 11, 33, -33, 0, 0, 0, 0, 0], [-6, -2, 0, 4, -2, 8, -4, 0, -2, 2, -4, -6, -4, -4, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -9, 29, 0, 3, 3, 9, -4, -3, -6, -6, -5, -5], [95, -22, 52, 52, 10, -22, -95, -9, 5, -1, 19, 0, 0, 0, 0, 0], [0, -3, 28, -7, 0, -3, -28, -3, -3, -3, 0, 0, -4, 4, 3, -3], [0, -32, 0, 0, 0, 23, 0, 0, 0, 19, -17, -21, -4, -4, -2, -2], [-19, -11, 45, -19, 11, -26, 0, -15, -11, -15, 11, 15, 15, 11, 30, 0], [82, 13, 95, 82, 0, 13, 0, 25, 1, 12, -1, 0, -12, 12, -13, 13], [0, 18, 0, 0, 0, 1, 0, 0, 0, 19, -23, -5, 18, 18, 4, 4], [0, 24, 10, -41, 4, 8, -10, 24, 24, 12, 0, 12, 1, 14, -12, 10], [46, 0, 41, 41, 0, 0, -46, 2, -2, 8, 8, 0, 0, 0, 0, 0], [0, 22, 0, 0, 24, -30, 0, 7, 7, -24, 16, 38, 31, 31, 8, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 10, 0, 62, 0, 10, -62, -8, 28, 0, -10, 0, 28, -28, -10, 10]]}