# Stored data for newform 225.4.k.b, downloaded from the LMFDB on 24 April 2024. {"label": "225.4.k.b", "space_label": "225.4.k", "level": 225, "weight": 4, "hecke_orbit": 2, "hecke_orbit_code": 4504286889246945, "dim": 8, "is_polredabs": true, "nf_label": "8.0.303595776.1", "trace_hash": 1397102744137819071, "analytic_rank": 0, "is_twist_minimal": false, "hecke_cutters": [[2, [1296, 0, -756, 0, 405, 0, -21, 0, 1]]], "qexp_display": "q+(\\beta _{1}-\\beta _{7})q^{2}+(\\beta _{4}-\\beta _{5}-\\beta _{7})q^{3}+\\cdots", "char_order": 6, "char_parity": 1, "char_degree": 2, "char_conductor": 45, "char_orbit_label": "k", "char_is_real": false, "Nk2": 3600, "analytic_conductor": 13.275429751291638, "hecke_ring_index": 9, "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": "0.508", "prec": 17}, "analytic_rank_proved": true, "char_values": [225, 6, [101, 127], [2, 3]], "hecke_ring_generator_nbound": 7, "inner_twist_count": 4, "level_primes": [3, 5], "conrey_indexes": [49, 124], "field_disc": 303595776, "field_disc_factorization": [[2, 8], [3, 4], [11, 4]], "field_poly": [81, 0, 45, 0, 16, 0, 5, 0, 1], "hecke_ring_index_factorization": [[3, 2]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.059", "prec": 17}, {"__RealLiteral__": 0, "data": "8.042", "prec": 17}, {"__RealLiteral__": 0, "data": "6.472", "prec": 17}, {"__RealLiteral__": 0, "data": "538.435", "prec": 24}, {"__RealLiteral__": 0, "data": "1144.371", "prec": 27}, {"__RealLiteral__": 0, "data": "60650.873", "prec": 30}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 4, "sato_tate_group": "3.2.3.c6", "trace_display": [0, 0, 0, 0], "traces": [0, 8, 0, 0, 10, 0, 18, 0, 0, -90, 0, -132, 0, 0, 120, 0, 14, 0, 0, 308, 0, 42, 0, 0, 198, 0, -1056, 0, 0, -102, 0, -86, 0, 0, 594, 0, -450, 0, 0, -1518, 0, -264, 0, 0, 924, 0, -1056, 0, 0, -1026, 0, 594, 0, 0, -2430, 0, -132, 0, 0, 1596, 0, -878, 0, 0, 2908, 0, -1980, 0, 0, -1782, 0, 5472, 0, 0, 1632, 0, 3058, 0, 0, -1606, 0, -1134, 0, 0, -1284, 0, -66, 0, 0, 1584, 0, -3124, 0, 0, 4200, 0, 2160, 0, 0, -594, 0, 3306, 0, 0, 132, 0, 5184, 0, 0, -8008, 0, 5496, 0, 0, 3366, 0, -312, 0, 0, 1584, 0, -1144, 0, 0, 512, 0, -5400, 0, 0, -528, 0, -1254, 0, 0, -10284, 0, 5940, 0, 0, 2192, 0, 2394, 0, 0, 2988, 0, -6654, 0, 0, -3894, 0, 418, 0, 0, -3696, 0, -4620, 0, 0, 16200, 0, -3300, 0, 0, 10362, 0, 1092, 0, 0, -1242, 0, 4554, 0, 0, -2556, 0, -1914, 0, 0, -17568, 0, -14240, 0, 0, 1716, 0, 2232, 0, 0, -6858, 0, -2166, 0, 0, 5574, 0, 486, 0, 0, -3688, 0, -9498, 0, 0, 19008, 0, -10992, 0, 0, 2046, 0, 2074, 0, 0, -2106, 0, -14256, 0, 0, -25422, 0, 5544, 0, 0, 7704, 0, -3216, 0, 0, -22, 0, -6534, 0, 0, 3564, 0, 9474, 0, 0, 14106, 0, -14432, 0, 0, 16400, 0, 15642, 0, 0, -4878, 0, 23364, 0, 0, -4524, 0, 6254, 0, 0, 4996, 0, -1998, 0, 0, -7326, 0, 11748, 0, 0, -12744, 0, -18128, 0, 0, -19722, 0, -5148, 0, 0, 3870, 0, 462, 0, 0, 21096, 0, 10032, 0, 0, 14356, 0, 4416, 0, 0, 21006, 0, -6864, 0, 0, 9042, 0, -22, 0, 0, 8558, 0, 32076, 0, 0, -4110, 0, -28506, 0, 0, -55752, 0, 53152, 0, 0, -3630, 0, 12798, 0, 0, -37260, 0, -16536, 0, 0, 11802, 0, 25582, 0, 0, -29832, 0, 5568, 0, 0, 9018, 0, 4092, 0, 0, 3102, 0, -33792, 0, 0, 30470, 0, 24354, 0, 0, -22446, 0, -5148, 0, 0, -11160, 0, 5100, 0, 0, -10736, 0, -2916, 0, 0, 11880, 0, 15660, 0, 0, 2970, 0, -10164, 0, 0, 24068, 0, 11652, 0, 0, 1386, 0, -35508, 0, 0, 4938, 0, 2376, 0, 0, -12096, 0, -33858, 0, 0, 9636, 0, 15216, 0, 0, -1488, 0, -5592, 0, 0, 25520, 0, -23472, 0, 0, 8316, 0, -9108, 0, 0, -7242, 0, -5846, 0, 0, -14256, 0, 26568, 0, 0, 49500, 0, 23112, 0, 0, 3372, 0, 16720, 0, 0, -24718, 0, 46170, 0, 0, 13584, 0, -6600, 0, 0, 1764, 0, -34320, 0, 0, 14190, 0, -37818, 0, 0, 42768, 0, -38994, 0, 0, 60, 0, -45630, 0, 0, -26900, 0, -89850, 0, 0, 17820, 0, 43560, 0, 0, 22398, 0, 26180, 0, 0, -49412, 0, -14580, 0, 0, -11976, 0, 19896, 0, 0, -90552, 0, -2384, 0, 0, 2300, 0, -29106, 0, 0, -16632, 0, 14256, 0, 0, 10758, 0, -20872, 0, 0, 38748, 0, 11814, 0, 0, -5742, 0, 55548, 0, 0, 49368, 0, -45672, 0, 0, -35666, 0, -71820, 0, 0, -3564, 0, -30426, 0, 0, 78804, 0, 85768, 0, 0, -44550, 0, -13860, 0, 0, 11430, 0, -3036, 0, 0, -64680, 0, -8450, 0, 0, -4004, 0, 12474, 0, 0, -65268, 0, -28920, 0, 0, 11832, 0, -43988, 0, 0, 33792, 0, 12726, 0, 0, -58542, 0, 4902, 0, 0, 46068, 0, 26160, 0, 0, -5984, 0, 42984, 0, 0, 44550, 0, 93588, 0, 0, 9966, 0, -3740, 0, 0, 115544, 0, 71244, 0, 0, -7614, 0, -98868, 0, 0, -47454, 0, 7788, 0, 0, -5440, 0, 40986, 0, 0, -3300, 0, 46266, 0, 0, -22176, 0, 18592, 0, 0, 88440, 0, 55728, 0, 0, -18792, 0, -85416, 0, 0, -5808, 0, 79002, 0, 0, -177144, 0, 6528, 0, 0, 8712, 0, -60060, 0, 0, -62238, 0, 22810, 0, 0, 30360, 0, 53352, 0, 0, 15510, 0, -56694, 0, 0, 134220, 0, 26460, 0, 0, 15250, 0, -13860, 0, 0, -137610, 0, 66396, 0, 0, 79596, 0, 24286, 0, 0, 136164, 0, -5148, 0, 0, -162270, 0, 133056, 0, 0, -7590, 0, 65406, 0, 0, 71294, 0, 55638, 0, 0, 78408, 0, 6552, 0, 0, -74160, 0, -24724, 0, 0, -92644, 0, -26532, 0, 0, -29160, 0, -6930, 0, 0, 38460, 0, 8316, 0, 0, -80476, 0, -93192, 0, 0, 4752, 0, -5808, 0, 0, -3132, 0, 9850, 0, 0, 12540, 0, -71712, 0, 0, 85932, 0, 11094, 0, 0, 192120, 0, -194376, 0, 0, 27842, 0, 16470, 0, 0, 2970, 0, -20922, 0, 0, -97482, 0, -52272, 0, 0, 5292, 0, 74448, 0, 0, -63954, 0, -420, 0, 0, -77424, 0, -74108, 0, 0, 66528, 0, -39204, 0, 0, -34422, 0, 14124, 0, 0, 122148, 0, 16756, 0, 0, 39072, 0, -10098, 0, 0, 14652, 0, 55110, 0, 0, -68772, 0, -11976, 0, 0, -66112, 0, -68970, 0, 0, -52290, 0, -105204, 0, 0, 54978, 0, 94922, 0, 0, -83296, 0, -189000, 0, 0, 2370, 0, 35508, 0, 0, 2760, 0, -110484, 0, 0, 14612, 0, 37620, 0, 0, 75330, 0, -2442, 0, 0, -134574, 0, 43538, 0, 0, -62040, 0, -25248, 0, 0, -23814, 0, 6768, 0, 0, -92268, 0, 69078, 0, 0, -9368, 0, 74844, 0, 0, 171468, 0, 53196, 0, 0, 4188, 0, -85536, 0, 0, -81444, 0, 124740, 0, 0, 47358, 0, -66354, 0, 0, 50718, 0, -148148, 0, 0, 338888, 0, -177078, 0, 0, -18612, 0, -17688, 0, 0, -98310, 0, -20790, 0, 0, -124686, 0, -5940, 0, 0, 20856, 0, -42726, 0, 0, -69612, 0, -14124, 0, 0, -115280, 0, -3762, 0, 0, 180792, 0, -108228, 0, 0, -51342, 0, 117018, 0, 0, -277684, 0, -9108, 0, 0, 143154, 0, 128304, 0, 0, 87492, 0, -28112, 0, 0, -45144, 0, 170280, 0, 0, 155232, 0, 13068, 0, 0, -13332, 0, 205216, 0, 0, 120096, 0, 104040, 0, 0, -21600, 0], "weight_parity": 1, "has_non_self_twist": 1, "level_is_prime": false, "level_is_prime_power": false, "level_is_square": true, "level_is_squarefree": false, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 5, 2, 1, 2, 0], [1, 1, 9, 3, 1, 3, 0], [1, 1, 45, 10, 1, 6, 0]], "char_orbit_index": 11, "prim_orbit_index": 10, "minimal_twist": "9.4.c.a", "char_is_minimal": false, "conrey_index": 49, "id": 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], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, -1, 0, -1], [-3, 0, 3, -4, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 3, -18, 0, 0, -3, 0], [0, 2, 0, 0, 0, 0, 0, -3], [0, 16, 0, 0, -1, 16, 0, 0], [-27, 0, 3, -24, 0, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-29, 0, -8, -29, 0, 0, 0, 0], [0, 30, 0, 0, 5, 52, 0, -2], [0, 15, 0, 0, 15, 13, 0, 15], [-8, 0, 8, -34, 0, 0, -8, 0], [0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -9, 8, 0, 0, 0, 0], [0, -54, 0, 0, -9, -54, 0, 0], [0, 72, 0, 0, 27, 72, 0, 18], [25, 0, 0, 0, 0, 0, -27, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-27, 0, 7, -53, 0, 0, -8, 0], [0, 21, 0, 0, 21, -6, 0, 21], [0, 19, 0, 0, 19, -7, 0, 19], [18, 0, 15, -24, 0, 0, 18, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-118, 0, 0, 0, 0, 0, 28, 0], [0, 135, 0, 0, 18, 117, 0, 36], [0, 92, 0, 0, 18, 92, 0, 0], [-25, 0, -1, -25, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -3, 23, 0, 0, 3, 0], [0, -9, 0, 0, -9, -216, 0, -9], [0, 159, 0, 0, 21, -6, 0, 66], [180, 0, -63, 180, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-108, 0, 15, -156, 0, 0, 60, 0], [0, -2, 0, 0, -54, -2, 0, 0], [0, 241, 0, 0, 0, 0, 0, -79], [-135, 0, 17, 107, 0, 0, 11, 0], [0, 0, 0, 0, 0, 0, 0, 0], [98, 0, -98, 115, 0, 0, 98, 0], [0, 150, 0, 0, 60, 162, 0, 18], [0, -47, 0, 0, 0, 0, 0, -6], [155, 0, 0, 0, 0, 0, 79, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-126, 0, 0, 0, 0, 0, 12, 0], [0, 154, 0, 0, 0, 0, 0, -91], [0, 57, 0, 0, -17, -88, 0, -7], [-21, 0, 21, 246, 0, 0, -21, 0], [0, 0, 0, 0, 0, 0, 0, 0], [162, 0, -63, 180, 0, 0, -36, 0], [0, -358, 0, 0, 0, 0, 0, 54], [0, -54, 0, 0, -162, -54, 0, 0], [-324, 0, 81, -54, 0, 0, 54, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-56, 0, 46, -56, 0, 0, 0, 0], [0, 405, 0, 0, -2, 164, 0, -79], [0, 24, 0, 0, 24, 42, 0, 24], [136, 0, -136, -331, 0, 0, 136, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-167, 0, -105, -167, 0, 0, 0, 0], [0, -70, 0, 0, -26, -70, 0, 0], [0, 216, 0, 0, 78, 192, 0, 57], [287, 0, 0, 0, 0, 0, -153, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-189, 0, 48, 174, 0, 0, -33, 0], [0, 66, 0, 0, 66, 527, 0, 66], [0, -171, 0, 0, -171, -432, 0, -171], [-171, 0, 45, 159, 0, 0, -33, 0], [0, 0, 0, 0, 0, 0, 0, 0], [612, 0, 0, 0, 0, 0, -144, 0], [0, -333, 0, 0, 99, 0, 0, 27], [0, -106, 0, 0, 243, -106, 0, 0], [436, 0, -56, 436, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-183, 0, 183, -856, 0, 0, -183, 0], [0, 71, 0, 0, 71, -47, 0, 71], [0, -420, 0, 0, -90, -78, 0, 174], [-247, 0, -309, -247, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 135, 216, 0, 0, 135, 0], [0, -1014, 0, 0, -213, -1014, 0, 0], [0, -460, 0, 0, 0, 0, 0, -107], [-324, 0, 110, -328, 0, 0, 56, 0], [0, 0, 0, 0, 0, 0, 0, 0], [35, 0, -35, 34, 0, 0, 35, 0], [0, 84, 0, 0, 24, 42, 0, 51], [0, -693, 0, 0, 0, 0, 0, -165], [234, 0, 0, 0, 0, 0, 72, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-356, 0, 0, 0, 0, 0, 69, 0], [0, -430, 0, 0, 0, 0, 0, -2], [0, -87, 0, 0, -43, -71, 0, -17], [-336, 0, 336, -1218, 0, 0, -336, 0], [0, 0, 0, 0, 0, 0, 0, 0], [81, 0, 198, 144, 0, 0, -423, 0], [0, 317, 0, 0, 0, 0, 0, -102], [0, -324, 0, 0, -225, -324, 0, 0], [216, 0, 279, 801, 0, 0, -81, 0], [0, 0, 0, 0, 0, 0, 0, 0]]}