# Stored data for newform 56.3.k.d, downloaded from the LMFDB on 24 April 2024. {"label": "56.3.k.d", "space_label": "56.3.k", "level": 56, "weight": 3, "hecke_orbit": 4, "hecke_orbit_code": 13511486127210552, "dim": 12, "is_polredabs": true, "trace_hash": 487741803812536408, "analytic_rank": 0, "is_twist_minimal": true, "hecke_cutters": [[3, [1225, 665, 466, 13, 28, 3, 1]], [5, [117649, 0, -64141, 0, 25022, 0, -4737, 0, 654, 0, -29, 0, 1]]], "qexp_display": "q-\\beta _{3}q^{2}+(\\beta _{8}-\\beta _{9}-\\beta _{10})q^{3}+(\\beta _{2}+\\cdots)q^{4}+\\cdots", "char_order": 6, "char_parity": -1, "char_degree": 2, "char_conductor": 56, "char_orbit_label": "k", "char_is_real": false, "Nk2": 504, "analytic_conductor": 1.5258894804176626, "hecke_ring_index": 16, "level_radical": 14, "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": "0.506", "prec": 17}, "analytic_rank_proved": true, "char_values": [56, 6, [15, 29, 17], [3, 3, 4]], "hecke_ring_generator_nbound": 3, "inner_twist_count": 4, "level_primes": [2, 7], "conrey_indexes": [11, 51], "field_disc": 177010447855401216, "field_disc_factorization": [[2, 8], [3, 6], [7, 2], [373, 4]], "field_poly": [4, -20, 60, -116, 164, -166, 155, -163, 86, 3, -4, -3, 1], "hecke_ring_index_factorization": [[2, 4]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.021", "prec": 17}, {"__RealLiteral__": 0, "data": "11.761", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "1010.276", "prec": 27}, {"__RealLiteral__": 0, "data": "275.595", "prec": 24}, {"__RealLiteral__": 0, "data": "146123.999", "prec": 34}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 6, "sato_tate_group": "2.2.3.c6", "trace_display": [2, -6, 0, 0], "traces": [0, 12, 2, -6, -4, 0, -56, 0, 8, -40, -6, 30, 32, 0, 52, 0, 16, 30, -16, 78, 48, 0, 24, 0, -76, -92, -128, 156, -4, 0, -16, 0, 112, -78, 76, -222, -248, 0, 80, 0, 44, -232, 132, -200, 132, 0, -156, 0, 176, 372, 48, 10, 132, 0, -36, 0, -112, 332, 4, -110, 84, 0, -96, 0, -160, -32, -138, 434, 96, 0, -188, 0, -328, 102, -34, -60, -168, 0, 720, 0, -256, -82, -24, -536, -624, 0, 240, 0, -204, 214, 440, -8, 160, 0, -16, 0, 48, -152, 382, 504, 16, 0, -112, 0, 624, -136, -652, 102, 436, 0, -84, 0, 1096, -680, -470, -94, -372, 0, -328, 0, -568, 240, -90, -584, -252, 0, -1076, 0, -96, -60, 316, -214, -360, 0, 448, 0, -132, 510, 98, 424, 188, 0, 340, 0, 224, 272, 732, 198, 952, 0, 484, 0, -12, 320, 144, 540, 1148, 0, 396, 0, -640, -422, -1140, -166, 748, 0, 292, 0, 868, -1684, -74, 604, -320, 0, -952, 0, 144, -694, -1034, -402, 292, 0, -1096, 0, -972, 242, 822, -378, -1944, 0, -304, 0, -912, -74, -420, -88, -704, 0, 636, 0, -216, 626, 916, 736, -248, 0, 672, 0, -384, 972, 926, 1928, -212, 0, -224, 0, 2056, -1158, 12, 78, 456, 0, 676, 0, 928, 2064, -260, -658, -532, 0, -1472, 0, 368, 246, -1548, -298, 256, 0, 4, 0, -424, -314, 172, -1408, -440, 0, 1112, 0, -852, -28, -374, -128, -600, 0, 700, 0, 1024, -274, 1480, -1338, 1416, 0, 1288, 0, -516, 932, 518, -3884, 256, 0, -628, 0, -64, 2000, 764, 228, 1640, 0, 44, 0, 236, -1272, -1662, -170, -88, 0, -168, 0, -3168, 1376, -184, 1144, 756, 0, -1420, 0, -604, -1074, -422, 2372, -1664, 0, 304, 0, -632, -1030, 404, 1336, -1956, 0, 648, 0, 944, -802, 316, 636, 912, 0, 1216, 0, -272, -3556, 1490, 484, 888, 0, -620, 0, 2528, 1802, 948, -798, -288, 0, -784, 0, 1072, 232, -1218, -120, 348, 0, -1160, 0, -1680, -834, 106, 946, 332, 0, -264, 0, -96, 2174, 386, -584, -680, 0, 408, 0, -360, 272, 36, -840, -2516, 0, 500, 0, -248, -1880, -188, -2262, 456, 0, 540, 0, 1284, 1200, 2938, -1024, 408, 0, -960, 0, 2688, -744, -3756, 2760, -228, 0, -156, 0, 664, 3308, -532, 874, 252, 0, -1656, 0, -928, -922, -1946, -132, 512, 0, -276, 0, -228, 302, -540, 2490, 768, 0, 1888, 0, -752, -1652, -702, 2536, -464, 0, 588, 0, -52, 196, -2200, 2950, 2456, 0, 760, 0, 456, 264, 912, 584, 1272, 0, -144, 0, 936, -1712, -632, -2114, 816, 0, -700, 0, -224, 296, 32, -2592, 980, 0, -624, 0, -1464, -874, -370, -710, -732, 0, -1896, 0, -1072, -1470, -82, -1294, -2528, 0, 656, 0, -732, 660, 434, -3112, -1228, 0, 616, 0, 928, -8, 3716, -3476, 1048, 0, 164, 0, -1980, -1332, -18, -1600, 1364, 0, 1292, 0, -2544, -504, 3264, -1246, -736, 0, 464, 0, 1456, -220, 990, 3938, -2136, 0, 432, 0, 3968, 1022, 1642, -214, -2240, 0, 388, 0, 1352, -1546, 68, 2926, 2272, 0, -3696, 0, 864, 1788, 166, 3592, 2212, 0, -36, 0, 356, 2234, 3212, 3282, -2304, 0, 296, 0, -32, 452, -652, 4256, -1216, 0, 540, 0, -4008, -1494, -2572, -770, 1352, 0, 2316, 0, 104, 804, -1524, -262, -812, 0, -232, 0, -32, 2774, -270, -2026, 180, 0, 1604, 0, -1408, 2198, 36, -5926, -1864, 0, -80, 0, -1308, 3144, 952, -8648, 1084, 0, -312, 0, -1104, 470, -3732, -548, -448, 0, -912, 0, -1784, 1976, -3780, -5736, -2172, 0, -1592, 0, 4352, -1496, -4220, 820, -348, 0, 1240, 0, -756, -4344, 1654, 2082, -96, 0, 2888, 0, 3232, -334, -1158, 2094, 436, 0, 948, 0, 2420, -1484, 1938, 1600, 6096, 0, -576, 0, 672, 830, -286, -2680, 596, 0, -152, 0, -1940, -438, 4480, 7470, -816, 0, -2324, 0, -368, 5472, -1346, 6904, 384, 0, 464, 0, -2224, -1078, -240, 604, 508, 0, 388, 0, -3968, 1624, 3680, -2420, -660, 0, 2880, 0, 1416, -1622, 1422, -3538, -84, 0, 320, 0, -1360, -3632, -266, -1910, -4760, 0, -3256, 0, 1984, -1892, -1444, 2492, 1136, 0, -3980, 0, -2160, 1990, -2620, 1758, -2640, 0, 236, 0, -252, -5004, -622, -4440, -2228, 0, 2156, 0, 1504, 2946, 7320, 1126, -1592, 0, -1372, 0, -384, -1492, -918, 820, 680, 0, -2032, 0, 4640, -2070, 3692, 1142, 824, 0, -1040, 0, -108, 3338, -726, 4416, 2112, 0, 1736, 0, 1128, 908, 2972, 1340, 5192, 0, -2488, 0, 636, -2138, -6744, 606, 2680, 0, -2388, 0, 3264, -7768, -2010, 9132, 2096, 0, -1120, 0, -1216, -2984, -1484, -2296, -2500, 0, 736, 0, -1600, -2780, -218, -4430, 4980, 0, 4960, 0, 2016, 3320, 1950, -1382, -724, 0, 2380, 0, 2464, -1288, 154, -2574, 4512, 0, -936, 0, -588, 142, -482, 7288, 2460, 0, 804, 0, 408, -3100, -1716, -14360, -4952, 0, -2168, 0, 1492, 4632, 692, -9368, -1900, 0, 1116, 0, -8704, 2694, -2032, 2364, -1176, 0, -3352, 0, -424, 4556, -344, 2896, -656, 0, 3768, 0, -6336, 50, 32, 2454, -3312, 0, 4940, 0, 2660, 4182, -2496, 3918, 1520, 0, 3640, 0, 3904, -1550, -2340, -2464, -1164, 0, -1176, 0, 2068, 832, -5668, 2220, -4368, 0, -2372, 0, 1752, 4056, -3404, 3864, -836, 0, 1164, 0, -5592, 2872, 1892, -2418, -2784, 0, -4208, 0, -2496, -592, 3472, 876, 1864, 0, -4224, 0, 2560, -2224, 4862, 1770, 340, 0, 3504, 0, 2544, 60, -3670, -2610, 7408, 0, -2808, 0, 1276, -5412, 1860, 7184, -2148, 0, 1032, 0, 720, 430, -1404, -3990, -3312, 0, -4116, 0, 2624, 2728, 228, -11564, -1632, 0, -2420, 0, 432, 50, 6840, 2634, -3504, 0, 1924, 0, -1484, 3032, 5760, 190, 740, 0, -2200, 0, 416, -1356, 3144, 5484, 2276, 0, 5276, 0, 1528, 622, -40, 1506, 3488, 0, 3968, 0, 3760, -5578, 1450, 3324, 864, 0, -2824, 0, -3896, -2056, 368, 4948, -3016, 0, -120, 0, -2784, 1100, -196, 7212, -3592, 0, 5196, 0, -1928], "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, 7, 3, 1, 3, 0], [1, 1, 8, 4, -1, 2, 0], [1, 1, 56, 11, -1, 6, 0]], "char_orbit_index": 11, "prim_orbit_index": 11, "minimal_twist": "56.3.k.d", "char_is_minimal": true, "conrey_index": 11, "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], [1, 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, -1, -1, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0], [-5, -1, 1, 1, 0, 2, 0, 0, 1, 0, 2, -1], [0, 0, -2, -1, -2, -1, 0, 0, 2, 0, 2, -1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2], [-6, 0, 0, -1, 0, 1, -4, -6, 0, 4, 0, 0], [0, 0, 1, 0, 2, 0, 0, 1, -1, -1, 0, 2], [0, 0, 0, 0, 0, 0, 0, -6, 0, -3, 0, 0], [5, 1, 0, 6, 0, 4, 3, 5, 0, -3, 0, 0], [0, 2, -2, -5, 0, -5, 0, 0, -5, 0, -5, -2], [4, 0, -1, 1, -1, -2, 1, -1, 2, 2, 0, -4], [0, 0, 0, -2, 0, -2, 0, 0, -2, 0, -2, 3], [4, -4, 0, -4, -4, -8, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, -1, -2, 1, 0], [0, 0, 1, 0, 4, 0, 0, 5, -10, 5, -8, 4], [16, 0, 0, 4, 0, -4, -1, 16, 0, 1, 0, 0], [5, -3, 3, 0, 0, 0, -3, 0, 0, 0, 0, 2], [0, -2, -1, 7, 5, 7, 0, 0, 7, 0, 7, -3], [0, 0, 0, -3, 0, 6, -3, 0, -3, 0, 6, -3], [0, 0, 0, 5, -7, 5, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, -2, 0, 0, 10, 8, -8, 8, -2], [0, 0, 0, 0, 0, 0, 0, 16, -2, -2, 2, 0], [-19, 9, 0, 2, 4, 0, -5, -19, 0, 5, 0, 0], [10, 0, 0, 2, 0, -2, 13, 0, 2, 0, -2, 0], [-10, 6, -1, -4, 2, 0, 8, -15, -2, -5, -4, 0], [0, -8, 8, -3, 0, -3, 0, 0, -3, 0, -3, 2], [0, -4, 0, -3, -3, -6, -5, 0, 0, 5, 0, 0], [0, 0, 0, 0, 9, 0, 0, 0, -6, 0, -6, 9], [0, 0, -4, 0, -8, 0, 0, -20, 8, 4, 0, -8], [-15, 0, 0, -6, 0, 6, -6, -15, 0, 6, 0, 0], [5, 1, -1, -3, 0, 4, -3, 0, -3, 0, 4, -2], [-16, 0, 0, 1, 0, -1, 3, 4, 3, -1, -3, 0], [-14, 2, -2, -4, 0, -16, -8, 0, -4, 0, -16, 10], [0, 9, 0, 4, -1, 4, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 1, 0, 0, -20, 15, -3, -2, 1], [0, 0, 14, 0, -8, 0, 0, 0, 17, 0, 17, -8], [4, -4, 0, -4, -2, 8, -2, 4, 0, 2, 0, 0], [-26, 0, 0, -1, 0, 1, 18, 0, -1, 0, 1, 0], [15, -1, 10, 3, 6, 4, 9, 5, -5, -17, -16, 2], [-20, 0, 0, -10, 0, 10, -10, 0, -10, 0, 10, 0], [21, 9, 0, 6, 6, 12, -3, 21, 0, 3, 0, 0], [0, 0, 4, 0, 2, 0, 0, 0, 9, 0, 9, 2], [0, 0, -19, 0, -7, 0, 0, 29, 7, 2, 14, -7], [0, -16, 0, -2, 5, -2, 0, 0, 0, 0, 0, 0], [20, -4, 4, 20, 0, 8, -4, 0, 20, 0, 8, -16], [41, 0, 0, -3, 0, 3, -2, 30, -9, 10, 9, 0], [10, 2, -2, 18, 0, 4, -4, 0, 18, 0, 4, -2], [0, 0, 0, -4, 0, 4, -3, 0, 0, 3, 0, 0], [0, 0, 6, 0, 18, 0, 0, -10, -28, 8, 0, 18], [0, 0, 3, 0, 7, 0, 0, 0, -20, 0, -20, 7], [-10, -2, 0, -23, -13, -26, 15, -10, 0, -15, 0, 0], [0, 12, -12, 3, 0, 3, 0, 0, 3, 0, 3, -9], [-14, 2, 6, -8, 0, -8, 0, 0, -12, 0, 16, 2], [45, 0, 0, 20, 0, -20, -12, 0, 20, 0, -20, 0], [-5, -1, 0, -2, -10, 12, 3, -5, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 5, -9, -5, 0], [0, 0, -3, 0, -2, 0, 0, -15, -2, 7, -12, -2], [0, -15, 0, -10, -25, -10, 0, 0, 0, 0, 0, 0], [0, -12, 12, -9, 0, -18, -15, 0, -9, 0, -18, 9], [0, 0, -6, -17, 8, -17, 0, 0, -36, 0, -36, 4], [-24, 8, -8, -16, 0, 0, 16, 0, -16, 0, 0, -8], [-4, 0, 0, 11, 0, -11, 18, -4, 0, -18, 0, 0], [0, 0, 6, 0, 6, 0, 0, 30, -21, 12, -12, 6], [0, 0, 0, 0, 0, 0, 0, -70, -3, 1, 3, 0], [15, 7, 0, 0, 6, 8, -5, 15, 0, 5, 0, 0], [0, 31, -31, 13, 0, 13, 0, 0, 13, 0, 13, 3], [-15, -3, 2, 18, -3, -4, 5, -5, 5, -4, 2, -1], [0, 6, -6, 22, 0, 22, 0, 0, 22, 0, 22, -8], [-40, -16, 0, 12, 0, -8, -24, -40, 0, 24, 0, 0], [0, 0, 0, 0, 0, 0, 0, -15, -12, -18, 12, 0], [0, 0, -6, 0, 8, 0, 0, 14, 1, 6, 20, 8], [10, 0, 0, 18, 0, -18, -24, 10, 0, 24, 0, 0], [-21, -17, 17, -18, 0, -4, 7, 0, -18, 0, -4, -6], [0, -24, 3, -3, -9, -3, 0, 0, 6, 0, 6, 3], [35, -1, 1, 8, 0, 44, 39, 0, 8, 0, 44, 6], [0, 16, 0, -5, 25, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 40, 4, 4, -8, -4], [0, 0, 0, 0, 0, 0, 0, -9, 21, -26, -21, 0], [5, 1, 0, 8, -18, -36, 17, 5, 0, -17, 0, 0], [-50, 0, 0, -14, 0, 14, -12, 0, -14, 0, 14, 0], [-55, 1, 4, -4, -6, 32, -5, 5, 16, -17, 40, -2], [0, 11, -11, 4, 0, 4, 0, 0, 4, 0, 4, -11], [50, 10, 0, 30, 10, 20, -20, 50, 0, 20, 0, 0], [0, 0, 8, 0, -8, 0, 0, 0, -23, 0, -23, -8], [0, 0, 6, 0, 18, 0, 0, 30, 12, 0, 0, 18], [13, 0, 0, -36, 0, 36, -4, 13, 0, 4, 0, 0], [31, -13, 13, -2, 0, 4, 11, 0, -2, 0, 4, 6], [-6, 0, 0, 3, 0, -3, -26, -30, 23, 46, -23, 0], [35, 7, -7, 34, 0, -28, -3, 0, 34, 0, -28, -24], [0, -15, 0, -54, -21, -54, 0, 0, 0, 0, 0, 0], [0, 0, 12, 0, -11, 0, 0, -16, -5, -19, -42, -11], [0, 0, 10, 0, 1, 0, 0, 0, -11, 0, -11, 1], [-20, 12, 0, 0, 16, 48, 36, -20, 0, -36, 0, 0], [0, 0, 0, 15, 0, -15, -8, 0, 15, 0, -15, 0], [45, 9, -6, -19, 2, -16, -1, 15, 20, 5, -20, 10], [24, 0, 0, -21, 0, 21, 12, 0, -21, 0, 21, 0], [-6, -14, 0, -4, 8, 8, 14, -6, 0, -14, 0, 0]]}