# Stored data for newform 336.4.b.e, downloaded from the LMFDB on 23 October 2025. {"Nk2": 5376, "analytic_conductor": 19.824641761928845, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 28, "char_degree": 1, "char_is_minimal": true, "char_is_real": true, "char_orbit_index": 2, "char_orbit_label": "b", "char_order": 2, "char_parity": 1, "char_values": [336, 2, [127, 85, 113, 241], [1, 2, 2, 1]], "cm_discs": [], "conrey_index": 223, "dim": 8, "field_disc": 7532245815174144, "field_disc_factorization": [[2, 10], [3, 4], [301349, 2]], "field_poly": [1232100, 0, 180672, 0, 8461, 0, 158, 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": [[5, [146313216, 0, 11183616, 0, 195588, 0, 828, 0, 1]], [19, [-5935552, -500320, -10380, 28, 1]]], "hecke_orbit": 5, "hecke_orbit_code": 18014467296067920, "hecke_ring_generator_nbound": 11, "hecke_ring_index": 884736, "hecke_ring_index_factorization": [[2, 15], [3, 3]], "hecke_ring_index_proved": true, "inner_twist_count": 2, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 28, 4, 1, 2, 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": true, "label": "336.4.b.e", "level": 336, "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": [2, 3, 7], "level_radical": 42, "minimal_twist": "336.4.b.e", "prim_orbit_index": 4, "qexp_display": "q-3q^{3}+\\beta _{5}q^{5}-\\beta _{2}q^{7}+9q^{9}+(-\\beta _{1}+\\cdots)q^{11}+\\cdots", "related_objects": [], "relative_dim": 8, "rm_discs": [], "sato_tate_group": "3.2.3.c2", "self_twist_discs": [], "self_twist_type": 0, "space_label": "336.4.b", "trace_display": [0, -24, 0, -4], "trace_hash": 1917847667163777208, "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "7.321", "prec": 17}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "287.154", "prec": 24}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "16355.149", "prec": 30}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.508", "prec": 17}, "traces": [0, 8, 0, -24, 0, 0, 0, -4, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, -56, 0, 12, 0, 0, 0, -656, 0, -216, 0, 240, 0, 320, 0, 0, 0, 600, 0, 392, 0, 0, 0, 0, 0, 0, 0, 0, 0, -816, 0, -16, 0, 0, 0, 288, 0, -456, 0, 168, 0, -1824, 0, 0, 0, -36, 0, -816, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1968, 0, -2064, 0, 0, 0, 648, 0, 1680, 0, 2568, 0, -720, 0, 0, 0, -864, 0, -960, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3584, 0, -1800, 0, 0, 0, -2624, 0, -1176, 0, 4704, 0, -4632, 0, 0, 0, 3528, 0, 4720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3792, 0, -1688, 0, 0, 0, -7872, 0, -2464, 0, 2448, 0, -5232, 0, 0, 0, 48, 0, -144, 0, 0, 0, 0, 0, 0, 0, 0, 0, -864, 0, 3216, 0, 0, 0, 1368, 0, -6576, 0, -2104, 0, -504, 0, 0, 0, 13540, 0, 5472, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12312, 0, 108, 0, 0, 0, -3928, 0, 2448, 0, 1152, 0, -16520, 0, 0, 0, 3720, 0, -5784, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13296, 0, -352, 0, 0, 0, 11280, 0, -248, 0, -5904, 0, -17856, 0, 0, 0, 6192, 0, -8592, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1944, 0, -3216, 0, 0, 0, -5040, 0, -13584, 0, 3912, 0, -7704, 0, 0, 0, 11600, 0, 2160, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3424, 0, 2592, 0, 0, 0, 31064, 0, 2880, 0, 0, 0, -32264, 0, 0, 0, 14664, 0, 12112, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42480, 0, -12312, 0, 0, 0, -15552, 0, -8216, 0, -10752, 0, -31872, 0, 0, 0, 5400, 0, -29232, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7872, 0, 32496, 0, 0, 0, 3528, 0, 12960, 0, -26240, 0, -14112, 0, 0, 0, 9284, 0, 13896, 0, 0, 0, 0, 0, 0, 0, 0, 0, 46872, 0, -10584, 0, 0, 0, -11784, 0, -14160, 0, 4176, 0, -30872, 0, 0, 0, 9144, 0, -8144, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6960, 0, 18888, 0, 0, 0, 10752, 0, -3192, 0, -11376, 0, -38784, 0, 0, 0, 5064, 0, 13440, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23616, 0, 32784, 0, 0, 0, 7392, 0, -47424, 0, -36904, 0, -7344, 0, 0, 0, 13968, 0, 15696, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9880, 0, -144, 0, 0, 0, 28824, 0, 432, 0, 10080, 0, 23928, 0, 0, 0, 64752, 0, 26752, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29616, 0, -32712, 0, 0, 0, -23040, 0, -47464, 0, 2592, 0, -47136, 0, 0, 0, -9648, 0, -86640, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4104, 0, -3072, 0, 0, 0, 19728, 0, 61392, 0, -1992, 0, 6312, 0, 0, 0, 39600, 0, 1512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 49280, 0, -40620, 0, 0, 0, -38384, 0, -16416, 0, 90576, 0, -97176, 0, 0, 0, -6384, 0, 28376, 0, 0, 0, 0, 0, 0, 0, 0, 0, -52176, 0, 10920, 0, 0, 0, 52656, 0, 54912, 0, -36936, 0, 39984, 0, 0, 0, -324, 0, -37296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11784, 0, -3840, 0, 0, 0, -7344, 0, -6864, 0, 28288, 0, -3456, 0, 0, 0, -51816, 0, 49560, 0, 0, 0, 0, 0, 0, 0, 0, 0, -44120, 0, -11160, 0, 0, 0, -9152, 0, 17352, 0, 61632, 0, -35104, 0, 0, 0, -3336, 0, 92816, 0, 0, 0, 0, 0, 0, 0, 0, 0, 71616, 0, -35040, 0, 0, 0, -98640, 0, -65960, 0, -39888, 0, -19200, 0, 0, 0, 1056, 0, -143424, 0, 0, 0, 0, 0, 0, 0, 0, 0, -33840, 0, 73392, 0, 0, 0, 744, 0, 5856, 0, -38200, 0, 17712, 0, 0, 0, 71568, 0, 53568, 0, 0, 0, 0, 0, 0, 0, 0, 0, 74912, 0, -18576, 0, 0, 0, -42072, 0, 25776, 0, 27696, 0, 109120, 0, 0, 0, -88056, 0, 76336, 0, 0, 0, 0, 0, 0, 0, 0, 0, -84384, 0, -41152, 0, 0, 0, -29472, 0, 99776, 0, 5832, 0, 32784, 0, 0, 0, 9648, 0, -31920, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15120, 0, 65376, 0, 0, 0, 40752, 0, -117936, 0, -19472, 0, -11736, 0, 0, 0, -90536, 0, 23112, 0, 0, 0, 0, 0, 0, 0, 0, 0, -228416, 0, -34800, 0, 0, 0, -7080, 0, -6480, 0, 194304, 0, 67040, 0, 0, 0, 13224, 0, -23040, 0, 0, 0, 0, 0, 0, 0, 0, 0, 123312, 0, -95016, 0, 0, 0, -72336, 0, -55168, 0, 10272, 0, 94560, 0, 0, 0, -7776, 0, -17664, 0, 0, 0, 0, 0, 0, 0, 0, 0, -93192, 0, 15984, 0, 0, 0, -8640, 0, 72720, 0, -38728, 0, 0, 0, 0, 0, 2188, 0, 96792, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100360, 0, -43992, 0, 0, 0, 155928, 0, -36336, 0, -24432, 0, 89952, 0, 0, 0, -140160, 0, -41312, 0, 0, 0, 0, 0, 0, 0, 0, 0, -168432, 0, -34200, 0, 0, 0, -39840, 0, 94008, 0, -127440, 0, 148800, 0, 0, 0, 36936, 0, -257232, 0, 0, 0, 0, 0, 0, 0, 0, 0, 46656, 0, 97152, 0, 0, 0, 24648, 0, -8880, 0, 58504, 0, 32256, 0, 0, 0, -136712, 0, 95616, 0, 0, 0, 0, 0, 0, 0, 0, 0, -269784, 0, -16200, 0, 0, 0, 95904, 0, 87696, 0, 118176, 0, 191112, 0, 0, 0, 111912, 0, 144840, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55584, 0, 30992, 0, 0, 0, -34656, 0, -38136, 0, -23616, 0, 22032, 0, 0, 0, -97488, 0, -169728, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10584, 0], "weight": 4, "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], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-3, 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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -1, -1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 2, 2, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 1, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 2, 2, 3, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-77, 0, -5, 5, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [-27, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [32, 0, -2, 2, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [40, 0, 0, 0, 0, 0, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 3, 3, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [74, 6, 4, 2, 3, -10, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0], [52, 0, -3, 3, 0, 0, -1, -3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 3, 3, 3, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 8, 8, 6, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -7, -1, -1, 6, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-90, 0, -12, 12, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, -9, -2, 4, -1, 1, 2, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, -6, -6, -6, 0, 9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [38, 0, -2, 2, 0, 0, 2, -5], [0, 0, 0, 0, 0, 0, 0, 0], [-52, 0, -5, 5, 0, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0], [24, 0, -3, 3, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 0], [-218, 0, -10, 10, 0, 0, -1, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 3, 3, -1, 7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-82, 0, -20, 20, 0, 0, -3, -8], [0, 0, 0, 0, 0, 0, 0, 0], [0, 13, -9, -9, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -6, -6, -6, -9, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -30, 6, 6, 3, -32, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 28, 18, 18, -2, -16, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [231, 0, 15, -15, 0, 0, 3, -3], [0, 0, 0, 0, 0, 0, 0, 0], [-258, -2, 4, 4, 6, -13, 2, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 1, 1, 6, 24, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [81, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [210, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0], [306, 0, 15, -15, 0, 0, 1, 9], [0, 0, 0, 0, 0, 0, 0, 0], [-96, 0, 6, -6, 0, 0, -3, -3], [0, 0, 0, 0, 0, 0, 0, 0], [0, -38, 4, 4, 6, -19, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-112, 35, 8, 0, 0, -42, 0, 7], [0, 0, 0, 0, 0, 0, 0, 0], [-120, 0, 0, 0, 0, 0, -6, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 54, 30, 30, 6, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 22, 14, 14, -10, 40, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -9, -9, -9, 0, 9, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]]}