# Stored data for newform 77.2.f.b, downloaded from the LMFDB on 19 April 2024. {"label": "77.2.f.b", "space_label": "77.2.f", "level": 77, "weight": 2, "hecke_orbit": 2, "hecke_orbit_code": 4503943258308685, "dim": 16, "is_polredabs": true, "trace_hash": 370242156897546411, "analytic_rank": 0, "is_twist_minimal": true, "hecke_cutters": [[2, [25, 200, 700, 925, 1850, 2300, 2135, 1175, 640, 245, 245, 145, 86, 32, 14, 3, 1]]], "qexp_display": "q+(-\\beta _{5}-\\beta _{6})q^{2}+(\\beta _{9}-\\beta _{11}+\\beta _{13}+\\cdots)q^{3}+\\cdots", "char_order": 5, "char_parity": 1, "char_degree": 4, "char_conductor": 11, "char_orbit_label": "f", "char_is_real": false, "Nk2": 308, "analytic_conductor": 0.614848095563859, "hecke_ring_index": 5, "level_radical": 77, "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": "0.005", "prec": 17}, "analytic_rank_proved": true, "char_values": [77, 5, [45, 57], [5, 1]], "hecke_ring_generator_nbound": 7, "inner_twist_count": 2, "level_primes": [7, 11], "conrey_indexes": [15, 36, 64, 71], "field_disc": 19573955532720947265625, "field_disc_factorization": [[5, 14], [1790809, 2]], "field_poly": [25, -200, 700, -925, 1850, -2300, 2135, -1175, 640, -245, 245, -145, 86, -32, 14, -3, 1], "hecke_ring_index_factorization": [[5, 1]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "14.237", "prec": 20}, {"__RealLiteral__": 0, "data": "13.716", "prec": 20}, {"__RealLiteral__": 0, "data": "709.323", "prec": 24}, {"__RealLiteral__": 0, "data": "2492.157", "prec": 27}, {"__RealLiteral__": 0, "data": "70295.674", "prec": 30}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 4, "sato_tate_group": "1.2.3.c5", "trace_display": [-3, -2, -5, -4], "traces": [0, 16, -3, -2, -11, -5, 3, -4, -5, -12, 12, -3, 18, -7, 2, -18, 17, -5, 11, 19, 1, 8, -33, 32, -35, 7, -27, 10, 4, 3, -2, -7, 32, -26, -24, -5, 52, 4, -5, 11, -10, -10, 3, -8, -38, 70, -42, -23, -36, -4, 52, -29, 33, 4, 60, -12, 0, -11, 20, 17, -30, -7, 79, -2, 7, -8, 8, -38, -2, 10, -18, -14, 0, -35, -29, 9, 52, -3, -58, 15, -87, -14, 19, 5, 8, 6, -52, -72, 55, 74, -14, 13, -55, 32, -24, 32, -42, 20, 2, 16, 22, 14, 64, 33, -25, 7, -7, 34, -35, -50, 2, -90, 17, -10, 60, 49, 59, -56, -65, -5, 50, -29, 134, 74, -4, 33, -19, -4, 10, 60, 43, -84, 41, -26, 16, -55, 105, -34, -108, -17, -24, 103, -56, 79, -144, -16, 28, -2, 0, -27, -57, 33, -25, 10, -3, -102, 19, -99, -40, 77, -23, 2, -83, -9, -118, -30, -72, 84, 45, 23, -74, -93, -10, 46, 85, 32, 81, 72, 25, -20, 103, -6, 33, -13, 30, -72, 46, 39, -66, 10, 85, 49, 46, 14, 111, 40, 4, -82, 111, -12, 35, -27, -21, -12, 15, -36, 58, -34, 44, -19, -72, 11, -144, 145, -34, 9, -25, -2, -35, -72, -50, -24, 59, 33, -18, -69, -32, -7, -78, 80, -32, -16, -120, 18, -96, 7, -19, -15, 36, 36, 35, -28, 12, 8, -4, -5, -39, 59, -30, 33, -48, -100, -28, -10, 66, 42, 131, -85, -63, -11, 120, 9, -56, 56, -70, 70, -5, 109, 14, 55, 60, -54, -16, 11, 186, 156, 186, -7, -110, -61, 5, 57, -36, -20, 20, -118, 3, -20, 36, -69, 65, 51, -21, -59, -2, 56, 80, -35, -10, -5, 56, 2, -86, -127, -34, -107, -22, 52, 62, -94, -136, -22, -60, -42, 176, 20, -110, -25, -68, -83, 60, -90, 38, -109, 39, -86, 28, 55, -5, 32, -122, -4, -41, 67, 56, 22, 39, 110, 61, -32, -104, 73, 50, -4, -70, 17, 43, -59, -61, 15, 52, 40, 22, 240, -115, 72, -108, 16, -75, 8, 135, 59, 94, 8, -82, -12, 51, -34, 10, 180, -79, -41, 14, 78, -119, -80, 15, 43, -35, 82, 41, -113, 14, -137, -10, -32, 18, -54, -12, -19, 82, 22, -5, -80, 46, 40, -59, 66, 70, 9, -99, -84, -46, -3, 124, 140, -30, 65, 60, -122, 44, -18, 23, -18, 49, 13, -97, 258, -70, -62, -30, 38, 9, 81, 135, 59, -14, -7, 160, -258, 128, 77, -5, 28, -61, 124, -30, -93, 62, -178, -195, 28, 44, -46, 67, -88, -57, 28, -3, 2, -214, 26, 24, -113, 11, -8, 10, 78, -30, 130, -14, -44, 18, -54, -53, -113, -107, -88, 9, 32, 116, -53, 130, 5, 10, -44, -7, 37, 40, 16, -47, -15, -106, 40, -101, 96, -57, 29, -115, -111, 12, -17, -5, 32, -100, 44, 58, -14, 62, -33, 0, -65, -11, 11, 0, -134, -12, 14, -4, 42, -21, 60, -23, 185, 121, -173, 106, 2, 66, -166, 65, -26, 40, 49, 175, 4, -57, 204, 98, -16, -37, 61, -8, -101, 65, 78, -75, 50, 130, -3, -30, 96, -106, -112, 6, 30, -28, -87, -9, 54, 87, -14, 15, -15, -9, 149, -26, 62, -118, 55, 8, 15, -16, 128, -89, 88, 163, 6, -155, -81, 85, 150, 81, 68, 19, -10, -24, -71, -6, -33, -3, 5, -36, -82, -100, -4, -52, -84, -17, 110, -115, -11, 65, -22, 10, 6, -11, 68, -98, 30, -170, 29, 48, -64, -194, -15, 126, -145, 75, -37, 104, -102, -91, -66, 6, 18, -55, 182, 132, -3, 146, -140, -161, -26, 62, -84, -182, -24, 166, -81, 16, 80, -45, -60, -59, 29, 7, -12, 140, -192, 40, -37, 129, -127, 20, -28, 90, 21, -70, -7, -212, 32, 57, -36, -20, -48, 76, -145, -24, 140, -11, 0, 34, 150, 130, -53, -142, -61, -55, 171, 146, 133, -12, 90, -44, -95, -23, 121, -83, -35, -5, 64, 204, -60, 96, -76, 2, 95, -60, 152, 92, 93, -248, -34, -74, 44, 65, -23, 60, 151, -33, 58, 25, 24, -46, -62, 8, 9, 111, 63, 129, 30, -50, -128, -56, -70, 120, -54, -85, 27, 59, -37, 2, -164, 144, 142, 8, -144, -25, -102, 93, -53, -116, -137, 46, 7, -132, 32, -98, 17, -118, 214, -17, -96, 80, -16, -7, -170, 165, 34, 228, 101, 77, -171, -65, 128, 55, 3, -100, -192, -40, 64, -176, 45, -36, -2, 33, -83, -247, -80, 67, -114, 106, 94, 129, 83, -70, 15, -70, 99, -54, 60, -176, 90, -18, -96, 1, -13, 52, 74, 15, 100, 55, -38, 1, -147, -16, 100, -15, -64, 22, -58, -426, 197, -94, 49, -184, 0, -75, 0, -78, 122, 59, -140, 6, -434, -20, -23, 130, 79, -66, -96, -36, -28, -50, 273, 130, -81, 70, 130, 68, 149, 6, 10, 0, 58, -32, -10, 280, 40, 65, -107, 109, 126, 255, 80, 68, 1, 238, 68, 81, 29, 8, -149, -81, 101, -50, 338, 257, -256, 71, -56, 264, -33, 160, 151, 58, 25, -34, -55, -190, 113, -10, -65, -55, -72, -10, -22, 115, 84, -70, 0, -19, 141, 35, -74, 13, 3, 20, -4, -190, 52, -212, 65, -65, -170, 20, 182, -80, -65, 26, -160, -206, -120, 25, -89, -144, 218, -78, -73, 8, -16, -174, 102, -114, -53, -305, 6, 40, 13, 15, -162, -66, -140, -39, 230, -122, 224, 75, -219, -159, 19, 38, 102, -264, -87, 50, -33, -89, 156, -26, -106, -131, 18, 68, -20, -7, 148, 5, -42, 20, -164, -30, -136, 199, 11, -62, -58, 230, -34, -76, -75, 19, -218, 78, -195, -108, -110, 110, -44, -149, 25, 33, -7, 26, 117, -237, -110, 542, -49, 46, -5, 74, -53, 45, 280, 90, -62, -83, 112, 51, 136, 54, -180, 24, -61, 50, -139, 40, 125, 340], "weight_parity": 1, "has_non_self_twist": 1, "level_is_prime": false, "level_is_prime_power": false, "level_is_square": false, "level_is_squarefree": true, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 11, 3, 1, 5, 0]], "char_orbit_index": 6, "prim_orbit_index": 3, "minimal_twist": "77.2.f.b", "char_is_minimal": true, "conrey_index": 15, "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, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0], [1, -2, 0, 1, 1, 1, -1, 0, 0, 0, -1, 1, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0], [0, -1, 1, 0, 0, 1, 1, 0, -1, 0, 1, 0, -1, -1, 0, 1], [0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0], [1, 2, -1, 1, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, -1, 0, 0, -1, 1, 1, 0, 0, -1, -1, 0], [-1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, -1, 1, 0, 0, 0], [0, 1, 0, 1, -1, 0, 0, -2, 1, -1, -1, 0, 0, -1, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -2, -1, 1, -1, 0, 2, -1, -2, 0, 0, 2, 0, 1, -1], [1, -1, 0, 0, 1, 1, 0, -1, 0, 0, -2, -1, 0, 0, -1, -1], [-1, 0, 0, -1, -1, -1, -1, 2, 0, 0, 0, 0, 0, 1, -1, -1], [-1, 0, -2, 1, 0, 2, 0, -1, 0, -1, 1, -1, 1, 0, 1, 1], [2, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 0, -1], [0, 1, 0, 1, -1, -1, -1, 1, 3, 2, 1, 0, 1, 0, -1, 0], [1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 1, -2, -2, -1, 0, 1, -1, -1, -1, 2, 0, 1, -1, -2], [0, 2, -1, -2, -2, 0, 2, 1, -1, 0, 0, -1, 1, 1, 1, 0], [0, 1, 0, -2, -1, 0, 0, -4, 1, 0, -5, 0, 1, 1, 1, -1], [2, -3, 0, 0, 0, 3, 0, 0, 2, -1, -5, 1, 0, -1, 0, -1], [-1, 0, 0, 0, 2, 1, 0, 1, 1, 1, -1, -1, -1, 0, 0, 0], [3, -2, 1, 1, 2, 2, 0, -2, 0, 1, -3, 2, 0, -1, -1, -1], [0, 0, -1, 0, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0, 1, -3, 2, 4, -1, -2, 0, 0, 1], [-3, 2, -1, 0, 0, -2, 2, 0, -2, 1, 7, -1, 1, 2, 0, 1], [0, -3, 0, 2, 3, 1, 1, -1, 1, -1, 1, 0, 0, -2, 0, 2], [3, -2, 0, 1, 2, 0, -2, 0, 0, 0, 0, 1, -1, -1, -1, 0], [-3, -2, -1, 2, 1, -2, -3, 1, -3, -1, 5, -1, -1, -1, 1, 3], [0, 0, 2, 1, 0, 0, 0, 2, -2, 0, 0, 1, -1, 1, 1, 0], [0, 1, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0], [5, 2, 1, 0, 0, -2, -1, 0, 4, 1, 2, -1, -1, 0, 0, 2], [0, 0, 1, 1, -1, -1, 0, 1, -1, 1, 2, 0, -1, 0, -2, 1], [0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 2, 0, 0, -1, 2, 2], [3, 4, 2, -1, -1, -1, 3, -4, 0, 2, 1, -1, 0, 1, 2, 2], [1, 0, 3, -2, -3, -3, 0, 2, -1, 0, -1, 3, 0, 0, 0, -2], [-2, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 1, 0, 0], [0, -1, 0, 1, 1, -1, -1, 1, -1, 0, 1, 0, -1, 0, -1, 0], [-2, 2, -2, 1, -2, 0, 2, -1, 1, 0, 0, -1, 1, 1, 1, 0], [-2, 1, 1, -1, 1, 2, 3, 1, -4, 0, -1, 0, -1, 1, 2, 0], [6, -2, 4, 2, 1, 0, -1, 5, -5, 0, 0, 4, -4, -3, -3, 0], [0, 0, 0, 1, 0, -3, -3, -2, 1, 1, -3, 0, -2, 1, -2, -1], [-2, 1, -2, 0, 0, -1, -2, 0, 0, -2, -4, 2, 2, 0, 0, -1], [-2, 0, 0, 1, 1, 1, 0, 2, 1, 2, 0, -2, -2, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [5, -3, 0, -1, 2, 2, -1, -4, 0, 0, -4, -4, 0, 1, 1, 1], [-1, 0, 2, -2, 1, 2, 0, 3, 0, 0, -2, 2, 0, 0, -1, -2], [0, -2, -1, 0, 0, 2, -1, 0, 1, 0, 3, 0, 1, -2, 0, -1], [0, 0, 0, -2, 0, -1, -1, 2, 5, 0, 0, 0, 3, 2, 0, -2], [4, 1, 1, -3, -1, 0, 1, -2, 2, 0, 0, -1, 1, 0, 0, 0], [-1, 1, -2, -1, -2, 1, -1, -4, 3, 2, 1, -4, 1, 2, 2, 0], [0, 0, -1, 0, 1, 0, -1, -1, 1, 0, 0, -1, 1, 1, 1, 0], [0, -1, 0, 0, 1, 2, 2, 0, 1, 4, -3, 0, 1, 3, -3, -3], [0, -2, -2, 0, 0, 2, 0, 0, 2, -2, -2, 2, 2, -2, 0, -1], [1, 0, 0, -1, -3, 0, 0, -1, 0, -1, -1, 1, 1, 0, 1, -1], [-4, 3, -1, -1, -3, -3, 0, 1, 0, -1, 4, -1, 0, 1, 1, 1], [0, -2, -1, 0, 3, 3, 1, -2, 0, -1, -2, 1, 0, 0, -1, -1], [0, 0, -5, -1, 0, -1, 0, -5, -1, -3, 0, -2, 3, 0, -1, -1], [-1, 0, -1, 0, 0, 0, 0, 0, 0, 1, -2, -1, 1, 0, 0, -1], [0, 2, 0, -4, -2, 0, 0, 1, 0, -1, -2, 0, 1, 3, 1, -3], [3, -4, 3, 1, 2, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0], [-6, 1, -1, 0, 1, 1, 2, 7, -6, -2, 2, 1, 3, -1, 0, -1], [-4, 2, 0, 0, 1, 0, -1, 2, -2, 0, 0, 0, 0, -1, -1, 0], [0, -4, 0, 2, 4, 5, 5, -1, 0, -1, 2, 0, -1, -3, 1, 3], [3, 4, 3, 0, 0, -4, 0, 0, 0, 1, 1, -1, -3, 0, 0, 1], [-1, 0, 1, 0, -2, 0, 0, 2, 0, 1, 0, 0, -1, 0, -1, 0], [-3, 1, -3, 3, -1, -1, 0, -3, 0, -3, 3, -1, 0, -3, 3, 3], [-4, 2, -2, 3, -4, -4, -2, 5, 0, -2, 2, 1, 0, -3, 0, 0], [-1, 0, -2, 2, 1, 0, 0, -1, 6, 0, -4, -2, 0, 0, -1, 2], [0, 0, 1, 0, 0, 0, 0, 0, -1, -2, -5, 2, -1, 0, 0, -2], [0, 5, 0, 2, -5, 3, 3, -1, 1, 1, -1, 0, -2, 0, -2, 0], [5, 0, 3, -3, -2, 0, 2, 2, -2, 0, 0, 2, -2, 0, 0, 0], [1, 1, 1, 1, 0, 0, 1, -1, 1, 0, 1, 0, -1, 0, 0, 1], [-3, -2, -1, 2, 1, 0, -1, -1, 1, 0, 0, 3, -3, -4, -4, 0], [0, -1, 0, 4, 1, -2, -2, 1, -2, 1, 5, 0, -1, -4, 0, 4], [-8, 1, 2, 0, 0, -1, 4, 0, -10, -1, 6, 1, -2, -1, 0, 1], [-4, 0, -3, 1, -4, 0, 0, 1, -1, 0, 2, -3, 0, 0, 3, 1], [1, 6, 2, -1, -3, -3, 3, 0, 0, 2, 1, -1, 0, 1, 0, 0], [-2, 0, -1, 2, -1, -1, -1, 4, 0, -1, -1, 2, 0, -2, -2, -2], [1, 0, 1, 0, 1, 1, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0], [0, 3, -1, 0, 0, -3, -3, 0, 1, -1, -1, 1, 1, -1, 0, 0], [0, 1, 0, 0, -1, 1, 1, -5, 2, 1, -4, 0, -1, -1, 1, 1], [-2, -3, -1, -3, 1, 0, -1, -4, 4, 0, 0, -1, 1, 2, 2, 0], [3, -3, -1, 0, 5, 4, 3, -3, 1, 1, 2, -2, 1, 1, -1, 1], [3, -4, -3, 0, 1, 0, -1, -1, 1, 0, 0, -3, 3, 2, 2, 0], [0, -4, 0, -3, 4, 2, 2, -2, 0, -5, -2, 0, 1, 0, 3, 0], [1, -1, -1, 0, 0, 1, 0, 0, 2, 0, -1, 0, 1, 1, 0, 0], [-3, 0, 0, 2, 2, 3, 0, 3, 2, -2, 0, 2, 2, 0, 2, 2], [5, -5, 0, 0, 1, 1, -4, -3, 0, 0, -7, 1, 0, 0, -2, -2], [-2, 4, 4, -2, 0, 0, 4, 6, 0, 4, 7, -1, 0, 2, 1, 1], [1, 0, 3, -2, 3, 1, 0, 2, -7, 2, 5, 1, -2, 0, -2, -2], [-5, -4, -3, 0, 0, 4, -2, 0, -2, 1, -1, -1, 3, 2, 0, -1], [0, -2, 0, -2, 2, 3, 3, 3, -2, -1, 4, 0, -1, -1, 3, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, -3, 3, -3, 2, 3, -3, -10, 7, -1, -8, 3, 2, 0, 0, -3], [2, -2, -3, 1, 6, 0, -6, -3, 3, 0, 0, -2, 2, 1, 1, 0]]}