# Stored data for newform 513.2.f.h, downloaded from the LMFDB on 17 September 2025. {"Nk2": 2052, "analytic_conductor": 4.096325623691683, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 19, "char_degree": 2, "char_is_minimal": true, "char_is_real": false, "char_orbit_index": 6, "char_orbit_label": "f", "char_order": 3, "char_parity": 1, "char_values": [513, 3, [191, 325], [3, 2]], "cm_discs": [], "conrey_index": 163, "dim": 12, "field_disc": 26502427891609569, "field_disc_factorization": [[3, 10], [31, 2], [21611, 2]], "field_poly": [9, 9, 45, -12, 135, 9, 91, -7, 41, -1, 8, -1, 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, [9, 9, 45, -12, 135, 9, 91, -7, 41, -1, 8, -1, 1]], [7, [3, 33, 98, 34, -27, -1, 1]]], "hecke_orbit": 8, "hecke_orbit_code": 31525541022532097, "hecke_ring_generator_nbound": 4, "hecke_ring_index": 9, "hecke_ring_index_factorization": [[3, 2]], "hecke_ring_index_proved": true, "inner_twist_count": 2, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 19, 3, 1, 3, 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": "513.2.f.h", "level": 513, "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, 19], "level_radical": 57, "minimal_twist": "513.2.f.f", "prim_orbit_index": 3, "qexp_display": "q+(-\\beta _{1}-\\beta _{6})q^{2}+(-\\beta _{1}+\\beta _{3}-\\beta _{8}+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 6, "rm_discs": [], "sato_tate_group": "1.2.3.c3", "self_twist_discs": [], "self_twist_type": 0, "space_label": "513.2.f", "trace_display": [1, 0, 5, 2], "trace_hash": 799156255655209606, "trace_moments": [{"__RealLiteral__": 0, "data": "0.110", "prec": 17}, {"__RealLiteral__": 0, "data": "11.882", "prec": 20}, {"__RealLiteral__": 0, "data": "5.334", "prec": 17}, {"__RealLiteral__": 0, "data": "627.608", "prec": 24}, {"__RealLiteral__": 0, "data": "872.976", "prec": 24}, {"__RealLiteral__": 0, "data": "65932.146", "prec": 30}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.003", "prec": 17}, "traces": [0, 12, 1, 0, -3, 5, 0, 2, -12, 0, 1, -4, 0, -5, 2, 0, 3, 10, 0, -9, -2, 0, -4, 3, 0, -5, -2, 0, -2, -6, 0, -4, 8, 0, 4, 3, 0, 26, -10, 0, -6, -13, 0, -3, -20, 0, -10, 18, 0, 26, 6, 0, -22, -1, 0, -6, 48, 0, 28, 11, 0, -13, 22, 0, 0, -72, 0, 9, -10, 0, -30, -6, 0, -15, 43, 0, -33, -58, 0, -13, 1, 0, 4, -16, 0, -3, -7, 0, -30, 29, 0, 22, 21, 0, 36, 22, 0, 20, -59, 0, 44, -10, 0, 16, 54, 0, -34, 56, 0, -5, 36, 0, -5, 16, 0, 10, 24, 0, -4, -2, 0, -16, -108, 0, -32, -84, 0, 12, -18, 0, -6, 42, 0, 49, -4, 0, 21, 24, 0, -25, 54, 0, 51, -11, 0, 2, -20, 0, -2, 24, 0, -80, 18, 0, -1, -38, 0, 4, -70, 0, 80, 55, 0, 50, 16, 0, 10, -13, 0, -7, 84, 0, -48, -5, 0, -4, 43, 0, 88, -110, 0, 25, -74, 0, 3, 43, 0, 37, 60, 0, -52, -54, 0, 15, -9, 0, 25, -32, 0, -47, -33, 0, 12, 59, 0, -20, 42, 0, 68, 7, 0, -13, -77, 0, -47, -53, 0, -108, 75, 0, -42, 56, 0, -47, 58, 0, -38, 102, 0, -64, 23, 0, -21, 14, 0, -42, -188, 0, 4, -108, 0, -23, -9, 0, -35, 106, 0, -13, -90, 0, -12, -13, 0, -5, 100, 0, 9, -51, 0, 24, 18, 0, -20, 43, 0, 18, 13, 0, 21, -43, 0, 33, -82, 0, 24, 56, 0, -48, -24, 0, 81, 39, 0, 8, 60, 0, 70, 0, 0, 14, 76, 0, -42, -138, 0, -9, -60, 0, 111, -1, 0, -23, 32, 0, 105, -58, 0, 30, -32, 0, 17, 56, 0, 61, -28, 0, -20, 40, 0, -46, 23, 0, -152, 92, 0, -29, -92, 0, 12, -51, 0, 94, 28, 0, 2, 154, 0, 61, 70, 0, 24, 102, 0, -68, -18, 0, 14, -23, 0, 130, -208, 0, -11, -210, 0, 30, 52, 0, 86, 109, 0, -87, -36, 0, 11, 32, 0, 7, 18, 0, -70, -1, 0, 64, 7, 0, -84, 28, 0, 134, -16, 0, -27, -16, 0, -144, -110, 0, -116, -3, 0, 14, -12, 0, -75, 21, 0, 13, 200, 0, 20, -2, 0, -9, 64, 0, -136, -162, 0, 9, -10, 0, -76, 59, 0, -29, 35, 0, -32, -144, 0, -9, 10, 0, 33, 16, 0, -8, -5, 0, 38, 70, 0, 55, 6, 0, 80, 69, 0, 44, -105, 0, 110, -28, 0, -8, 104, 0, -112, 32, 0, 0, 77, 0, 27, 150, 0, 62, 11, 0, 8, -3, 0, -60, -186, 0, -65, -200, 0, 64, -27, 0, -66, 46, 0, 188, -40, 0, 39, 14, 0, -60, 142, 0, 97, -70, 0, 60, -9, 0, 68, 23, 0, -170, -22, 0, -20, -90, 0, -4, -105, 0, 176, 44, 0, -22, -20, 0, -12, 34, 0, 59, 136, 0, -116, -35, 0, 69, 72, 0, 120, -92, 0, 66, -54, 0, -11, 0, 0, 57, 27, 0, -103, -42, 0, 37, -36, 0, 17, 60, 0, -79, -9, 0, 20, 85, 0, -3, 96, 0, 140, 153, 0, 32, -89, 0, -149, 50, 0, -148, 87, 0, -98, 148, 0, -102, 84, 0, -39, -42, 0, -100, -4, 0, -24, -37, 0, -108, -72, 0, -20, -264, 0, -103, -42, 0, -91, 105, 0, -26, -48, 0, -128, 9, 0, 54, 42, 0, -61, -30, 0, 10, -59, 0, 17, 62, 0, -2, 13, 0, -5, 45, 0, 93, -135, 0, -18, 7, 0, -80, -166, 0, 136, -15, 0, -44, 78, 0, 182, -72, 0, -20, 51, 0, -176, -130, 0, -99, 26, 0, 141, 124, 0, 21, 29, 0, 128, -120, 0, 59, -68, 0, 43, -86, 0, 83, 10, 0, -24, 100, 0, 18, 37, 0, -222, 30, 0, -69, -86, 0, 68, -17, 0, 140, 36, 0, 208, 60, 0, 7, 54, 0, 60, 180, 0, 48, 85, 0, -15, -23, 0, 84, -40, 0, 94, -82, 0, 27, 20, 0, 100, 97, 0, -128, 80, 0, -32, 1, 0, -34, 132, 0, -111, -44, 0, -90, -100, 0, -36, -22, 0, 326, 4, 0, -49, -32, 0, -65, -87, 0, -230, -19, 0, 126, -120, 0, -86, -56, 0, 26, 26, 0, 140, -4, 0, -54, 72, 0, 14, -48, 0, 52, 6, 0, -203, -52, 0, -65, -82, 0, 30, 88, 0, -43, 90, 0, -63, -128, 0, 13, -21, 0, -78, 37, 0, 118, -33, 0, 60, 31, 0, 42, -58, 0, 109, 8, 0, -2, 68, 0, -108, 298, 0, -7, 51, 0, 71, -76, 0, 78, 30, 0, 62, -79, 0, 168, -90, 0, -51, -250, 0, 13, -11, 0, 0, 139, 0, 60, -68, 0, 32, 33, 0, 42, 168, 0, 140, 52, 0, 40, -25, 0, 1, 63, 0, -354, 5, 0, 70, 72, 0, 6, -61, 0, 190, 13, 0, 58, -110, 0, 1, -29, 0, -21, -80, 0, -198, 11, 0, -31, 35, 0, 196, -108, 0, 50, 170, 0, -24, -31, 0, 104, -9, 0, -161, -174, 0, 26, -78, 0, -36, 18, 0, -122, -77, 0, 98, 52, 0, -136, 59, 0, 158, 78, 0, -22, -18, 0, -212, -22, 0, -146, 24, 0, 276, 178, 0, -42, 12, 0, -117, -20, 0, -90, 26, 0, 22, 2, 0, -300, 72, 0, 51, -30, 0, -125, -7, 0, -70, 28, 0, -53, 94, 0, -69, 6, 0, 8, -20, 0, -105, -59, 0, -8, -11, 0, 46, -39, 0, -50, -129, 0, 33, -21, 0, 153, 21, 0, -216, -96, 0, -80, -318, 0, 152, -28, 0, -135, 168, 0, 58, -18, 0, -8, 104, 0, 4, 42, 0, -36, 200, 0, 178, -2, 0, 7, -86, 0, 250, -12, 0, -37, -82, 0, -39, -148, 0, 157, 108, 0, -258], "weight": 2, "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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0], [-1, 0, 0, 1, 0, 0, 1, 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, -1, 0, 0], [-1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, -1, 1, 0, 1, -1, 0, 0, 0], [1, 0, -1, 0, -1, 0, 0, 1, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, -1, 0, 1, 0, 2, 0, 0, 1, 1, 0], [1, 1, -1, 0, 1, 0, 1, 1, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 1], [0, 0, 0, 1, 0, -1, 1, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1], [0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -2, 0, 2, 0, 2, 2, 0, 0, 0], [0, 0, -2, 2, 0, 0, 4, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 1, 1, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, -1, 0, 1, -2, 0, 0, 0, -1, 1], [0, 0, 0, -1, -1, -1, 0, -1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 1, -1, 0, -1, 2, 1, 0, 0], [1, -1, 3, 0, 2, 0, -1, 1, 2, -3, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 1, 1, 0, -1, -3, 0, 0, 0, -1, -1], [-1, -1, -1, 2, -2, 1, 2, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 0, 1, 0, 1, 0, 2, -2, 1, 0], [-2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 0, 0, 0, -1, -1, -1, 1, 0, -2], [0, 0, 0, -1, 0, 0, 0, 4, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, -1, -2, 0, 0, 0, -2, -1], [0, 2, 0, -2, 1, 1, 0, -2, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -1, 0, 0, -3, 4, 0, 0, 0, 0, -3], [0, 0, -2, -1, 0, 0, -3, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 6, -1, 0, 2, 0, 6, -3, 3, 1, 2, 0], [0, -2, 0, 3, 1, 0, 0, -2, -3, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -2, 0, -3, 0, 1, -1, -2, 2, -3, 2], [5, 0, -1, 1, 0, -2, 1, 0, 0, 0, -1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, -2, 0, -2, -3, 0, 0, 0, -3, -2], [0, -2, 1, 0, 1, 0, -2, 0, -4, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0], [2, -3, 0, 0, 0, 0, -3, 2, -3, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 1, 0, -1, 1, 0, 0, 0, 0, -1], [-6, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, -2, 0, -2, 1, 2, 0, 0], [-1, 0, 0, -1, 0, -1, -3, 0, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, -1, 3, 0, 6, 1, 2, 0, 0], [0, 3, -1, 0, -2, 0, 3, 0, 0, 1, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -3, 3, 0, -1, 0, -3, -2, -1, -3, -1, 1], [6, -4, 2, 0, 0, 0, -4, 6, -4, -2, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 5, -1, 1, 1, 1, 5, 0, 2, 0, 0, 2], [-6, 0, 1, -2, 0, 4, -1, 0, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -4, 0, 0, 1, 0, -4, -3, 1, 0, 1, 0], [0, 3, 0, 0, -2, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 1, 0, 1, -2, 0, 0, 0, -5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -2, 1, 0, 2, 0, 0, 0, 0], [0, -2, 0, 1, 0, -1, 0, 1, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3], [0, 4, 0, -1, 2, 1, 0, -5, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -1, 2, -2, 0, -4, 1, -3, 0, 0], [3, 0, 1, 0, 0, 0, 0, 3, -2, -1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -3, 0, 1, -8, 0, 0, 0, -1, 1], [0, -2, 1, 3, 1, -2, 1, -7, -2, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 0, 1, 0, -1, 2, -2, 1, 1, -2], [-8, 1, 3, 0, 1, 0, 1, -8, 4, -3, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, -3, 2, 0, 1, 0, -3, 7, -1, -2, 1, -2]]}