# Stored data for newform 690.2.e.b, downloaded from the LMFDB on 25 April 2024. {"label": "690.2.e.b", "space_label": "690.2.e", "level": 690, "weight": 2, "hecke_orbit": 2, "hecke_orbit_code": 4503874538832562, "dim": 16, "is_polredabs": true, "trace_hash": 1378425090351160396, "analytic_rank": 0, "is_twist_minimal": true, "hecke_cutters": [[11, [68, 920, 1560, -1704, 93, 202, -29, -6, 1]]], "qexp_display": "q+\\beta _{9}q^{2}+\\beta _{2}q^{3}-q^{4}+q^{5}+\\beta _{1}q^{6}+\\cdots", "char_order": 2, "char_parity": 1, "char_degree": 1, "char_conductor": 69, "char_orbit_label": "e", "char_is_real": true, "Nk2": 2760, "analytic_conductor": 5.509677739468347, "hecke_ring_index": 8, "level_radical": 690, "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": "0.515", "prec": 17}, "analytic_rank_proved": true, "char_values": [690, 2, [461, 277, 511], [1, 2, 1]], "hecke_ring_generator_nbound": 11, "inner_twist_count": 2, "level_primes": [2, 3, 5, 23], "conrey_indexes": [551], "field_disc": 2245030614384775328060932096, "field_disc_factorization": [[2, 24], [17, 2], [197, 2], [3454109, 2]], "field_poly": [6561, -4374, 2187, -2916, 1215, -108, 405, -198, -32, -66, 45, -4, 15, -12, 3, -2, 1], "hecke_ring_index_factorization": [[2, 3]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "12.638", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "948.865", "prec": 24}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "112903.199", "prec": 34}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 16, "sato_tate_group": "1.2.3.c2", "trace_display": [0, 0, 16, 0], "traces": [0, 16, 0, 0, -16, 16, 2, 0, 0, 2, 0, 12, 0, 0, 12, 0, 16, 0, 8, 0, -16, 4, 0, -4, -2, 16, 0, 24, 0, 0, 2, 4, 0, 28, 0, 0, -2, 0, 16, -8, 0, 0, 0, 0, -12, 2, -4, 0, 0, -4, 0, 2, 0, 8, -26, 12, -12, -28, -8, 0, 0, 0, 0, 0, -16, 0, -10, 0, 0, -22, 12, 0, -8, -16, 24, 0, 0, 0, -12, 0, 16, 22, -16, 40, -4, 0, 40, 20, 0, -80, 8, 0, 4, -4, -24, 0, 2, 0, 0, 8, -16, 0, 16, 0, 0, 4, 0, 0, -24, 0, 0, -40, 0, -32, 20, -4, 0, 20, 8, 0, -2, 12, -48, -20, -4, 16, -18, -32, 0, -28, 0, 0, -28, -24, 16, 24, 0, 8, 40, -64, 0, 32, 0, 8, 2, 0, 0, -92, 0, -20, 2, 36, -16, -52, 48, 4, 8, 0, -8, 52, 0, 24, -16, -24, 0, 28, 0, 0, 0, 108, 0, 106, 0, 0, -4, 0, 12, -40, 0, 0, -2, 0, -88, -120, 4, 0, 20, -80, 0, 2, 16, -48, 0, 72, -20, -8, 4, 0, -4, 0, 0, 0, 24, 24, -2, 0, 12, -40, 0, 0, 0, 0, -8, -8, 0, 0, 26, 0, 24, -24, -12, 40, -60, -32, 12, 2, 0, -48, 28, 0, -4, 106, 8, 0, -26, 0, 0, 0, 64, 0, 0, 0, 0, -20, 0, -4, -26, 0, 0, -112, 0, -68, 0, 16, 0, 2, 16, 0, -36, -64, 0, -36, -24, -80, 10, 8, 0, -56, 0, 0, -26, -76, 0, 92, 0, 12, 22, -96, 0, 66, -12, -16, 12, 0, 0, -28, 0, 144, 8, 92, -8, -2, 16, -16, 40, 0, -24, 132, 0, 92, 0, -16, 0, 16, 0, 0, 40, -88, 0, 150, 0, 0, 12, 0, 72, 0, 0, 0, 24, 0, -16, -76, -16, 0, -22, 0, 0, 80, 16, 0, -10, 72, -40, 24, -16, 0, 4, 0, 0, 8, 0, 216, 20, 0, -40, -22, 36, 0, -20, -16, 12, -42, 0, 0, 44, 0, 80, 52, 8, 16, -8, 60, 112, 40, 0, -16, 42, 0, -4, 24, 24, 0, 4, 0, 0, 0, 24, 0, 76, 0, 0, -64, 0, 16, -2, 0, 0, -32, 0, -100, -12, 44, 0, -76, 44, 0, -8, 168, -56, -14, 16, -160, -88, 40, 0, 22, 0, 0, -16, -132, -16, 92, 0, 72, 66, 40, 0, 4, 32, -96, -4, 0, 0, -84, 0, 0, 30, -88, 0, -98, 40, 128, 24, 0, 20, 20, 0, 96, -32, -12, 0, -42, 0, 0, 40, -80, 0, 80, 0, 0, 8, 0, 32, 144, 0, 0, -20, 0, -32, -168, 4, 0, 72, 120, 0, -4, -80, 16, -20, 64, -24, -28, -8, 0, 68, 0, 0, -72, -152, 88, 2, 0, -64, -56, -12, 0, -68, -16, 48, -28, 0, 0, 20, 0, 52, 8, 4, 88, -116, -152, -16, 36, 0, -200, 18, 0, 48, 76, 32, 0, 16, 0, 0, 72, 40, 0, 28, 0, 0, -14, 0, -56, -108, 0, 0, 4, 0, -208, 28, -40, 0, 16, 24, 0, -8, 0, -16, 124, -24, -216, -24, -128, 0, -72, 0, 0, -182, 256, -8, 84, 0, 88, -40, 16, 0, -40, 64, -80, -48, 0, 0, 32, 0, -40, -32, -32, -160, 32, 0, -8, 20, 0, -8, 112, 0, -4, -2, 72, 0, -100, 0, 0, 56, 16, 0, 20, 0, 0, 92, 0, 8, -6, 0, 0, 72, 0, 20, -180, -128, 0, -2, -196, 0, 92, -36, 12, -44, -32, 16, 96, -48, 0, 52, 0, 0, -20, -48, 112, 52, 0, -4, -106, 32, 0, -8, 16, 36, -28, 0, 0, -18, 0, 8, -108, -28, -32, -52, 184, 144, 72, 0, -224, 112, 0, -24, -28, -80, 0, 16, 0, 0, 86, 24, 0, 88, 0, 0, 68, 0, 192, -28, 0, 0, -136, 0, -24, 36, 40, 0, -96, 16, 0, 0, 56, -68, 24, -108, -64, -144, 208, 0, -16, 0, 0, -106, 8, -144, 88, 0, 224, 40, -112, 0, 0, 12, -64, 4, 0, 0, -40, 0, -172, 80, -96, -12, 32, -24, 152, 40, 0, 0, 48, 0, 216, -66, 8, 0, 40, 0, 0, 2, -152, 0, 164, 0, 0, 38, 0, 88, 122, 0, 0, 120, 0, -80, -92, -4, 0, -96, -40, 0, 42, 48, -32, -20, -20, -32, -24, 80, 0, 2, 0, 0, -108, 80, 36, -2, 0, -88, -50, -16, 0, -88, -112, 48, -52, 0, 0, 0, 0, 48, -100, -72, -88, -224, 4, 20, -132, 0, -16, 8, 0, -88, -140, -4, 0, -20, 0, 0, 110, -8, 0, 4, 0, 0, 52, 0, 24, -68, 0, 0, -192, 0, -96, 0, 24, 0, 96, -24, 0, -16, -88, -24, 68, -88, -24, 2, -16, 0, -144, 0, 0, -164, 32, -12, 28, 0, 392, 40, 24, 0, 112, 0, -104, 112, 0, 0, 60, 0, -152, 0, 40, -96, 28, 0, 108, -112, 0, 8, 20, 0, 16, 8, 128, 0, 106, 0, 0, 108, 88, 0, 102, 0, 0, -26, 0, 252, -200, 0, 0, -4, 0, -24, -60, -20, 0, 24, 40, 0, 148, 12, -88, 24, 112, -40, -40, -52, 0, 60, 0, 0, -78, 32, 216, 44, 0, -12, -316, 64, 0, -2, 184, 16, 140, 0, 0, 166, 0, 48, -8, -88, 72, -28, -288, 208, -120, 0, -288, 68, 0, 4, 98, -24, 0, -106, 0, 0, 72, -8, 0, 20, 0, 0, -112, 0, -80, 26, 0, 0, -8, 0, -300, 60, 8, 0, 2, -16, 0, 0, 8, 16, 12, -64, -80, -128, -48, 0, -36, 0, 0, 0, 156, -64, 56, 0, 72, 0, -48, 0, 142, -20, 36, 20, 0, 0, -8, 0, -120, 88, 240, 4, 2, 72, 48, 26, 0, -72, 216, 0, 160, -4, -12, 0, 24, 0, 0, 112, -80, 0, 100, 0], "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, 69, 3, 1, 2, 0]], "char_orbit_index": 5, "prim_orbit_index": 3, "minimal_twist": "690.2.e.a", "char_is_minimal": true, "conrey_index": 551, "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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 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], [1, 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, 0, 1, 0, -1, 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, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 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, -1, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 1, 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], [1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, -1, 0, -1, -1, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0], [0, -1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, -1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, 0, -1, -1, -1, -1, 1, 0, 1, 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, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1], [1, 0, 1, 0, 1, 0, 0, 0, -1, 1, -1, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, -1, 0, -1, 2, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, -1, 0, -1, 1, 1, 0, -1, -1, 0, 0, 1], [0, -1, 0, -1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 0, 0, 0, 1, 1, -1, 1, 0, 0, 1, 0, 0], [0, 1, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0], [-1, -1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 2, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 1, 0, -2, -1, 2, 1, 0, -1, -1, 1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 1, -1], [0, 1, 1, 1, -1, 1, 0, -1, -2, -2, 1, 0, -1, -2, 1, -1], [-1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1, -1], [0, 1, -1, 1, 1, -1, 0, 0, -1, 0, -1, 0, 1, -1, -1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -2, -1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1, 0, 2, -2], [0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0], [0, 1, 1, -1, -1, 1, 0, 0, -1, 0, -1, 0, 1, 1, -1, 0], [-1, 1, 0, 1, 0, -1, 1, 0, 1, 1, 0, 0, 1, 0, -1, 0], [1, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-1, -2, -1, 0, -1, 0, 0, 0, 1, -1, -2, 0, -1, -1, 0, -1], [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 1, -1, 3, 3, 0, 0, -1, 0, 0], [0, 1, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, -2], [0, -1, 0, -2, 0, -1, 0, 1, 2, 1, 0, 0, 0, 1, 1, 1], [-1, 0, 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, -1, 0], [-1, 1, 1, 0, 0, 0, 1, -1, -1, 2, 0, 1, -1, -1, 0, 0], [0, 1, 1, 1, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, -1, 1], [-1, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0], [-2, 0, 0, -2, 2, 0, -1, 0, 0, -3, 1, 0, 1, 1, 1, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, -1, -2, 2, 0, -1, -1, 2, 0, 0, -2, -1, 2, -3], [-1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], [0, -1, -1, 1, -1, -1, 0, 0, 1, 0, 1, -2, -1, -1, -1, 0], [1, 1, -1, -1, 0, -1, 0, 0, 0, 0, 1, -1, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, 1, 0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0], [0, -3, -2, -3, 0, 0, 0, 2, 1, -4, -2, 0, 0, 1, 0, 2], [-1, 0, 1, 0, 1, -1, 1, 2, 1, -1, 0, 1, 0, 1, 0, 1], [0, -1, -1, -1, 0, 0, 0, 1, 1, 1, -1, 0, 0, 1, 0, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 2, 1, -1, 0, -1, 0, 0, 2, 1, -2, -1, 1, 0, 0], [-2, 1, 0, -1, 1, 1, 0, 0, -1, 0, 0, 2, 1, 1, 1, 0], [3, -3, -3, 3, 0, 1, -2, 0, 0, 0, 3, 1, 1, 0, 0, 0], [-1, 1, 0, 0, 1, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, -1, 0, -1, -1, 0], [2, 1, -1, -1, 1, 2, -1, 0, -1, 0, 1, 1, 2, 1, 1, 0], [2, 1, -1, 3, 1, 0, -1, 0, 1, 0, 1, -1, 2, 1, -1, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1], [-5, 2, 0, -2, -1, 1, 3, 0, -1, 0, 0, 0, 1, 1, -1, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0], [0, 4, -1, 4, 1, -1, 0, -2, -1, 4, -1, 0, 1, -1, -1, 2], [0, 0, 1, 1, 1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, -2, 1, 0, 1, 0, 0, -2, 1, -1, 0, -2, -1, -1, 0, 3], [0, -1, -1, 1, -1, 1, 0, 0, 1, 0, 1, 0, 1, -1, -1, 0], [0, -1, 1, -1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 2, -2, 0, -1, 1, 0, -1, 0, 2, 1, -2, 1], [0, -2, -1, -2, 0, 0, 0, 2, 0, -1, -1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 2, -1, -2, 0, -1, -1, -3, -2, 1, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]}