# Stored data for newform 784.2.x.j, downloaded from the LMFDB on 19 April 2024. {"label": "784.2.x.j", "space_label": "784.2.x", "level": 784, "weight": 2, "hecke_orbit": 10, "hecke_orbit_code": 40533977227854608, "dim": 16, "is_polredabs": true, "trace_hash": 2295765574847212802, "analytic_rank": 0, "is_twist_minimal": false, "hecke_cutters": [[3, [10000, 4000, 800, 6720, -3056, -4176, 832, -1344, 1708, -136, 32, -32, -44, 8, 0, 0, 1]], [5, [16, -32, 32, -128, 80, 208, -64, 240, 108, -248, 32, -184, 100, 56, 8, 4, 1]]], "qexp_display": "q+(\\beta _{9}+\\beta _{12})q^{2}+(\\beta _{8}-\\beta _{13})q^{3}+(-1+\\cdots)q^{4}+\\cdots", "char_order": 12, "char_parity": 1, "char_degree": 4, "char_conductor": 112, "char_orbit_label": "x", "char_is_real": false, "Nk2": 3136, "analytic_conductor": 6.260271518468382, "hecke_ring_index": 4, "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": "0.273", "prec": 17}, "analytic_rank_proved": true, "char_values": [784, 12, [687, 197, 689], [12, 3, 8]], "hecke_ring_generator_nbound": 9, "inner_twist_count": 4, "level_primes": [2, 7], "conrey_indexes": [165, 373, 557, 765], "field_disc": 302713551298448326656, "field_disc_factorization": [[2, 24], [3, 8], [229, 4]], "field_poly": [256, -256, 256, 128, -208, 256, -16, -68, 121, -34, -4, 32, -13, 4, 4, -2, 1], "hecke_ring_index_factorization": [[2, 2]], "trace_moments": [{"__RealLiteral__": 0, "data": "0.050", "prec": 17}, {"__RealLiteral__": 0, "data": "14.018", "prec": 20}, {"__RealLiteral__": 0, "data": "5.412", "prec": 17}, {"__RealLiteral__": 0, "data": "1088.189", "prec": 27}, {"__RealLiteral__": 0, "data": "1674.310", "prec": 27}, {"__RealLiteral__": 0, "data": "169351.128", "prec": 34}], "related_objects": [], "self_twist_discs": [], "cm_discs": [], "rm_discs": [], "relative_dim": 4, "sato_tate_group": "1.2.3.c12", "trace_display": [-2, 0, -4, 0], "traces": [0, 16, -2, 0, -4, -4, 32, 0, -8, 0, 12, 0, -12, 0, 0, -16, -8, 24, -6, -12, 16, 0, -8, 0, 0, 0, 8, -24, 0, -32, -20, 16, 28, -24, 24, 0, 48, -16, -16, 0, 28, 0, 0, -64, 32, -8, -20, 24, -40, 0, -28, -8, 32, 8, -16, 0, 0, 0, -12, -28, 28, 28, -40, 0, -64, 48, -16, 0, 28, -88, 0, 0, -44, 0, 4, 28, 48, 0, 24, 24, -12, -40, 4, 0, 0, -80, 0, 0, 40, 0, 32, 0, 72, -16, 28, -16, 8, 64, 0, 96, -16, -8, 52, 0, 24, 0, 8, 8, -20, 40, 0, 0, 0, 128, 20, 4, -20, -20, 104, 0, 56, 0, 4, -8, -36, -56, 0, 32, 40, -16, -28, -24, 48, 0, 48, 0, -40, 0, 4, -72, 0, -112, 36, 40, -40, -16, -8, 0, 168, 24, -36, 0, 32, 0, 0, 80, 52, 24, -60, -64, -112, 0, -52, 32, -68, -40, -16, 0, 0, 0, -60, -8, -48, -8, -136, 0, -144, -48, 28, -16, -16, 72, 0, 0, -28, 0, 12, 8, -104, 0, -96, -16, 40, 16, -60, 56, 0, 32, -76, 0, -12, 0, 64, 0, -32, -8, 4, 72, 4, -80, 0, -64, 4, -48, -20, 0, 176, 0, 48, 40, 8, -48, -52, 16, 0, -80, -16, -12, -16, 48, 80, 0, -104, 0, -12, 24, 36, 64, 0, -128, 48, -8, 46, 16, -8, 0, 136, 0, 40, 0, -44, -48, 0, 144, 88, -72, -16, -56, -176, 0, -128, -32, 40, 0, 40, 0, 0, 48, -56, 0, -12, 72, -80, 0, -40, -32, -36, -8, 84, 0, 0, 0, 76, -64, 0, 36, -176, 0, 24, 8, 16, -72, 24, 112, 0, 0, -28, 0, 80, -36, -152, 0, -104, -112, 52, 0, 20, -16, 0, 32, 4, 0, 100, 0, -96, 0, -64, -24, -4, 32, -64, 80, 0, -80, 12, 76, -4, 0, 304, 0, 0, -48, -12, 48, -44, 128, 0, 0, 18, 36, 88, -40, 72, 0, 288, 0, -32, 16, 12, 8, 0, 144, 40, -104, -56, 80, 96, 0, 64, 0, -4, 0, -4, 256, 0, 0, 48, 112, -72, 144, -32, 0, -64, 56, 32, 0, -24, 0, 0, 144, -20, 100, 32, 48, -256, 0, 8, 104, -68, -24, 80, 0, 0, 0, -28, 112, -88, -68, 24, 0, 48, -80, 88, 112, -68, 80, 0, 0, 40, 0, -60, -24, -232, 0, -184, 8, 48, 64, 24, -192, 0, 208, -64, 0, -28, 0, 256, 0, -96, 104, -32, 32, 0, -160, 0, -64, -12, 12, 64, 0, 32, 0, -24, -8, 76, 56, -80, -144, 0, -160, 26, -136, 32, -28, 192, 0, 72, 0, 32, -80, 4, 200, 0, -128, 40, 40, -92, -36, 32, 0, 40, 0, 16, 0, 96, 32, 0, 32, 96, -40, -32, 48, -104, 0, -96, 16, -56, 0, 20, 0, 0, -48, 0, -16, -8, -24, -96, 0, 0, -128, -20, 136, -4, 0, 0, 0, -112, -52, 4, -60, -288, 0, -80, -32, -12, 32, 80, -112, 0, 0, -76, 0, -28, -12, -40, 0, -104, 48, 72, -24, -8, 128, 0, 0, -128, 0, 88, 0, 160, 0, -48, -64, 8, -104, -104, -48, 0, 144, -16, -24, -20, 0, 320, 0, 64, 104, -44, -80, -36, 96, 0, -256, -80, 92, 80, 4, 56, 0, -16, 0, -52, -120, 40, -32, 0, -176, 8, -56, 154, -60, -8, 0, 104, 0, 40, 0, 72, 24, 0, 32, 28, 160, -96, 24, 16, 0, -40, -64, -80, 0, -80, 0, 0, 64, 44, -28, -48, -24, -112, 0, 64, -80, 60, -24, 112, 0, 0, 0, 68, 8, -64, 24, -224, 0, -176, 88, -108, -32, -56, -144, 0, 0, -112, 0, 32, -92, 8, 0, -32, -56, 56, 48, 24, -24, 0, -16, -4, 0, 40, 0, 40, 0, -144, -40, -4, -40, -128, 0, 0, -112, -32, -40, 24, 0, 0, 0, -56, 8, -44, -104, 96, 16, 0, -96, 224, -36, 0, 76, 240, 0, 304, 0, -80, 144, -56, -96, 0, 64, 8, 32, -44, 56, 112, 0, 208, 0, -80, 0, -76, -144, 0, 0, -12, -200, 48, 8, -104, 0, 336, 32, -80, 0, -96, 0, 0, 64, 72, 28, -132, 0, -40, 0, -68, 96, 40, -96, -52, 0, 0, 0, 24, -128, 28, 28, 0, 0, 64, 64, 140, -48, -4, -152, 0, 0, -128, 0, 0, 12, 144, 0, 56, 160, -88, -32, -136, -56, 0, 80, -48, 0, 76, 0, 56, 0, 80, 16, 100, -104, 88, -112, 0, -48, -40, -12, 120, 0, 32, 0, 168, -136, -12, 0, 40, 144, 0, 416, 104, -36, -60, -88, 176, 0, 112, 0, 148, -48, -68, 0, 0, 192, 140, 72, -76, 48, 48, 0, 136, 0, -108, 0, 80, 0, 0, -16, 32, 152, -40, -48, -184, 0, 32, 64, -88, 0, 256, 0, 0, -64, 76, -36, -20, -168, -64, 0, -112, 72, 64, 48, -48, 0, 0, 0, -88, 272, -32, -24, -216, 0, -208, 144, -20, 64, 152, 128, 0, 0, -16, 0, 112, -24, -16, 0, 48, 128, -96, -48, 92, -176, 0, -192, -20, 0, -44, 0, -56, 0, -16, 24, 80, 160, 8, 16, 0, 224, -112, -84, 160, 0, 208, 0, 208, 0, -24, 104, 28, -416, 0, 288, -136, -120, -12, -40, 288, 0, 8, 0, -12, -24, -8, -56, 0, -224, 0, -24, 104, -4, 104, 0, -144, 0, -48, 0, 68, -304, 0, -192, 40, -272, -112, -80, 208, 0, 160, -64, 48, 0, 20, 0, 0, -528, -8, -56, 84, 120, -56, 0, 96, -72, 32, 8, 92, 0, 0, 0, -68, -64, 4, 24, -184, 0, -208, -8, -184, -32, 76, 72, 0, 0, -24, 0, 28, 180, 200, 0, 8, -264, -12, -48, 56, 192, 0, -112, 80, 0, -112, 0, 16, 0, -152, 56, -24, 40, 8, 64, 0, 208, -24, 4, 36, 0, 112], "weight_parity": 1, "has_non_self_twist": 1, "level_is_prime": false, "level_is_prime_power": false, "level_is_square": true, "level_is_squarefree": false, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 7, 3, 1, 3, 0], [1, 1, 16, 5, 1, 4, 0], [1, 1, 112, 23, 1, 12, 0]], "char_orbit_index": 24, "prim_orbit_index": 23, "minimal_twist": "112.2.m.c", "char_is_minimal": false, "conrey_index": 165, "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, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 0], [2, 0, 0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1], [0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1], [2, 0, 0, -1, 0, 0, 0, 1, -1, 0, 2, 0, 0, -1, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, -1, 1, 0, 1, 0, 2, 1, 1, 2, 0, -1, -1], [-1, 2, 0, 0, 1, -1, 1, 0, 0, -2, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, 0, 1, 0, 1, 1, 0, 0, -1, 0, 0, 0, 1, 1, 1], [0, -1, 1, 0, 1, 1, -1, -1, 1, -1, 1, 1, -1, 0, 1, -1], [4, 0, 0, 0, 0, 0, 0, 0, -1, 0, 4, -1, -1, -1, 0, 0], [-1, 0, 1, 1, 0, 0, 0, 1, 1, 0, -1, 1, 0, -1, 0, 0], [0, 1, 2, 0, 2, -2, 2, -2, -2, -1, 2, -1, -1, 0, 2, 0], [2, 0, 0, 0, 2, -2, -1, 0, 0, 1, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2], [0, -1, -1, 0, 0, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, 1], [-1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1, 3, -1, 1, 0, 0], [0, 0, 1, 1, 0, 0, 0, -1, 2, 0, 0, -2, -2, 2, 0, 0], [0, 2, 2, 0, 3, 1, 1, -3, -1, 0, 0, -2, 0, 0, 2, 1], [-2, -1, 0, -2, -2, 1, 1, 0, 0, -1, 0, 0, 0, 1, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, 0, 0, 1, 2, -2, 0, 0, 2, 0, 0, 0, 2, -2, 0], [0, 2, 0, 0, 0, 2, 1, 0, -1, -1, 4, -2, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 1, 1, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -2, -2, 2, 0, 0], [0, 2, -2, 0, 0, 0, 2, 0, -2, 0, 4, -2, 0, 0, -2, -2], [3, -1, 0, -1, -1, -1, -1, 0, 0, 3, 0, 0, 0, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, -1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 2, 1, 0], [0, 2, 0, 0, -3, 2, 2, 3, -2, 2, 3, -2, 2, 0, 0, 4], [-1, 0, -1, -2, 0, 0, 0, 0, 1, 0, -1, -3, 0, -2, 0, 0], [0, 0, -3, -3, 0, 0, 0, 0, -1, 0, 0, 1, 3, -3, 0, 0], [0, -1, -1, 0, 1, -1, -3, -1, 3, 1, -5, 1, 1, 0, -1, -1], [0, -1, 0, -4, 0, -3, 1, 0, 0, -3, 0, 0, 0, -3, 4, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -2, 0, -4, -4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -4], [0, 0, 0, 0, -2, 2, -2, 2, 2, 0, -4, 0, 0, 0, 0, -2], [-1, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 0, 0, -1, 0, 0], [-2, 0, 2, 0, 0, 0, 0, 4, -1, 0, -2, 0, 1, -2, 0, 0], [0, -3, 0, 0, 0, 1, -3, 0, 3, -1, -4, 3, -1, 0, 0, 0], [-2, 0, 0, 2, 6, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -3, 0, 1, -7, 3, -1, 0, 0, 2, 0, 0, 0, 3, -1, 1], [0, 2, 2, 0, 2, 2, -2, -2, 2, -2, 2, -2, -2, 0, 2, 0], [4, 0, -2, 0, 0, 0, 0, -2, 3, 0, 4, -2, 1, -2, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, -2, 0, 0, 0], [0, 1, 1, 0, -1, -1, 3, 1, -3, -1, 3, -1, -1, 0, 1, 3], [0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, -2, -2, -2, 0, 0, -2, 0, 0, 0, -2, 1, -1], [0, -4, 0, 0, -2, -2, -3, 2, 3, -3, 0, 4, -3, 0, 0, -2], [-2, 0, 0, -2, 0, 0, 0, -2, -3, 0, -2, 0, 0, -3, 0, 0], [5, 0, 1, -1, 0, 0, 0, -3, -1, 0, 5, -1, -3, -1, 0, 0], [0, 1, 0, 0, -3, 0, 0, 3, 0, -1, -3, -1, -1, 0, 0, 0], [-3, 1, 0, 3, 3, 3, 1, 0, 0, 3, 0, 0, 0, 3, 1, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -2, 0, 0, -4, 0, -2, 0, 0, 4, 0, 0, 0, 0, -2, 0], [0, -2, -1, 0, 0, -2, -2, 0, 2, 2, -8, 2, 2, 0, -1, -1], [-2, 0, 2, -4, 0, 0, 0, -4, 0, 0, -2, 2, -2, 0, 0, 0], [-2, 0, -4, 0, 0, 0, 0, 2, 4, 0, -2, 4, 4, -4, 0, 0], [0, -4, -2, 0, 0, -2, -2, 0, 2, 2, -6, 4, 2, 0, -2, 0], [-6, 1, 0, -4, -6, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 10, -6, 0, 0, 0, -6, 0, 0, 0, -6, 2, -2], [0, 1, -1, 0, -1, -1, 0, 1, 0, 2, 5, -1, 2, 0, -1, -1], [0, 0, -2, -2, 0, 0, 0, 6, 2, 0, 0, -2, 2, -2, 0, 0], [2, 0, -2, 2, 0, 0, 0, 2, -1, 0, 2, 2, -1, -6, 0, 0], [0, -4, 0, 0, 6, -5, -5, -6, 5, -4, -6, 4, -4, 0, 0, -2], [3, 1, 0, 3, -5, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 2, 6, -4, 3, 0, 0, -1, 0, 0, 0, -4, 0, 2], [0, -2, 2, 0, 0, -4, -2, 0, 2, 4, -6, 2, 4, 0, 2, 2], [-4, 0, 0, 2, 0, 0, 0, -2, 2, 0, -4, 2, 2, 4, 0, 0], [-5, 0, 1, -1, 0, 0, 0, 0, -2, 0, -5, -2, 2, 2, 0, 0], [0, 7, -1, 0, 1, -3, 3, -1, -3, 1, 1, -7, 1, 0, -1, 5], [0, 0, 0, 0, 0, 1, -1, 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], [-8, 0, 0, 2, -8, 6, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2], [0, 2, 2, 0, 0, -4, 6, 0, -6, -4, 2, -2, -4, 0, 2, 4], [0, 0, 2, 2, 0, 0, 0, 0, -1, 0, 0, 1, -7, 7, 0, 0], [4, 0, 0, 0, 0, 0, 0, -4, -2, 0, 4, 0, 2, 4, 0, 0], [0, 0, -4, 0, -6, 6, 0, 6, 0, 6, 0, 0, 6, 0, -4, 4], [2, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, -1, 0, -1, 1, -1, -3, 0, 0, -3, 0, 0, 0, -1, 3, -1], [0, -4, -6, 0, 0, 4, -4, 0, 4, 4, 0, 4, 4, 0, -6, 0], [3, 0, 1, 3, 0, 0, 0, 3, -1, 0, 3, -1, 1, 3, 0, 0], [-4, 0, 1, -1, 0, 0, 0, 0, 1, 0, -4, 1, 3, 3, 0, 0], [0, -2, 2, 0, 0, 0, -6, 0, 6, -4, -2, 2, -4, 0, 2, -4], [4, -3, 0, -2, 0, 3, -3, 0, 0, -3, 0, 0, 0, 3, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, -6, 4, -4, 0, 0, 0, 0, 0, 0, 4, 2, 0], [0, -3, -1, 0, -6, 0, 5, 6, -5, -2, 3, 3, -2, 0, -1, -2]]}