# Stored data for newform 78.2.e.a, downloaded from the LMFDB on 12 August 2026.
{"Nk2": 312, "analytic_conductor": 0.622833135765987, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 13, "char_degree": 2, "char_is_minimal": true, "char_is_real": false, "char_orbit_index": 5, "char_orbit_label": "e", "char_order": 3, "char_parity": 1, "char_values": [78, 3, [53, 67], [3, 1]], "cm_discs": [], "conrey_index": 55, "dim": 2, "field_disc": -3, "field_disc_factorization": [[-1, 1], [3, 1]], "field_poly": [1, -1, 1], "field_poly_is_cyclotomic": true, "field_poly_is_real_cyclotomic": false, "field_poly_root_of_unity": 6, "has_non_self_twist": 1, "hecke_cutters": [[5, [-3, 1]]], "hecke_orbit": 1, "hecke_orbit_code": 274911461454, "hecke_ring_generator_nbound": 2, "hecke_ring_index": 1, "hecke_ring_index_factorization": [], "hecke_ring_index_proved": true, "inner_twist_count": 2, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 13, 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": "78.2.e.a", "level": 78, "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": true, "level_primes": [2, 3, 13], "level_radical": 78, "minimal_twist": "78.2.e.a", "nf_label": "2.0.3.1", "prim_orbit_index": 3, "qexp_display": "q+(1-\\zeta_{6})q^{2}+(-1+\\zeta_{6})q^{3}-\\zeta_{6}q^{4}+\\cdots", "related_objects": [], "relative_dim": 1, "rm_discs": [], "sato_tate_group": "1.2.3.c3", "self_twist_discs": [], "self_twist_type": 0, "space_label": "78.2.e", "trace_display": [1, -1, 6, -2], "trace_hash": 93643654084952683, "trace_moments": [{"__RealLiteral__": 0, "data": "0.010", "prec": 7}, {"__RealLiteral__": 0, "data": "2.008", "prec": 14}, 0, {"__RealLiteral__": 0, "data": "12.072", "prec": 17}, 0, {"__RealLiteral__": 0, "data": "109.070", "prec": 20}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.014", "prec": 7}, "traces": [0, 2, 1, -1, -1, 6, 1, -2, -2, -1, 3, -6, 2, -7, -4, -3, -1, 3, -2, -2, -3, 4, 6, 6, 1, 8, -5, 2, -2, -3, 3, -8, 1, -6, 6, -6, -1, 7, -4, 5, -6, 3, 2, 10, 12, -3, -6, 12, -1, 3, 4, -6, 2, 6, 1, -18, 2, 4, 3, 0, 6, 7, -4, -2, 2, -21, -12, 10, 3, 6, -12, -6, 1, -26, -7, -4, -2, 24, -2, -8, -3, -1, -3, -12, -2, 9, 20, -3, 6, -18, -6, 4, -12, 4, 6, -6, -2, -14, -3, 12, -4, -15, -3, 28, 7, 12, 3, 6, -1, 28, 18, 7, 4, 3, 2, 18, 6, 2, 0, 6, 3, -25, 14, 3, 4, -6, 2, 4, 1, -20, -15, 0, -6, -4, -10, 6, -3, -9, 12, 4, -6, -6, -12, 30, 2, -9, -13, 3, -14, 9, 4, -20, 2, 3, 12, -24, -7, 10, -4, -3, 3, -24, 1, 4, -6, -18, -6, 0, -4, 23, 18, -2, 10, 6, -6, -8, -6, 0, 18, -6, -3, -14, 14, -14, -6, 21, -4, -36, -6, -2, -12, 12, -1, -23, -28, 15, -6, -6, 6, 10, -8, 10, 15, 12, 3, 9, 14, -12, 5, 24, 6, 16, -3, 12, -6, 30, -2, 8, 14, 13, 36, -6, 14, -8, 2, -4, 6, -18, -2, -44, -18, -12, 3, -12, 7, 36, 0, 4, -6, 12, -3, 1, -50, -1, 7, 9, 6, 4, 8, 6, -3, 12, 4, 36, -4, -18, -1, 3, -10, -28, 6, 6, 0, 6, 6, 18, 4, -18, -20, -18, 3, 16, -6, -14, -18, -24, 6, -17, 8, 4, 6, 18, 6, -14, -6, 12, -12, -12, 1, 8, 9, 28, 13, 21, 6, 0, -7, -6, 18, -30, 8, 20, -10, -15, 4, 21, -3, -20, -12, -14, -12, -60, -5, -20, 5, -6, 4, 6, 3, -18, 6, 6, -12, 6, 2, -28, 8, -14, -3, -12, -36, 4, 6, -14, 0, 30, -2, 46, 22, -6, 9, 24, 2, -40, -10, 18, 12, 30, -3, 10, -16, -7, -12, 15, 0, -18, 36, 6, 6, 12, 3, 15, -7, 50, 10, -78, -7, -2, 6, -6, -21, -6, -8, -29, -18, 3, -12, 15, -4, -20, -6, 4, 24, -24, 1, 72, 23, 10, -14, 78, -6, 36, -3, 0, 6, -24, -6, -14, 20, -4, -4, 3, 20, 28, 30, -3, 6, 42, 6, 1, -9, 18, -14, 0, -6, -36, -2, -8, 12, -24, -6, 58, -16, -6, -6, 12, 6, 14, -12, 12, 60, 6, -1, 13, 16, -9, -14, -24, -13, -14, 18, -6, -21, -72, 7, -54, 8, -18, -2, -18, -8, 18, 3, 10, -36, 12, -4, -11, -22, 3, -36, -15, 12, 76, -3, 12, -6, -36, 5, -40, 18, -5, 0, -120, -4, -8, -12, -3, 6, 0, -6, -35, 2, 12, -25, -42, -2, -2, -7, -8, -9, 18, 3, -18, 14, 36, 4, -12, -6, 64, 3, 0, 24, -6, 2, -45, 72, -22, -8, -3, -9, 26, -2, -2, -3, 84, 10, -36, -14, -12, 21, 66, 3, 34, 0, 16, -6, -12, 12, -13, 9, 0, 8, -15, -36, 18, -10, -6, -36, 18, -3, 58, -16, 7, -3, 84, -10, -68, -9, 7, 24, 12, -6, 8, -34, 21, 4, -3, 8, -20, 12, 18, 9, 0, 12, 9, 14, 4, 6, -6, 6, -44, -42, -24, -6, 24, -1, 22, -8, -23, 18, 12, 14, -18, 26, 6, 42, 0, 3, 8, 0, -6, 7, 18, 6, 18, 9, -20, 12, 48, 4, 37, -20, -20, 10, -75, -30, -32, 2, -6, 42, -42, -6, 31, -10, 9, -24, 15, 14, 16, 12, 6, -30, 72, 2, -58, -10, -12, -5, 42, 6, -20, 8, 16, 3, 12, 6, -15, -36, -6, 3, 3, 12, 16, 12, -60, -6, -24, 1, 0, -20, -16, 4, -42, 14, 0, 3, 13, -24, -24, -18, -5, 8, 21, 12, -12, -7, 18, 0, -8, -30, -84, 2, 13, 23, 8, -1, 36, -3, -28, -9, 36, -24, 48, 4, -27, -20, 22, -20, -21, 36, -26, 6, -12, 60, 12, 3, 18, -10, 6, -8, -60, -5, -28, -6, -18, -15, 60, 0, -5, -36, 4, 18, -24, 12, 90, 12, -6, 6, 0, 6, -28, -15, -2, 7, -12, 25, 28, -4, 2, -39, -30, 7, -62, 2, 9, 12, 60, -3, 16, -42, -14, -12, 36, -4, 27, -58, 6, 18, -24, -3, -14, -6, -24, -6, -60, -2, 34, 20, -72, 6, 30, 8, -28, 12, 9, -48, 0, 2, -14, 36, 3, 46, 30, -10, -32, 14, 14, 39, -12, -21, 72, 18, -3, 3, 30, 0, 28, 12, 6, -12, 6, -12, -14, -28, -9, 10, -30, -8, 18, 4, 36, -3, 78, 10, -72, 20, 36, 15, 51, 3, -8, -6, 16, 84, 12, 3, 20, 2, 10, -18, -18, 9, 40, -28, -24, 0, -96, 6, -17, -18, 34, -7, 18, -4, 0, -12, -8, 24, 12, -12, 20, 29, -9, -32, 69, -12, -50, -3, -14, 24, -42, -6, -38, -14, -6, -6, 42, 42, 52, 30, 6, -6, 36, 1, 18, 26, 8, 8, 24, -18, -50, -28, -14, -12, 6, -26, -41, 14, -42, -18, -33, -3, 16, -15, 0, -36, 48, -7, -16, 54, -6, 16, -12, -9, -18, -4, -12, -36, 12, -4, 9, 36, 20, -3, -42, -10, -44, -18, 30, 42, -48, -2, 36, 11, -42, 22, 0, 6, 16, -18, 10, -30, 12, 24, 28, 38, -14, -6, -33, -12, -12, 6, 30, -18, -108, -2, 94, -20, 10, -18, 84, 5, -18, 0, -6, -60, -24, -8, 91, -16, -3, -6, -54, -6, 36, -6, 36, 0, -18, -3, -30, 14, -12, 1, -69, -12, -44, 25, 6, -84, -60, -1, 8, -4, 20, -14, 3, -4, -108, -18, -14, -18, -72, -3, -18, -9, 24, 10, 120, 18, -38, -4, -8, 12, 30, -12, -5, 32, 7, 6], "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, "projective_field": null, "projective_field_label": null, "projective_image": null, "projective_image_type": null, "qexp": [[0, 0], [1, 0], [1, -1], [-1, 1], [0, -1], [3, 0], [0, 1], [0, -2], [-1, 0], [0, -1], [3, -3], [-6, 6], [1, 0], [-3, -1], [-2, 0], [-3, 3], [-1, 1], [0, 3], [-1, 0], [0, -2], [0, -3], [2, 0], [0, 6], [6, -6], [1, -1], [4, 0], [-4, 3], [1, 0], [-2, 2], [-3, 3], [0, 3], [-4, 0], [0, 1], [0, -6], [3, 0], [0, -6], [-1, 1], [7, -7], [-2, 0], [4, -3], [-3, 0], [3, -3], [2, -2], [0, 10], [6, 0], [0, -3], [0, -6], [6, 0], [0, -1], [3, -3], [4, -4], [-3, 0], [-1, 4], [3, 0], [1, -1], [-18, 18], [0, 2], [2, 0], [0, 3], [0, 0], [3, 0], [0, 7], [-4, 4], [-2, 2], [1, 0], [-9, -3], [-6, 0], [10, -10], [3, -3], [0, 6], [-6, 0], [0, -6], [0, 1], [-13, 0], [0, -7], [-4, 4], [-2, 2], [12, 0], [1, -4], [-4, 0], [-3, 3], [-1, 1], [0, -3], [-6, 0], [0, -2], [0, 9], [10, 0], [0, -3], [6, -6], [-18, 18], [-3, 0], [-2, 8], [-6, 0], [4, -4], [6, -6], [0, -6], [-1, 0], [0, -14], [0, -3], [6, 0], [0, -4]]}