# Stored data for newform 137.2.a.a, downloaded from the LMFDB on 20 September 2025.
{"Nk2": 548, "analytic_conductor": 1.0939505076915412, "analytic_rank": 1, "analytic_rank_proved": true, "atkin_lehner_eigenvals": [[137, 1]], "atkin_lehner_string": "+", "char_conductor": 1, "char_degree": 1, "char_is_minimal": true, "char_is_real": true, "char_orbit_index": 1, "char_orbit_label": "a", "char_order": 1, "char_parity": 1, "char_values": [137, 1, [3], [1]], "cm_discs": [], "conrey_index": 1, "dim": 4, "field_disc": 725, "field_disc_factorization": [[5, 2], [29, 1]], "field_poly": [1, 1, -3, -1, 1], "field_poly_is_cyclotomic": false, "field_poly_is_real_cyclotomic": false, "field_poly_root_of_unity": 0, "fricke_eigenval": 1, "has_non_self_twist": 0, "hecke_cutters": [[2, [-1, -4, 0, 3, 1]]], "hecke_orbit": 1, "hecke_orbit_code": 33554569, "hecke_ring_generator_nbound": 2, "hecke_ring_index": 1, "hecke_ring_index_factorization": [], "hecke_ring_index_proved": true, "inner_twist_count": 1, "inner_twists": [[1, 1, 1, 1, 1, 1, 1]], "is_cm": false, "is_largest": true, "is_maximal": true, "is_polredabs": true, "is_rm": false, "is_self_dual": true, "is_self_twist": false, "is_twist_minimal": true, "label": "137.2.a.a", "level": 137, "level_is_powerful": false, "level_is_prime": true, "level_is_prime_power": true, "level_is_prime_square": false, "level_is_square": false, "level_is_squarefree": true, "level_primes": [137], "level_radical": 137, "minimal_twist": "137.2.a.a", "nf_label": "4.4.725.1", "prim_orbit_index": 1, "qexp_display": "q+(-1+\\beta _{1})q^{2}+(-1-\\beta _{1}-\\beta _{2}+\\beta _{3})q^{3}+\\cdots", "related_objects": [], "relative_dim": 4, "rm_discs": [], "sato_tate_group": "1.2.3.c1", "self_twist_discs": [], "self_twist_type": 0, "space_label": "137.2.a", "trace_display": [-3, -5, -2, -13], "trace_hash": 114817131469201998, "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "3.815", "prec": 17}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "38.950", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "578.982", "prec": 24}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.003", "prec": 17}, "traces": [0, 4, -3, -5, 1, -2, 1, -13, -3, 5, -5, 1, -4, -8, 14, -8, -1, -4, 0, -10, 13, 11, 1, -1, 20, 8, 18, -5, -9, 11, 33, -17, 6, 6, 5, 10, 10, -4, 7, -1, -16, -7, -4, -13, 7, 5, -21, -11, -9, 21, -27, 19, -10, -2, 2, -27, 6, 24, 14, 2, -35, 7, 20, -15, -3, -2, -28, -6, -24, 4, 9, 8, -30, -27, 17, 3, -23, -19, -20, 3, 24, -16, 31, -3, 22, -9, -16, -8, -2, 14, -40, 33, 33, 42, -12, 4, -16, -7, -44, 20, 36, 2, -14, -43, -9, 43, 2, 18, -8, 10, 24, -12, 15, 11, -3, 27, -45, 10, 5, 20, 1, 33, -18, -21, -17, 9, 5, -32, 16, 46, 1, 7, 3, 42, 20, 29, 38, -4, 48, -5, -46, 22, 4, 10, -5, -36, -11, 16, 3, 11, 26, -22, 40, -45, 14, -42, 29, -13, -44, 11, 1, -11, 27, -18, -21, 5, 18, -20, -72, -8, 51, -55, 16, 19, -50, -52, -51, -42, -3, 30, 45, 7, -71, -8, -8, -10, -62, -44, 22, 32, 46, 12, 24, 4, 49, 64, 37, -4, 30, 16, -6, -39, -42, -16, 45, 5, 23, 0, -1, -63, -126, 32, -15, 0, 29, 5, 5, 40, -34, 15, -8, -9, -27, -50, -25, 15, 6, 7, 41, 38, -34, -28, 23, -17, 30, 25, 2, -21, -27, 29, -1, 0, 3, 35, 9, -42, -26, -2, 39, -41, 12, 2, -35, 3, 42, 16, -25, -2, 21, 12, -36, -15, 20, -4, 48, -3, -43, -10, 19, -18, -19, -45, 18, 10, 3, 56, -73, -2, -38, -25, 80, -53, 37, 46, -8, 18, 27, 46, 35, 4, 48, 25, 57, -3, 2, 57, -32, -78, -53, -30, -71, 19, 56, 49, 32, 26, 15, -5, -5, 19, 98, -50, -23, 18, -17, -35, 54, -15, -5, -35, -82, -25, 76, 83, -29, -45, 41, -28, -31, 23, 37, 62, 35, 45, -12, 29, 39, -4, -12, -9, -68, 73, 0, -23, 46, -85, -38, -20, 87, -19, 117, -2, 36, -17, 52, -44, -4, -69, -72, -70, 0, 32, -12, -2, 47, 81, 44, -24, 1, 30, -18, 12, 65, -34, -22, -60, 12, -3, -28, 13, -75, 51, -32, -29, -78, 114, -43, -55, -88, -85, -86, -26, 23, -47, -79, 105, -45, -79, 30, -73, -38, 18, 9, 43, 25, -12, -48, 63, -88, -23, -29, 5, 25, 25, -70, 8, 1, -11, -16, -31, 127, -28, 43, -25, 24, -52, -48, -18, -68, -81, 54, -40, 52, 30, -66, 67, 46, -17, 92, -34, -11, 5, -48, -53, 11, -37, 33, 23, -2, 48, -45, -11, -13, -21, 18, -19, -123, 36, 11, 48, 68, -37, 100, 22, 12, 63, 59, 112, -55, 30, -2, 64, -49, 6, 149, -69, 73, -30, -13, -77, -14, 54, 12, 8, -94, -74, -45, -5, 1, 20, 44, 61, 48, 63, -75, -25, -56, -36, -10, 44, 11, 91, 13, 65, 95, -7, -20, 34, -50, -8, -123, 74, -1, 73, -49, 83, -28, -8, -55, -65, 99, -11, 75, 60, -3, -9, -11, 11, 20, -15, 29, 25, 74, 72, -34, -59, -78, 36, 50, 107, -5, -73, 35, 59, -53, 62, 52, -31, -1, -15, -50, 44, 22, -4, -9, 64, 52, 90, 65, -32, -106, 75, 75, -11, -46, -42, 21, 62, 4, -69, -165, -22, -33, -16, -125, 74, 5, -14, 8, -10, -68, 3, -123, -35, -31, -75, -19, -41, -67, 14, -108, 56, -30, -33, 29, 46, 68, -1, 43, 23, 84, 20, 55, 80, -36, -76, 35, -26, -67, -15, -32, -14, -20, -30, 51, 117, -37, 7, 60, 54, -124, -2, 42, -53, 62, -12, 22, 81, 20, -65, 145, 18, -31, -91, 0, -18, 26, -95, 11, 20, 114, 77, -88, 62, -105, -77, -33, 29, 22, -98, 96, -137, -52, 59, 96, 2, 20, 0, 35, 14, 7, -65, -126, 3, -94, -9, 40, -95, -32, 131, -101, -61, 51, -13, -43, -64, 15, -81, -48, 49, 33, 29, -61, -16, -15, 2, 106, 9, -15, -9, 103, 10, 41, -25, -24, 109, -19, -15, 6, -85, -115, -19, -45, -74, 115, -59, 25, -42, -30, -95, 40, -15, -3, -23, 73, 64, 23, -145, 69, -62, 15, 148, 31, 59, 5, -85, -79, -84, 34, -22, -85, 35, -39, -34, 118, -155, 14, 39, 50, -46, -3, 32, -14, 2, -79, -10, 35, 115, 94, -26, 16, 3, 5, -41, -18, 20, 14, 26, 23, 47, 47, 92, -60, -34, 60, -10, 82, 17, 97, -19, -132, 4, 16, -33, -50, -1, 64, 43, 58, -7, 98, 72, 102, 24, -86, -32, -18, -40, 102, 37, -139, -39, -45, -45, 5, 6, -34, 74, 34, 5, 43, 25, -54, -13, -30, -84, -74, -32, 66, 57, 50, -6, 121, 60, 32, -21, -27, 57, 52, -20, -34, -57, -1, 72, -29, 39, -44, -7, 75, -28, -99, -91, 49, -49, 147, -14, 96, -56, 12, 30, -6, 55, -46, 58, -70, -7, -60, -72, 16, -24, 89, -23, 32, -42, 58, 0, 14, 117, 37, -9, -31, 74, 0, -25, -42, 108, -67, -63, 99, -136, -78, 79, -40, -50, 126, -8, -118, 63, 12, 30, 78, 34, -25, 68, 98, 22, 56, -31, 49, 117, 20, -9, -27, 29, 43, -29, -26, 71, 14, -31, 60, 49, 81, -129, -2, -122, -36, 41, -65, -70, 20, -51, -21, 125, 13, -65, -96, -5, 17, 52, -119, -48, 57, 4, -33, -1, 27, -10, -32, -7, -184, -98, 17, 144, -91, 42, 20, -34, -78, -29, 13, 13, -15, -69, 81, -124, -86, 7, -157, 24, 127, 80, -115, -41, 5, 8, 41, 27, 157, 13, 109, 59, -13, 5, -58, 113, -160, -130, 89, -184, 145, -163, 95, 22, 20, 7, 38, 15, -62, 56, 72, 20, -55, -31, -13, 43, -109, -43, 108, 74, -40, -11, -14, 1, 8, -134, -46, -23, -42, -59, -83], "weight": 2, "weight_parity": 1, "id": null, "artin_degree": null, "artin_field": null, "artin_field_label": null, "artin_image": null, "embedded_related_objects": null, "projective_field": null, "projective_field_label": null, "projective_image": null, "projective_image_type": null, "qexp": [[0, 0, 0, 0], [1, 0, 0, 0], [-1, 1, 0, 0], [-1, -1, -1, 1], [0, -1, 1, 0], [0, 0, 1, -2], [1, -1, 1, -2], [-3, 1, 0, -1], [0, -1, -2, 1], [0, 1, 2, 0], [-1, -1, -3, 3], [-2, -1, 1, 4], [-1, 2, -2, 1], [-3, 2, 1, 0], [4, -4, 0, 1], [-3, 4, 0, 0], [1, 1, 0, -3], [1, 0, -3, -1], [-1, 2, -1, 2], [0, 0, -3, -2], [3, -1, 3, -2], [2, 3, 4, -4], [0, 3, 2, -3], [0, -3, 1, 0], [3, 0, 3, 1], [1, -4, 1, 3], [4, -2, 1, 1], [1, 1, -1, -4], [-2, 3, -3, 1], [4, 3, -3, -1], [7, -3, 4, 0], [-6, -1, -1, 5], [0, 0, 2, 1], [5, -4, -4, -1], [2, -3, 2, -2], [1, -2, -5, 9], [4, -2, 1, -3], [-4, 2, 3, 2], [3, -5, 1, -1], [2, -1, 1, -5], [-5, 6, 0, -1], [-3, 5, 2, -2], [-3, 2, -5, 8], [-2, -5, -2, 2], [5, 1, -4, -3], [3, -5, 0, -1], [-4, 1, -4, 1], [-2, -3, 0, 0], [-4, 3, 2, 0], [4, -7, -1, 7], [-6, 5, -2, -2], [1, 1, 6, 1], [-1, 2, -4, 0], [1, 0, -2, -1], [1, -4, -2, 3], [-5, 3, 1, -6], [0, 4, 7, -6], [1, 4, 8, 0], [2, 0, 5, -2], [2, 4, 2, -7], [-8, 3, -7, 4], [4, -1, -1, -3], [6, -2, 5, -6], [-1, -3, -6, 2], [-4, 1, -1, 7], [0, -4, -6, 7], [-5, 0, -1, -3], [-5, 4, 6, -1], [-9, 2, -1, 6], [-1, 6, -2, 3], [2, 5, 12, -14], [0, 0, 0, 4], [-5, -2, -4, 0], [-7, -3, 4, -2], [3, 1, 1, 1], [1, 1, -6, 5], [-9, 3, -1, 6], [0, 5, 1, -13], [-4, -2, -7, 6], [5, -3, 1, -8], [5, -4, -1, 5], [-4, 2, -1, 0], [6, -3, 1, 4], [-7, 3, 8, 3], [6, -6, 7, -5], [-4, 5, -2, 3], [-1, -2, -1, -4], [-2, -8, 3, 1], [0, -8, -2, 5], [2, 0, 0, 3], [-8, 2, -6, 1], [10, -9, -3, 4], [9, -1, 4, -5], [12, -2, 3, -5], [-1, -2, -3, 0], [-2, 4, -3, 7], [-1, -2, -5, 0], [-1, 5, -5, 1], [-10, 10, 1, -8], [4, 8, 1, -3], [11, -2, 3, -6]]}