# Stored data for newform 1638.2.r.x, downloaded from the LMFDB on 20 January 2026.
{"Nk2": 6552, "analytic_conductor": 13.079495851085728, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 13, "char_degree": 2, "char_is_minimal": true, "char_is_real": false, "char_orbit_index": 18, "char_orbit_label": "r", "char_order": 3, "char_parity": 1, "char_values": [1638, 3, [911, 703, 379], [3, 3, 1]], "cm_discs": [], "conrey_index": 757, "dim": 4, "field_disc": 3249, "field_disc_factorization": [[3, 2], [19, 2]], "field_poly": [25, -5, -4, -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": [[5, [-14, 1, 1]], [11, [16, 4, 1]], [17, [9, -3, 1]]], "hecke_orbit": 24, "hecke_orbit_code": 103583959694181990, "hecke_ring_generator_nbound": 5, "hecke_ring_index": 3, "hecke_ring_index_factorization": [[3, 1]], "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": false, "label": "1638.2.r.x", "level": 1638, "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": [2, 3, 7, 13], "level_radical": 546, "minimal_twist": "546.2.l.j", "nf_label": "4.0.3249.1", "prim_orbit_index": 3, "qexp_display": "q-\\beta _{1}q^{2}+(-1-\\beta _{1})q^{4}+(-1+\\beta _{2}+\\cdots)q^{5}+\\cdots", "related_objects": [], "relative_dim": 2, "rm_discs": [], "sato_tate_group": "1.2.3.c3", "self_twist_discs": [], "self_twist_type": 0, "space_label": "1638.2.r", "trace_display": [2, 0, -2, -2], "trace_hash": 210824846295131262, "trace_moments": [{"__RealLiteral__": 0, "data": "0.008", "prec": 17}, {"__RealLiteral__": 0, "data": "3.800", "prec": 17}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "54.960", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "1399.604", "prec": 27}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.007", "prec": 17}, "traces": [0, 4, 2, 0, -2, -2, 0, -2, -4, 0, -1, -8, 0, 14, -4, 0, -2, 6, 0, 2, 1, 0, 8, -3, 0, 38, 10, 0, -2, 9, 0, 10, 2, 0, 12, 1, 0, 7, 4, 0, 2, 9, 0, 1, 16, 0, 3, -4, 0, -2, 19, 0, -4, -16, 0, 4, 2, 0, -9, -1, 0, 18, 5, 0, 4, -7, 0, 13, 6, 0, 2, 3, 0, 38, -7, 0, 2, 16, 0, -32, 1, 0, -9, 2, 0, -3, 2, 0, 8, 9, 0, -4, 6, 0, -2, 56, 0, -14, 2, 0, -19, 11, 0, 6, -14, 0, -8, -10, 0, 12, -4, 0, 4, 7, 0, 30, -18, 0, -2, 6, 0, -10, 36, 0, -5, -66, 0, -16, 2, 0, -5, 26, 0, 2, -13, 0, -6, 23, 0, -12, 1, 0, 6, -40, 0, -33, 19, 0, -14, -2, 0, -8, -2, 0, 8, 52, 0, -30, -16, 0, -1, 6, 0, 19, -18, 0, 1, 0, 0, 46, -6, 0, 1, 6, 0, -19, -8, 0, -9, 18, 0, -22, -14, 0, 3, 25, 0, -48, 2, 0, 112, 21, 0, 27, -28, 0, 4, -21, 0, -1, -38, 0, -11, -18, 0, -33, 3, 0, -10, -16, 0, -12, 8, 0, 10, -86, 0, -5, 6, 0, -8, 12, 0, -29, 2, 0, 14, -12, 0, 66, -30, 0, -9, -60, 0, -112, -1, 0, -6, 6, 0, 39, -20, 0, 18, 1, 0, 4, -10, 0, -33, 19, 0, -12, 16, 0, -2, 30, 0, -14, 2, 0, 13, 12, 0, -106, -2, 0, -26, -36, 0, -43, -12, 0, 46, -76, 0, -7, -24, 0, -1, 38, 0, 34, 3, 0, 16, -18, 0, 16, 33, 0, -19, -29, 0, 86, -7, 0, -4, -15, 0, 1, -4, 0, -4, -9, 0, -36, -8, 0, 26, -20, 0, -20, -15, 0, 16, 0, 0, 36, -2, 0, 3, -6, 0, 133, 38, 0, -9, 2, 0, -8, -1, 0, 0, 22, 0, -14, 44, 0, -3, -20, 0, 4, -1, 0, 12, 46, 0, 17, -38, 0, -16, -24, 0, -30, -18, 0, -18, -56, 0, -78, -11, 0, -10, 38, 0, -19, -3, 0, -25, 8, 0, -31, -24, 0, 4, 45, 0, 40, 56, 0, 42, -16, 0, -8, -27, 0, -14, 8, 0, -18, 2, 0, 21, 16, 0, -39, -2, 0, -19, 3, 0, 35, -22, 0, -9, 28, 0, 7, 33, 0, -3, -1, 0, -172, 4, 0, -8, -7, 0, 18, 12, 0, 16, 57, 0, 18, 20, 0, -172, -19, 0, 13, -10, 0, -6, 108, 0, 4, -4, 0, 42, -108, 0, 24, 29, 0, -2, -8, 0, 36, 7, 0, -24, 2, 0, 34, 33, 0, -60, -1, 0, -12, 9, 0, -30, 78, 0, -26, -56, 0, 1, -8, 0, -38, -12, 0, 3, 40, 0, 35, 78, 0, -10, -50, 0, -28, -18, 0, -1, 20, 0, 54, 14, 0, -5, 3, 0, -82, 33, 0, 38, -12, 0, 23, -24, 0, 32, -49, 0, -19, -4, 0, -30, -60, 0, 8, -7, 0, 7, 38, 0, -8, -13, 0, -12, 15, 0, 13, -53, 0, -4, 45, 0, -52, -13, 0, -72, -8, 0, 26, 43, 0, -6, 108, 0, 72, 23, 0, 76, -96, 0, 16, -14, 0, -12, -10, 0, 2, -2, 0, 19, 48, 0, -89, -34, 0, -3, 40, 0, -74, 56, 0, -9, -57, 0, -38, -16, 0, 66, -1, 0, 32, -38, 0, -58, -35, 0, 62, 172, 0, 7, -48, 0, -3, -2, 0, 6, 58, 0, 23, -1, 0, 4, 5, 0, -37, -2, 0, -18, -14, 0, -21, -18, 0, -16, 27, 0, 28, -26, 0, -10, -18, 0, 128, -10, 0, 15, 42, 0, 28, 32, 0, 0, 8, 0, -10, 72, 0, -1, 15, 0, -10, -3, 0, 6, 34, 0, 8, 95, 0, 19, -1, 0, 44, 9, 0, 4, -42, 0, -50, -16, 0, -2, 56, 0, 42, 0, 0, -22, -144, 0, 6, -7, 0, -2, 44, 0, -14, 3, 0, 20, -54, 0, -97, 2, 0, -2, -56, 0, -16, 6, 0, 92, -108, 0, 54, -17, 0, -19, 46, 0, 128, -8, 0, 24, -22, 0, 57, -60, 0, -9, 21, 0, 20, -36, 0, -28, -16, 0, -3, 78, 0, 11, 114, 0, 34, 4, 0, 19, -3, 0, -40, 19, 0, -6, 52, 0, 91, -50, 0, 16, -31, 0, 1, -62, 0, 24, 20, 0, 22, 2, 0, -18, -224, 0, 52, -40, 0, -56, -20, 0, -6, 21, 0, -32, -2, 0, -38, -4, 0, -54, -72, 0, 38, 14, 0, 4, -96, 0, -24, -9, 0, -2, -42, 0, -12, 42, 0, 8, 7, 0, 36, -78, 0, -1, 18, 0, -6, 19, 0, -3, -76, 0, -60, 25, 0, -11, 27, 0, -40, 9, 0, 56, -38, 0, 170, 14, 0, 66, 23, 0, 44, -6, 0, 1, 40, 0, 67, -86, 0, 14, -12, 0, 0, 8, 0, 7, 34, 0, -11, 9, 0, 24, -23, 0, -10, 8, 0, 114, -18, 0, -88, -18, 0, 10, 102, 0, -20, -86, 0, 19, 40, 0, 54, 26, 0, -5, 64, 0, 65, -12, 0, 54, 33, 0, 15, -4, 0, 4, -12, 0, 66, 30, 0, -54, 78, 0, 32, -24, 0, 58, -116, 0, 48, -4, 0, -16, -6, 0, -24, 72, 0, -7, 182, 0, 57, -12, 0, 7, 40, 0, -4, -34, 0, -33, -13, 0, -34, -30, 0, -2, 6, 0, 38, -6, 0, 18, 92, 0, -4, 30, 0, 39, 24, 0, 122, -13, 0, 56, -60, 0, 42, 2, 0, -4, 12, 0, 133, -76, 0, -6, 8, 0, 18, -3, 0, -40, 23, 0, -42, -14, 0, 39, -99, 0, -44, 10, 0, -100, -1, 0, -12, -56, 0, -36, 87, 0, 36, -2, 0, -20, -4, 0, 96, 27, 0, 10, -174, 0, -2, 5, 0, -3, 86, 0, 38, -41, 0, 66, 56, 0, 3, 19, 0, -24, -122, 0, -102, -23, 0, -12, 138, 0, 66, 16, 0, 49, 120, 0, -21, -38, 0, -2, 114, 0, -112, -60, 0, -30, -16, 0, -70, -8, 0, 7, -54, 0, -72, 5, 0, 19, -2, 0, -448, 8, 0, -26, 112, 0, 5, -24, 0, 30, 120, 0, 4, 26, 0, 53, -16, 0, -93, -2, 0, -18, 112, 0, -49, 52, 0, 13, -3, 0, 95, -36, 0, -16, 7, 0, -54, 13, 0, 86, -26, 0, -80, 6, 0, 54, -66, 0, -78, 36, 0, -23, 18, 0, 15, 152, 0, -48, 23, 0, -6, 32, 0, -7, -29, 0, 44, 12, 0, 10, -14, 0, -8, 7, 0, -1, -170, 0, 76, -19, 0, 96, -3, 0, -17, -178, 0, -68, -12, 0, 120, -6, 0, -40, 78, 0, 19, -37, 0, 40, 24, 0, 46, 9, 0, 57, 8, 0, -25, -19, 0, -32, 45, 0, -18, 33, 0, -2, -34, 0, -84, -32, 0, -19, 0, 0, 106, -29, 0, 35, 76, 0, -40, 124, 0, 86, 36, 0, -2, 14, 0, -24, 16, 0, -69, 3, 0, 2, -20, 0, -60, 21, 0, 29, -24, 0, -86, -23, 0, -2, 9, 0, -9, 8, 0, 10, 6, 0, 31, -74, 0, 2, 108, 0, -45, -9, 0, -10, -120, 0, -19, 21, 0, 18, 78, 0, 47, -8, 0, 54, 112, 0, -21, 14, 0, -52, 57, 0, -56, 10, 0, -9, 180, 0, -114, 64, 0, 10, -36, 0, -38, 30, 0, 21, -64, 0, -28, 56, 0, 16, 120, 0, 11, 0, 0, -8, -6, 0, 140, 4, 0, 36, 23, 0, 16, 1, 0, 30, 50, 0, -15, -20, 0, -6, 57, 0, -78, 12, 0, -34, -50, 0, -22, 4, 0, -38, -152, 0, -36, -19, 0, 1, -39, 0, 24, 22, 0, 18, 55, 0, -120, 2, 0, -84, 60, 0, 86, 50, 0, -8, -38, 0, 4, -1, 0, -56, -96, 0, -83, 84, 0, 0, 21, 0, 21, -44, 0, -72, -48, 0, 18, -6, 0, 7, 54, 0, -54, -46, 0, 22, 50, 0, 174, 14, 0, 6, 106, 0, 48, 40, 0, -108, -64, 0, 21, -194, 0, -2, -72, 0, 132, -1, 0, -40, 42, 0, 5, 16, 0, -6, 89, 0, 76, 46, 0, -216, -50, 0, -1, 27, 0, -34, -64, 0, -104, 19, 0, 23, -54, 0, -190, 64, 0, 8, -6, 0, 3, 48, 0, -11, 40, 0, 42, 114, 0, -30, 9, 0, 24, 9, 0, -21, 5, 0, -62, -8, 0, -18, -90, 0, 66, 28, 0, -32, 43, 0, -52, -6, 0, 156, -8, 0, 28, 22, 0, -114, -120, 0, -1, 17, 0, 14, -124, 0, 20, -19, 0, 3, 8, 0, 43, -20, 0, 38, 14, 0, 48, -3, 0, 104, -95, 0, 24, -91, 0, -25, 2, 0, 40, 8, 0, 31, -92, 0, 38, 2, 0, -31, -34, 0, 150, 48, 0, 10, -46, 0, -16, -22, 0, -2, -86, 0, -184, -63, 0, -112, -56, 0, 54, -52, 0, -80, -3, 0, -5, -112, 0, -40, -108, 0, 171, -12, 0, -21, 96, 0, -48, -16, 0, -7, -96, 0, 21, 38, 0, 4, -9, 0, 55, -27, 0, -144, 30, 0, -19, 19, 0, 28, -82, 0, -104, -4, 0, -48, 8, 0, 58, -12, 0, 9, -332, 0, -32, -4, 0, -21, 0, 0, -20, -24, 0, 21, -170, 0, -53, -8, 0, -7, -44, 0, -114, 126, 0, -39, -12, 0, -164, 1, 0, 36, -132, 0, 16, -12, 0, 38, 100, 0, -33, -6, 0, 76, -294, 0, 47, -30, 0, -10, 120, 0, -168, 11, 0, -27, 12, 0, -14, -20, 0, 18, -231, 0, -98, 28, 0, -76, 30, 0, -66, -170, 0, 7, -72, 0, 16, 33, 0, -23, -154, 0, 56, 88, 0, -3, 42, 0, -80, 2, 0, 20, 4, 0, 49, -67, 0, 86, 16, 0, 64, 10, 0, -6, -42, 0, 74, 0, 0, 16, -3, 0, 95, 14, 0, 68, 0, 0, 13, -22, 0, -9, -50, 0, 39, 12, 0, -22, 132, 0, -50, 10, 0, -8, -50, 0, 90, 57, 0, -36, 91, 0, -104, -44, 0, -36, -43, 0, -180, -10, 0, 51, -2, 0, 66, -10, 0, 86, 152, 0, -5, 38, 0, 20, 60, 0, 10, 108, 0, 13, 4, 0, -80, 5, 0, -64, -64, 0, 168, -26, 0, -6, -208, 0, 70, -54, 0, 66, 9, 0, 49, 30, 0, -8, 19, 0, 16, 8, 0, 12, 180, 0, 69, 33, 0, -12, 162, 0, -12, 54, 0, -78, 57, 0, 78, 16, 0, -48, 46, 0, 76, 29, 0, -232, 30, 0, 62, -48, 0, -2, -152, 0, -200, -8, 0, 6, -60, 0, -58, -48, 0, 36, -75, 0, -24, -14, 0, 91, -63, 0, 7, -57, 0, 12, 24, 0, -60, 5, 0, 20, 120, 0, 304, 4, 0, -68, 83, 0, -82, -66, 0, -26, 38, 0, 0, -68, 0, 30, -24, 0, 128, -1, 0, 21, 100, 0, -171, -38, 0, 6, 18, 0, 53, 9, 0, 184, -176, 0, -81, -2, 0, 60, 101, 0, 172, -39, 0, 12, 34, 0, 92, 61, 0, 13, -306, 0, -31, 112, 0, -30, -36, 0, -165, 84, 0, 1, 10, 0, 45, 4, 0, -12, -20, 0, 86, 95, 0, -38, -75, 0, -48, 6, 0, 16, 108, 0, -87, 36, 0, -6, -123, 0, -220, -80, 0, -23, 136, 0, -21, -21, 0, -49, 152, 0, 48, -39, 0, 99, -27, 0, -92, -22, 0, 20, -4, 0, -7, -50, 0, -2, -30, 0, -346, 12, 0, -28, -270, 0, 72, -18, 0, -87, 180, 0, -54, 72, 0, -1, -85, 0, -28, -40, 0, -2, -19, 0, 72, -96, 0, -27, -11, 0, -304, -4, 0, -87, 157, 0, -68, 2, 0, 10, 48, 0, 82, -6, 0, 172, 44, 0, -93, 76, 0, 41, 96, 0, -2, 33], "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, 0, 0], [1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [-1, -1, 0, 0], [-1, 0, 1, 0], [0, 0, 0, 0], [-1, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 0], [0, 1, 1, -1], [0, 4, 0, 0], [0, 0, 0, 0], [4, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [3, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 2], [1, 1, 0, -1], [0, 0, 0, 0], [4, 4, 0, 0], [0, 2, 1, -1], [0, 0, 0, 0], [10, 0, -1, 0], [1, -3, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, -5, -1, 1], [0, 0, 0, 0], [2, 0, 1, 0], [1, 1, 0, 0], [0, 0, 0, 0], [3, 0, 0, 0], [1, 1, 0, -1], [0, 0, 0, 0], [0, -3, 1, -1], [0, 0, 2, 0], [0, 0, 0, 0], [1, 0, -1, 0], [0, -5, -1, 1], [0, 0, 0, 0], [2, 2, 0, -3], [4, 0, 0, 0], [0, 0, 0, 0], [2, 2, 0, -1], [0, 0, -2, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, -10, -1, 1], [0, 0, 0, 0], [-3, -4, 0, 0], [-3, 0, -2, 0], [0, 0, 0, 0], [0, -4, -4, 4], [1, 1, 0, 0], [0, 0, 0, 0], [-5, -5, 0, 1], [-2, -2, 0, 3], [0, 0, 0, 0], [9, 9, 0, 0], [0, -2, 1, -1], [0, 0, 0, 0], [1, 0, 0, 0], [-4, -1, 3, 1], [0, 0, 0, 0], [0, -6, 1, -1], [0, -3, 0, 0], [0, 0, 0, 0], [1, 0, -1, 0], [2, 2, 0, -1], [0, 0, 0, 0], [9, 0, 1, 0], [-3, -3, 0, -1], [0, 0, 0, 0], [0, 0, 2, -2], [4, 0, 0, 0], [0, 0, 0, 0], [-8, 0, 0, 0], [0, -1, -1, 1], [0, 0, 0, 0], [-5, -5, 0, 1], [2, 0, -3, 0], [0, 0, 0, 0], [-3, -3, 0, 3], [2, 0, -3, 0], [0, 0, 0, 0], [0, -4, 0, 0], [0, -4, 1, -1], [0, 0, 0, 0], [-3, -4, 0, 0], [2, 0, -1, 0], [0, 0, 0, 0], [0, 0, -2, 2], [28, 28, 0, 0], [0, 0, 0, 0], [-6, -6, 0, -2], [1, 1, 0, 0], [0, 0, 0, 0], [-10, -10, 0, 1]]}