# Stored data for newform 209.2.e.b, downloaded from the LMFDB on 06 December 2024. {"Nk2": 836, "analytic_conductor": 1.66887340224476, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 19, "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": [209, 3, [134, 78], [3, 1]], "cm_discs": [], "conrey_index": 45, "dim": 18, "field_disc": -38250249325181609293057213418688, "field_disc_factorization": [[-1, 1], [2, 6], [3, 13], [210943, 2], [2902511, 2]], "field_poly": [9, 81, 639, 1230, 2904, 105, 4735, -115, 3955, -681, 2060, -276, 629, -92, 130, -11, 14, -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": [[2, [9, 81, 639, 1230, 2904, 105, 4735, -115, 3955, -681, 2060, -276, 629, -92, 130, -11, 14, -1, 1]]], "hecke_orbit": 2, "hecke_orbit_code": 4503874538832081, "hecke_ring_generator_nbound": 3, "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, 19, 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": "209.2.e.b", "level": 209, "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": [11, 19], "level_radical": 209, "minimal_twist": "209.2.e.b", "prim_orbit_index": 3, "qexp_display": "q+\\beta _{1}q^{2}+(\\beta _{4}-\\beta _{12})q^{3}+(-1+\\beta _{3}+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 9, "rm_discs": [], "sato_tate_group": "1.2.3.c3", "self_twist_discs": [], "self_twist_type": 0, "space_label": "209.2.e", "trace_display": [1, 0, 0, -6], "trace_hash": 716718813816261672, "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "15.659", "prec": 20}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "1018.444", "prec": 27}, {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "116953.543", "prec": 34}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.007", "prec": 17}, "traces": [0, 18, 1, 0, -9, 0, -3, -6, 0, -11, 21, -18, 16, 11, 1, 7, -13, 6, -8, 4, 4, 24, -1, -1, -3, -7, -4, 6, -2, 13, -64, -32, -14, 0, 8, -12, 10, -36, -20, 20, 32, 4, 11, 1, 9, 22, -14, -14, 37, -12, 30, -16, 57, -4, -33, 0, -34, -7, 8, 31, 9, 3, 5, 50, 64, -56, 3, -2, -96, 0, -58, -12, 5, -26, 35, 50, -36, 6, 3, 31, 31, -21, 25, 36, -102, -11, 41, -124, 0, -11, 51, -20, -33, 20, -86, 17, 60, 16, 4, 11, -4, 38, 90, -42, 23, 28, 70, -68, -58, 11, -21, 10, -30, 14, 70, 28, 48, 44, 52, 25, -26, 18, -6, -24, 85, -42, 24, -28, -6, 1, -88, -19, -16, 19, 104, 55, 60, -12, -14, 25, -30, -144, -46, -11, -230, 10, -41, -49, -40, 16, 27, -110, 72, 132, -1, -5, 13, 21, -43, 8, -26, -18, -34, -36, 166, -7, 82, 52, -75, 0, 58, 85, 26, 8, 21, 64, 13, 59, -96, -44, -7, 43, -114, 44, -68, -30, -10, -6, -23, -44, 12, 48, 78, 13, 5, 1, 2, 66, 8, -9, 10, -92, 28, -66, -10, 48, -66, -5, -168, -4, 248, 60, -50, -41, 1, 9, -234, -114, -80, 40, -4, 72, -4, -33, 12, -56, -45, -12, 13, -50, 16, -24, 88, 82, -106, 64, -64, 17, 33, -28, 84, 64, 1, 31, 19, -52, 2, 84, 132, 46, -30, -17, 81, 1, -48, -138, -51, -41, -124, 34, -28, 69, -66, 36, 3, -146, -82, -96, 23, 46, 83, -47, 126, 160, 48, 7, 23, 0, 152, -20, 79, -7, -42, -8, 154, -91, 4, 58, 96, 33, -127, 76, -46, -168, 59, 10, -150, -6, -44, 22, -104, 29, 48, -110, 82, 72, -92, -50, 2, 46, -44, 238, 34, -38, -8, 42, -176, -9, -52, -13, -2, -26, -50, 77, 82, 37, 154, -42, 42, -37, 64, 6, 45, 141, 124, 12, -79, 64, 29, -53, -29, 32, -187, -18, -47, 48, 90, -5, 53, 42, 120, 4, 14, -70, -44, 13, 34, 27, -38, 10, 244, 102, -144, 0, -22, -34, 76, 47, 20, -176, 5, -81, 10, 10, -8, -94, 68, -43, 76, -166, -74, -94, 1, -84, 3, 12, -101, -10, 126, 39, -83, -10, -150, 17, 82, -10, -10, -11, -4, 31, -98, -56, 56, -8, 11, 76, 52, 36, 69, -2, -274, -136, -11, -66, 56, 3, 61, 80, 20, -102, 24, -36, -19, 12, -101, -54, 148, -22, 184, -20, -82, 0, -44, 82, -67, -16, 52, -4, -56, -10, -32, -86, -119, 76, -350, -110, -33, -141, 238, -110, -23, -4, -1, 15, 105, 120, -2, -54, -43, 37, 100, 69, -11, 146, 88, -109, 24, 72, 196, 98, 0, -88, 107, -1, 66, 43, -70, 103, -105, -80, -184, 28, -64, 114, -9, -16, 280, 118, 60, -88, 22, 29, 41, -60, -121, -22, -7, 39, 10, -91, -57, 86, 76, 59, 7, -38, 14, -20, 67, -78, 53, 45, 118, 17, -52, 39, -66, 14, 16, 34, 290, -68, -30, 9, 64, -318, -18, -80, -37, 86, -27, -194, 22, -88, -63, -4, -179, 133, -171, 12, 104, -6, 12, 18, -338, 189, 107, -7, -81, -61, -30, -4, -90, -76, 70, -148, -72, 75, -388, 14, -22, 16, 242, -60, 86, 74, 7, 104, 78, 110, -137, 24, -57, -2, 40, 112, 57, -130, 418, 109, -92, -20, 576, 4, -88, 122, 55, -63, 186, -10, 258, 105, -271, 0, 33, -32, -308, 260, 84, 17, -4, -148, -42, -145, 23, 0, -30, -84, -200, 150, -44, 47, -156, -47, -27, 408, 34, -93, -117, -36, 12, 151, 94, 60, -45, 47, -108, 7, 8, 16, -294, 66, -142, -15, -396, -54, 82, 27, -8, 38, 54, -91, 101, 155, -17, 334, 148, 44, -201, -31, -70, -187, -117, -76, 242, 64, -247, 316, -36, 19, -9, 136, 55, -35, -262, 5, -188, 104, 117, -58, 77, -3, -22, -272, 138, -217, 266, -110, 46, -29, -201, 7, -5, -88, -35, -328, -95, -16, 31, -73, -80, 156, 296, -50, 144, 22, 244, 106, 123, -6, 116, -21, 32, 39, -64, -11, -10, -99, 65, 66, -256, 76, -76, -9, -226, 56, 93, -102, -173, 3, -106, -120, 52, 42, -121, -16, -3, -42, -95, 208, -96, 85, -101, 174, 70, 12, -34, 2, 14, 63, 208, -197, -350, -147, 223, 54, 101, -33, 96, 0, 128, 10, 352, -84, -32, -86, 40, 100, -3, 0, 17, 154, -233, -6, -12, 151, -36, 202, 376, -61, 58, 214, -162, 47, 93, 73, 142, -128, 270, 56, -226, 12, 79, -72, 235, 109, -82, -186, 84, -43, 390, -102, -5, -30, 0, 150, 63, 184, -289, -42, -55, -33, 130, 26, -195, -136, 188, 10, -80, 60, -332, 124, -93, -177, -35, -69, 44, -107, 90, -117, -173, -2, -266, -74, 382, -50, 47, -100, -250, -406, -17, -1, 162, -44, -60, -350, 36, -590, -8, -114, -71, 16, 65, -188, -132, -35, 92, -6, 216, 28, -32, -1, 42, 10, 130, 292, -198, -10, -3, 91, -136, 6, -374, 94, 75, -241, 344, 74, 1, -31, -190, 179, -182, 280, 163, -86, -152, 76, -4, 52, -31, -198, 202, -60, -268, 196, 84, 29, -98, 36, 37, 21, 128, -5, 86, 66, 38, -122, -177, 30, 306, -134, -25, -61, 510, 242, 29, -70, 12, 103, 146, 72, -151, -36, -273, 48, 206, 21, -109, -16, -128, 28, -5, -216, 102, 118, -22, -154, 90, 49, 404, 83, -394, -150, -238, 11, -20, 34, 516, -170, -115, 50, 298, -196, -2, -230, -41, 153, 5, -274, -303, 152, -23, 63, -108, -48, -98, 124, -168, 106, 194, 254, 268, 15, 261, 61, 22, 22, 0, 25, -36, -87, 71, -1, 18, 218, -556, 104, -79, 11, 142, -194, -85, 7, -114, -49, -9, 214, 15, -26, -51, -141, -40, 385, 43, -4, 92, -42, -122, -86, 406], "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, "nf_label": null, "projective_field": null, "projective_field_label": null, "projective_image": null, "projective_image_type": null, "qexp": [[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, 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, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, -1, 1, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, -1, 1, 0], [2, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, -1, -1, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, -1, 0], [0, -1, 0, 0, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0, -1, 1, -1, 0], [1, 2, 2, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 1, -1, -1, 0], [0, 1, 0, 0, -2, 0, -1, 0, 1, 1, -1, 0, 2, -1, 1, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, 0], [0, 0, 1, 1, -1, 2, -2, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, -1, 1, 1, 0, 0, 0, 0, -1, 0, 1, 0], [0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -1, 0, -3, 2, -1, -1, 2, 0, 0, 0, 2, 1], [0, -1, 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, 0, 1, 0, 0, 0, -1], [0, -1, 0, 0, -1, -2, 0, 0, 1, 0, 0, 1, 1, 0, 0, -2, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 1], [-1, 0, -3, 1, -1, -1, -1, 1, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, -1, 1, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, -1, 1, -1, 0], [1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, -2, -1, 0, 1, 0, 1], [-4, 0, 0, 0, 4, 1, 1, 1, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 1, 0, -2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-2, -1, -1, 1, 0, 0, 0, 2, -2, 0, 3, 0, 0, 1, 2, -1, -1, -1], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 2, 2, -2, 0, 0, 0, -1, 1, 0, -2, 0, 2, -2, -1, 1, 1, 1], [0, 3, 0, 0, -1, -2, 2, 0, 2, 0, 2, 1, 1, 1, 1, -2, 0, -1], [0, -1, 0, 0, 3, 0, 1, 0, -1, -1, 1, 1, -3, 2, 0, 0, -1, -1], [-3, 0, -1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -3, -1, 0, -1, 2, 0, -1, 0, 3, -1, 0, -1, 1, 1], [1, 0, 1, 0, 2, 1, -1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 2, 1, 1, 0, -4, 0, 1, 0, -2, 2, 0, 1, 0, 0], [0, -1, 0, 0, 0, 1, -1, 0, -1, 2, -1, 1, 0, -1, 0, 1, 2, -1], [2, 5, 5, 1, 0, 0, 0, 1, 2, 0, 3, 0, 0, 1, 1, -2, 0, 0], [0, -2, 0, 0, 0, 1, -1, 0, 0, 2, -1, 0, 0, -1, -2, 1, 2, 0], [1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [1, 0, -3, -1, 2, -2, 3, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, -2, 0, -1, 0, 0, 0], [4, -1, -1, -1, 0, 0, 0, 0, 4, 0, 1, 0, -2, -1, 0, 0, 1, -1], [0, 0, -1, 0, -2, 0, -1, -1, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0], [2, 0, 2, 0, 1, 2, -2, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -2, -2, -1, 0, 0, 0, -1, -2, 0, -1, 0, 1, -1, -1, 1, 0, 1], [0, -1, 0, 0, 1, -1, 2, 0, -6, -1, 2, 0, -1, 3, 0, -1, -1, 0], [-1, -2, -2, 1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 2, -2, -1], [0, 3, 0, 0, 2, 1, 2, 0, 4, 0, 2, -1, -2, 0, -1, 1, 0, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0], [-2, 0, 1, 0, -2, -1, 1, -1, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 1, 0, 2, -1, 0, 0, 0, -2, -1, 1, 0, 0, 3, 0, 1], [1, 0, -4, 1, 0, 0, 1, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, -1, 0, 0, -3, 2, 0, 1, 2, 0, -1, -1, 2, -1], [0, -3, 0, 0, 1, 2, -2, 0, -1, -2, -2, -1, -1, -1, -2, 2, -2, 1], [-1, 1, 1, 0, 0, 0, 0, 2, -1, 0, -1, 0, 1, 0, 2, 1, 1, 1], [0, -4, 0, 0, -1, -2, -1, 0, -1, 2, -1, 0, 1, 0, 1, -2, 2, 0], [5, 0, 0, -1, 0, 0, 0, 0, 5, 0, 2, 0, -4, -1, 0, 2, -1, 0], [4, 0, 2, -3, 3, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 4, -1, 1, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, -1, -1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, -4, 1, 0], [-5, 0, -3, 4, -2, 1, -3, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 1, 2, 1, -1, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0], [-7, 1, 1, 0, 0, 0, 0, 2, -7, 0, -1, 0, 5, 0, 2, -2, 3, 1], [0, 2, 0, 0, 1, 0, 1, 0, 1, -2, 1, 1, -1, 0, 2, 0, -2, -1], [2, 4, 4, -2, 0, 0, 0, -2, 2, 0, -2, 0, -1, -2, -2, -1, 2, 0], [0, 1, 0, 0, 0, 2, 0, 0, 3, -2, 0, 0, 0, 0, -1, 2, -2, 0], [0, -1, 0, 0, 0, 2, -4, 0, -4, 3, -4, -3, 0, 0, -3, 2, 3, 3], [2, 0, -1, -2, -4, -3, -2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -2, -1, 3, -2, 0, 0, 3, 0, 1, 3, 1, 1, 2, 3, -3, -1, -2], [0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 3, 2, -2, 0, 0, -2, -2, 1, -3, 0, -2, 2, -2, -1], [0, 2, 0, 0, 0, -1, 1, 0, -3, -1, 1, -2, 0, 0, 0, -1, -1, 2], [4, 1, 1, -2, 0, 0, 0, -4, 4, 0, -5, 0, 2, -2, -4, 3, 2, 1], [0, 3, 0, 0, 1, 3, 2, 0, 1, 0, 2, -2, -1, -2, 1, 3, 0, 2], [3, -3, -3, 1, 0, 0, 0, 2, 3, 0, 2, 0, 0, 1, 2, -2, -4, -3], [2, 0, -3, 1, 1, 0, 0, -1, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0], [-5, 0, 2, 1, 3, 2, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 1, 0, 0, 0, 1, -1, 0, 3, 0, -3, 1, 1, -1, -3, -1], [5, -1, -1, 0, 0, 0, 0, -1, 5, 0, 0, 0, -2, 0, -1, 0, -1, 0], [-8, 0, 0, -1, 3, -1, 1, 1, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, 1, -1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -2, -3, -1], [0, 1, 0, 0, -8, -3, -4, 0, -5, 3, -4, -1, 8, -1, 0, -3, 3, 1], [-2, 3, 3, 1, 0, 0, 0, 2, -2, 0, 1, 0, 3, 1, 2, -2, 0, -1], [0, 2, 0, 0, -2, 0, -2, 0, 4, 3, -2, 0, 2, -1, -1, 0, 3, 0], [0, -1, 0, 0, -3, -4, -2, 0, -1, -2, -2, 2, 3, 2, 1, -4, -2, -2], [-4, 0, 0, 1, -1, -1, 3, -3, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [3, -1, 0, 1, 3, 0, 1, 0, 5, -3, 1, 1, -3, 2, -1, 0, -1, -2], [2, 0, -6, -1, 1, -2, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -1, 0, 4, 0, -3, -3, 4, 0, 1, -1, 4, 0, -3, 0], [0, 1, 0, 0, -1, -3, 4, 0, 0, -1, 4, 1, 1, 2, 2, -3, -1, -1], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0], [0, 0, 0, 0, 4, 2, 0, 0, 0, -3, 0, -1, -4, 0, 0, 2, -3, 1]]}