# Stored data for newform 209.2.k.b, downloaded from the LMFDB on 21 February 2026.
{"Nk2": 836, "analytic_conductor": 1.66887340224476, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 209, "char_degree": 4, "char_is_minimal": true, "char_is_real": false, "char_orbit_index": 11, "char_orbit_label": "k", "char_order": 10, "char_parity": 1, "char_values": [209, 10, [134, 78], [7, 5]], "cm_discs": [], "conrey_index": 18, "dim": 8, "field_disc": 484000000, "field_disc_factorization": [[2, 8], [5, 6], [11, 2]], "field_poly": [121, 0, 66, 0, 16, 0, 1, 0, 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, [121, 0, 99, 0, 31, 0, -1, 0, 1]]], "hecke_orbit": 2, "hecke_orbit_code": 4504286855692497, "hecke_ring_generator_nbound": 4, "hecke_ring_index": 1, "hecke_ring_index_factorization": [], "hecke_ring_index_proved": true, "inner_twist_count": 4, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 11, 4, -1, 10, 0], [1, 1, 19, 2, -1, 2, 0], [1, 1, 209, 11, 1, 10, 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.k.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.k.b", "nf_label": "8.0.484000000.9", "prim_orbit_index": 11, "qexp_display": "q+(-\\beta _{1}-\\beta _{4})q^{2}+(-1-\\beta _{2}-3\\beta _{3}+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 2, "rm_discs": [], "sato_tate_group": "1.2.3.c10", "self_twist_discs": [], "self_twist_type": 0, "space_label": "209.2.k", "trace_display": [0, 0, 4, -10], "trace_hash": 203304659108411282, "trace_moments": [{"__RealLiteral__": 0, "data": "0.073", "prec": 17}, {"__RealLiteral__": 0, "data": "7.383", "prec": 17}, {"__RealLiteral__": 0, "data": "4.620", "prec": 17}, {"__RealLiteral__": 0, "data": "312.106", "prec": 24}, {"__RealLiteral__": 0, "data": "342.390", "prec": 24}, {"__RealLiteral__": 0, "data": "18527.275", "prec": 30}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.570", "prec": 17}, "traces": [0, 8, 0, 0, 6, 4, 0, -10, 0, -6, 0, 18, 0, 0, 0, 0, -22, -30, 0, -10, 8, 0, 0, -12, 0, 2, 4, 0, -20, 0, 0, 0, 0, 0, 0, -10, 18, 0, 32, 0, 0, 0, 0, 0, 56, 12, 0, 16, 0, -4, 0, 0, 0, 0, 0, -6, 0, 0, -24, 0, 0, 30, -40, 30, 26, 0, 0, 0, 30, 0, 0, 0, 0, 0, 60, 0, 0, -20, 0, 0, 14, -18, -40, -40, 0, -30, 0, 0, 0, 0, 0, 0, -4, 0, 0, -10, 0, 0, 0, -6, -76, -40, 0, 0, -100, 0, -20, 0, 0, 0, 0, 0, 40, 0, 0, -6, 0, 0, -100, 30, 0, -2, 0, 0, 0, 46, 0, 0, 0, 0, 32, 0, 0, 20, 80, 0, 0, 44, 0, 20, -20, 0, 0, 0, 24, 0, 0, 0, 0, 80, 0, 0, 40, 0, 0, 0, 0, 14, -60, 0, 0, 10, 0, 12, 0, 0, 0, 0, 0, 34, 0, 30, 160, 0, 0, 0, 18, 0, 60, 0, -6, 0, -40, 0, 0, 0, 0, -120, 52, 0, -24, -10, 0, 0, -100, 0, -48, 0, 0, 20, 0, 0, 0, 0, 0, 0, -160, -6, 0, -30, 0, 0, 0, 0, -32, 30, 0, 0, 24, 0, 28, 0, 0, 0, 0, 36, 80, 0, 0, -102, 0, 0, -40, -80, 12, 18, 0, 0, 0, -20, 0, 0, 0, 0, -110, -32, 0, 64, -20, 0, 0, -6, 0, -32, -8, 0, -22, 0, 0, 0, 0, 0, 0, 0, 0, 0, -60, 0, 0, 0, 0, 140, 120, 0, 0, 72, 0, 40, 0, 0, 0, 0, 0, -50, 0, 0, 144, 0, 0, 56, -52, 0, -40, 0, 0, 0, 0, 0, 0, 0, 0, 50, 172, 0, 70, 30, 0, 0, -110, 0, 20, -40, 0, -78, 0, 0, 0, 0, 0, 0, 68, 0, 0, 0, -126, 0, 0, 0, -160, 0, 0, 0, -20, 0, -12, 0, 0, 0, 0, 0, 0, 0, -24, -10, 0, 0, 144, -10, 0, -90, 0, 0, 0, 148, 0, 0, 0, 0, 240, -100, 0, 22, 0, 0, 0, 40, 0, 54, 18, 0, -20, 0, 0, 0, 0, 0, 0, -48, 0, 0, -20, 0, 0, 0, 0, -10, -56, -150, 0, -60, 0, 60, 0, 0, 0, 0, -102, -116, 0, 0, -58, 0, 0, -80, 70, 36, 0, 0, 0, 0, -80, 0, 0, 0, 0, -40, -36, 0, -28, -68, 0, 0, 88, -42, -120, 30, 0, -70, 0, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 0, 108, 180, 62, 0, 0, -160, 0, -10, 0, 0, 0, 0, 0, 128, 0, 0, -160, 0, 0, -2, 0, 0, 132, 0, 0, 0, -84, 0, 0, 0, 0, 0, -60, 0, 40, -30, 0, 0, -30, 0, 120, 168, 0, 126, 0, 0, 0, 0, 0, 0, -130, 0, 0, 0, 42, 0, 0, 0, 38, -78, 0, 0, 180, 0, 0, 0, 0, 0, 0, 0, 40, 0, 0, 280, 0, 0, 26, -120, 0, 20, 0, -12, 0, 130, 0, 0, 0, 0, -156, -100, 0, 30, -80, 0, 0, 140, 0, 0, -14, 0, 70, 0, 0, 0, 0, 0, 0, 248, 0, 0, 96, 0, 0, 0, 0, 30, 110, 120, 0, 0, 0, -168, 0, 0, 0, 0, 0, -200, 0, 0, 0, 0, 0, 20, -18, -102, 76, 0, 0, 0, 10, 0, 0, 0, 0, -84, -96, 0, -40, -40, 0, 0, 0, 0, 60, -80, 0, 24, 0, 0, 0, 0, 0, 0, -76, 0, 0, -28, 0, 0, 0, -180, 0, -172, 0, 0, -16, 0, -40, 0, 0, 0, 0, 0, -72, 0, 0, -52, 0, 0, -80, 180, 0, -60, 0, 0, 0, -164, 0, 0, 0, 0, 78, 10, 0, -180, 16, 0, 0, -264, 0, -46, -28, 0, -110, 0, 0, 0, 0, 0, 0, -140, 0, 0, 10, 60, 0, 0, 0, 0, 160, 0, 0, 72, 0, -162, 0, 0, 0, 0, 0, 40, 0, -90, 72, 0, 0, 30, -40, 0, -60, 0, 60, 0, 30, 0, 0, 0, 0, 140, 120, 0, 0, 146, 0, 0, 40, 0, -148, 140, 0, 160, 0, 0, 0, 0, 0, 0, 150, 12, 0, -200, 0, 0, 0, 0, 76, 160, -54, 0, -60, 0, 230, 0, 0, 0, 0, 360, 0, 0, 0, -60, 0, 0, 80, -24, 0, -90, 0, 0, 0, -24, 0, 0, 0, 0, -4, 0, 0, -40, 0, 0, 0, 280, -90, -120, -200, 0, 0, 0, 0, 0, 0, 0, 0, -160, 0, 0, 120, 0, 0, 0, 0, 86, 32, 0, 0, 280, 0, -40, 0, 0, 0, 0, 0, 30, 0, 0, -90, 0, 0, 0, 80, 0, 20, 0, 0, 0, 80, 0, 0, 0, 0, 100, -44, 0, -20, 64, 0, 0, 200, 0, -28, 0, 0, 200, 0, -12, 0, 0, 0, 0, -30, 0, 0, -150, 0, 0, 0, 0, 106, -220, 0, 0, 12, 0, 10, 0, 0, 0, 0, 0, -30, 0, 30, -40, 0, 0, -92, 70, 0, 100, 0, 0, 0, -280, 0, 0, 0, 0, 160, 180, 0, -28, -80, 0, 0, -364, 0, 104, 0, 0, -202, 0, 0, 0, 0, 0, 0, 0, 72, 0, -10, 0, 0, 0, 0, -240, -40, 192, 0, -20, 0, -320, 0, 0, 0, 0, -150, -40, 0, 0, -130, 0, 0, 96, -40, 0, 80, 0, 0, 0, 0, 0, 0, 0, 0, -24, -130, 0, 30, -310, 0, 0, -30, 240, 10, -80, 0, 26, 0, 0, 0, 0, 0, 0, 124, 0, 0, 88, 0, 0, 0, 60, -50, 0, 0, 0, -30, 0, -22, 0, 0, 0, 0, 0, 0, 0, 0, -60, 0, 0, -10, 60, 0, -240, 0, 0, 0, -44, 0, 0, 0, 0, 40, 60, 0, 208, 0, 0, 0, 60, 0, -20, 10, 0, 50, 0, 0, 0], "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, 0, 0, 0, 0], [1, 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, -1, -3, 0, 3, 0, 0], [1, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, -2, 0, 1, -2], [0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [4, 0, 2, 3, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, 0], [0, 2, 0, 0, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -1, -4, 0, 0, 0, 0], [-3, 0, 0, 3, 0, 0, 0, 0], [0, 3, 0, 0, 3, 0, -3, 0], [0, 0, 2, 1, 2, -2, 0, 2], [0, 0, -1, -1, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -2, 1], [-2, 0, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -4, 0, 1, 0, 0], [-6, 0, -6, -10, 0, 10, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 3, 5, 0, -6, 0, 0], [0, 0, 0, 0, 2, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 2, -2], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 3, 0, 0, 3], [-2, 0, -1, -1, 0, 1, 0, 0], [6, 0, 0, 9, 0, -6, 0, 0], [0, -2, 0, 0, 2, 0, 2, 4], [6, 4, 2, 6, 2, 0, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -2, 0, 2, 0], [0, -4, 0, 0, 2, 0, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -3, 4, 0, -7, 0, 0], [3, 0, -6, -2, 0, 8, 0, 0], [3, 0, 3, 0, 0, -3, 0, 0], [0, 0, 0, 0, 1, 0, 1, 1], [3, 0, 0, 1, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 5, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 0, -4, 0, 6, -4], [0, -2, 0, 0, 4, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 3, 4, 0, 0, 0, 0], [0, 4, 0, 0, 3, 0, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 6, 0, 2, 0, 0], [0, 2, 0, 0, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -4, -7, 0, 8, 0, 0], [-4, 0, 4, -6, 0, -6, 0, 0], [3, 0, 0, 0, 0, 3, 0, 0], [0, 0, -6, -6, 0, 1, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 4, 4], [6, 0, -3, 9, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0], [0, -4, 0, 0, -8, 0, 2, -2], [0, 0, 0, 0, 0, 0, -6, 3], [0, 0, 0, -4, 0, -4, 0, 0], [8, 0, 4, 2, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [5, -4, 2, 10, -2, -8, 0, 4], [-5, 0, -1, -4, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, 2, 0, -4, -4], [1, 0, 0, -4, 0, -1, 0, 0], [-9, 0, -9, -9, 0, 9, 0, 0], [-2, 0, 14, -2, 0, 0, 0, 0], [-7, 0, -3, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, -6, -3, 0, 0, 0, 0], [0, 2, 0, 0, 5, 0, -7, 7], [0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, -9, 0, 4, 0], [0, 2, 0, 0, 2, 0, 0, 2], [0, 3, 0, 0, 0, 0, -3, -3], [0, 4, 0, 0, 2, 0, 0, -2], [0, 0, 0, 1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 1], [-1, 0, 1, 0, 2, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 2, -2], [0, 4, 0, 0, -1, 0, 0, -4], [0, 0, 3, -6, 0, -6, 0, 0], [-5, 0, 9, 0, 0, -9, 0, 0]]}