# Stored data for newform 85.4.a.b, downloaded from the LMFDB on 11 July 2026.
{"Nk2": 1360, "analytic_conductor": 5.015162350487953, "analytic_rank": 1, "analytic_rank_proved": true, "atkin_lehner_eigenvals": [[5, 1], [17, -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": [85, 1, [52, 71], [1, 1]], "cm_discs": [], "conrey_index": 1, "dim": 1, "field_disc": 1, "field_disc_factorization": [], "field_poly": [0, 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, [-3, 1]], [3, [5, 1]]], "hecke_orbit": 2, "hecke_orbit_code": 4503599694479445, "hecke_ring_generator_nbound": 1, "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": false, "is_maximal": false, "is_polredabs": true, "is_rm": false, "is_self_dual": true, "is_self_twist": false, "is_twist_minimal": true, "label": "85.4.a.b", "level": 85, "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": [5, 17], "level_radical": 85, "minimal_twist": "85.4.a.b", "nf_label": "1.1.1.1", "prim_orbit_index": 1, "qexp_display": "q+3q^{2}-5q^{3}+q^{4}-5q^{5}-15q^{6}+\\cdots", "related_objects": [], "relative_dim": 1, "rm_discs": [], "sato_tate_group": "3.2.3.c1", "self_twist_discs": [], "self_twist_type": 0, "space_label": "85.4.a", "trace_display": [3, -5, -5, -22], "trace_hash": 625340152407498925, "trace_moments": [{"__RealLiteral__": 0, "data": "0.000", "prec": 17}, {"__RealLiteral__": 0, "data": "0.953", "prec": 17}, {"__RealLiteral__": 0, "data": "0.054", "prec": 17}, {"__RealLiteral__": 0, "data": "1.852", "prec": 17}, {"__RealLiteral__": 0, "data": "0.182", "prec": 17}, {"__RealLiteral__": 0, "data": "4.564", "prec": 17}], "trace_zratio": 0, "traces": [0, 1, 3, -5, 1, -5, -15, -22, -21, -2, -15, 60, -5, -31, -66, 25, -71, 17, -6, -61, -5, 110, 180, -78, 105, 25, -93, 145, -22, 69, 75, -31, -45, -300, 51, 110, -2, 56, -183, 155, 105, -6, 330, -538, 60, 10, -234, -465, 355, 141, 75, -85, -31, 723, 435, -300, 462, 305, 207, -753, 25, 35, -93, 44, 433, 155, -900, -322, 17, 390, 330, -99, 42, -1123, 168, -125, -61, -1320, 465, 488, 355, -671, -18, -852, 110, -85, -1614, -345, -1260, 1215, 30, 682, -78, 155, -1395, 305, 225, -601, 423, -120, 25, 450, -255, 1544, 651, -550, 2169, 276, 145, 1739, -900, -280, 1562, -657, 915, 390, 69, 62, -2259, -374, -525, 2269, 105, 30, -31, -125, 132, 761, 1659, 2690, 465, -1074, -300, 1342, -966, -725, -357, -1242, 1170, -952, 110, 2325, -297, -1860, 142, -345, -3369, -705, 56, -960, -375, 2174, 1281, -34, -3960, 155, 155, 866, 1464, -3615, 225, 1716, -2013, 1712, -6, 1500, -2556, 2202, -2310, -1236, -255, 122, -538, -414, -1035, -550, -4260, 3765, 3645, -4212, 10, 1238, 2046, -175, 1638, -280, 465, 1020, -465, -3190, 915, -4236, -2165, -274, -1803, -775, 141, -156, -360, 1775, -525, 1610, 1350, -1518, -85, 30, 4632, 156, 2201, -3660, -1650, 3158, 723, 495, 828, 2690, -3045, 682, 5217, 5615, -300, -527, -840, -3643, 990, -50, -1971, -5703, 305, -3166, 1170, 6600, -1449, 5367, 186, 2325, -753, -2440, -1122, -2394, -1775, -142, 6807, -560, 35, -705, 90, 1891, 651, 4260, -375, -5868, 44, -4680, 2283, 425, 1513, 4206, 8070, -1232, 155, -138, -3222, 7947, 6300, -3615, 4026, -6075, -322, -4737, -2175, -5116, -1207, -3410, -3726, 1500, 390, -7462, -2856, 62, -2310, 8619, 6975, -343, -99, -1525, -5580, 132, 90, 289, -1035, 3005, -1123, -3957, -2115, 3765, -1176, 8700, -2880, 2418, -125, 11836, 6522, -2250, 4331, -175, -102, -5884, -1320, -7720, 465, -8712, -3255, -1966, 2598, -220, 488, -6960, -10845, 4140, -2165, -1380, 5148, -1037, -671, -775, 5136, -8695, 126, 10230, 4500, 1949, -852, -112, 6606, 1610, -7810, -3643, -3708, 3285, -85, -1860, 366, 4444, 11298, -1950, -1242, -4941, -345, -4912, -1650, -4495, -2700, 6690, 11295, 495, 1215, 1870, -12636, 4530, -210, -3138, 3714, -11345, 682, 5615, -525, 1406, 5538, 12, -840, -15906, 155, 6518, 3060, 625, 9765, -2139, -9570, -8032, 305, -3805, -12708, -147, -8295, 6600, -822, 1076, -601, 5064, -2325, -1326, -2961, 5370, -468, -2440, -120, 9992, 5325, -6710, -1775, -3384, 4830, 961, 450, 3355, -4554, 3360, 1785, -10291, 90, 6210, 1544, 16566, 468, 4260, 1395, 4760, -10980, 8322, -550, 5132, 9474, 930, -15183, 425, 1485, -770, 276, 9300, 8070, 15672, -10295, 6806, 2046, 1725, 1739, 4758, 16845, 13640, 6300, -282, -1581, 17064, -280, -6075, -10929, 4800, -9526, 2676, -150, -360, -657, -10870, -17109, -3410, -6405, 9938, -9498, 2465, 390, -7302, 19800, -8929, -4899, -775, 16101, 2004, 62, 7084, 6975, -4330, 15813, -32280, -7320, -1525, -374, -1446, -7182, -4041, -1125, -1736, -426, -8580, 2269, 3005, -1680, -10930, -735, -8560, -2115, -20235, 30, 1173, 5673, 600, 2201, 2178, 12780, -1930, -125, -11010, -17604, 9504, -924, -2250, -14040, 6180, 761, -14148, 1275, 24706, -8733, -8845, 12618, -7720, 2690, -27900, -3696, 2070, -3255, -4044, -414, 1982, -1074, 2750, 23841, -527, 21300, -6083, -10845, 1506, 1342, 186, -18225, -1380, 6762, 21060, -14211, 8460, -725, 16346, -15348, -6190, -765, -8695, -10230, -16309, -1242, -70, 4500, -4209, -8190, -10736, -22386, 1400, -952, -22335, 186, 16678, -7810, -5100, 25857, 16458, 2325, 3285, -1029, 14762, 2079, -17877, -4575, 2438, -1860, 21180, 396, -1950, -866, 14612, 867, 1370, -345, 18744, 9015, 43380, 23583, -310, -11871, 9798, -705, 1891, 11295, 780, -3976, 9834, 26100, 1870, -960, -8875, 7254, -9606, 2625, 12980, 35508, 644, 2174, -11345, -6750, -25270, 2745, 7590, -525, 14415, -34, -7183, -17652, -150, 27720, -10377, -23160, 22538, 155, -11310, -26136, -26730, -11005, 625, -5898, 18300, 866, 952, -660, 13436, -10248, -15790, -20880, -3805, -3615, -4371, 12420, 198, -8295, 2610, -4140, 20756, 1716, -13450, -3111, 11061, 14091, -45180, -2325, -3410, 1712, 18858, -26085, 5370, 426, 2246, 30690, -6459, 1500, -11332, 5847, 2635, 17892, -6710, -336, -5382, 2202, 18215, 4830, 2100, -4950, -6799, -10929, 3625, -1236, -4350, 9855, 13222, 1785, 28515, -5580, -19785, 122, 6210, 13332, 15830, 38198, -22413, -5850, -29668, -414, 2640, -14823, 4760, 7245, -102, -14736, -26835, -550, -13428, -13485, -3416, 25980, -11625, 20070, -9900, 3765, 11729, 1485, -976, -25515, 2418, 5610, 9300, -4212, 11970, 13590, 31341, -710, -33968, -9414, 710, 1238, 1725, -34035, 10535, -14322, 20917, 16845, -9146, -175, -12070, 4218, 3525, 3510, -19320, 36, 2537, -280, -9455, -47718, -15114, -3255, 4800, 19554, 1704, 1020, -6072, 1875, -28285, 33015, 29340, -6417, -10870, -3190, 18269, -24096, 23400, -6405, -7122, -11415, -38258, -4236, 170, -441, 23343, -7565, -511, 19800, -21030, -274, 8262, 3228, -775, 12621, 6160, 15192, 366, -775, -5940, -3978, 10005, -10011, -4330, 16110, 30371, -156, -39735, -7320, 14454, 2520, -1085, 29976, 18075, 1775, 4290, -20130, -7905, -1125, -2430, -10152, -67380, 1610, -8580, 2883, 23685, -9450, -26166, 10065, -6250, -1518, 25580, 10080, -8560, 6035, 32818, -30873, -1364, 30, -16713, 18630, -12190, -32424, -7500, 49698, 30084, 156, -16516, 12780, 37310, -13423, 2397, 14280, -11010, -3660, -4495, 24966, 17055, 11550, -19628, 15396, -43095, 3158, 6180, 2790, -49918, -51333, 1715, 1275, -4368, 495, 6662, -2310, -610, -5796, 15, 27900, -11077, 2690, -660, 47016, 10488, -6525, 2070, 20418, -1445, 682, 29280, 5175, 9982, -36519, 1202, 14274, 2750, 5615, 17372, 40920, 19785, 21300, 14154, -846, -34846, -527, -18825, 51192, 42174, 5880, -16742, -18225, -40260, -3643, 28365, 14400, 21060, -36498, -12090, 8028, -2139, -50, 12291, -1080, -59180, 13797, -6190, -32610, 31589, -5703, -900, -10230, -9144, -21655, -51120, 29814, 875, -3166, 23628, 7395, -45574, -8190, 29420, -21906, 3069, 6600, 1400, -26787, -3088, -3105, 31290, -2325, -8601, 5367, 43560, 6012, -5100, -1302, -13420, 21252, 9830, 2325, 53361, -12990, 468, 53463, 15950, -96840, -6591, -2440, 34813, -4575, 34800, 7854, 20304, -4338, 21180, -2394, -20700, -12123, 27324, 10825, -28830, -5208, -552, -142, 1370, -25740, -6280, -47649, 5185, 9015, 32943, -560, 20944, -32790, 3875, -2485, -35424, -25680, 72900, -705, -3478, -60705, 40266, -630, 780, 3519, -51150, 1891, 41964, 1800, -22327, 1395, -9745, 6534, -8875, 4260, 50762, -5790, 8120, 2625], "weight": 4, "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], 1, 3, -5, 1, -5, -15, -22, -21, -2, -15, 60, -5, -31, -66, 25, -71, 17, -6, -61, -5, 110, 180, -78, 105, 25, -93, 145, -22, 69, 75, -31, -45, -300, 51, 110, -2, 56, -183, 155, 105, -6, 330, -538, 60, 10, -234, -465, 355, 141, 75, -85, -31, 723, 435, -300, 462, 305, 207, -753, 25, 35, -93, 44, 433, 155, -900, -322, 17, 390, 330, -99, 42, -1123, 168, -125, -61, -1320, 465, 488, 355, -671, -18, -852, 110, -85, -1614, -345, -1260, 1215, 30, 682, -78, 155, -1395, 305, 225, -601, 423, -120, 25, 450, -255, 1544, 651, -550, 2169, 276, 145, 1739, -900, -280, 1562, -657, 915, 390, 69, 62, -2259, -374, -525, 2269, 105, 30, -31, -125, 132, 761, 1659, 2690, 465, -1074, -300, 1342, -966, -725, -357, -1242, 1170, -952, 110, 2325, -297, -1860, 142, -345, -3369, -705, 56, -960, -375, 2174, 1281, -34, -3960, 155, 155, 866, 1464, -3615, 225, 1716, -2013, 1712, -6, 1500, -2556, 2202, -2310, -1236, -255, 122, -538, -414, -1035, -550, -4260, 3765, 3645, -4212, 10, 1238, 2046, -175, 1638, -280, 465, 1020, -465, -3190, 915, -4236, -2165, -274, -1803, -775, 141, -156, -360, 1775, -525, 1610, 1350, -1518, -85, 30, 4632, 156, 2201, -3660, -1650, 3158, 723, 495, 828, 2690, -3045, 682, 5217, 5615, -300, -527, -840, -3643, 990, -50, -1971, -5703, 305, -3166, 1170, 6600, -1449, 5367, 186, 2325, -753, -2440, -1122, -2394, -1775, -142, 6807, -560, 35, -705, 90, 1891, 651, 4260, -375, -5868, 44, -4680, 2283, 425, 1513, 4206, 8070, -1232, 155, -138, -3222, 7947, 6300, -3615, 4026, -6075, -322, -4737, -2175, -5116, -1207, -3410, -3726, 1500, 390, -7462, -2856, 62, -2310, 8619, 6975, -343, -99, -1525, -5580, 132, 90, 289, -1035, 3005, -1123, -3957, -2115, 3765, -1176, 8700, -2880, 2418, -125, 11836, 6522, -2250, 4331, -175, -102, -5884, -1320, -7720, 465, -8712, -3255, -1966, 2598, -220, 488, -6960, -10845, 4140, -2165, -1380, 5148, -1037, -671, -775, 5136, -8695, 126, 10230, 4500, 1949, -852, -112, 6606, 1610, -7810, -3643, -3708, 3285, -85, -1860, 366, 4444, 11298, -1950, -1242, -4941, -345, -4912, -1650, -4495, -2700, 6690, 11295, 495, 1215, 1870, -12636, 4530, -210, -3138, 3714, -11345, 682, 5615, -525, 1406, 5538, 12, -840, -15906, 155, 6518, 3060, 625, 9765, -2139, -9570, -8032, 305, -3805, -12708, -147, -8295, 6600, -822, 1076, -601, 5064, -2325, -1326, -2961, 5370, -468, -2440, -120, 9992, 5325, -6710, -1775, -3384, 4830, 961, 450, 3355, -4554, 3360, 1785, -10291, 90, 6210, 1544, 16566, 468, 4260, 1395, 4760, -10980, 8322, -550, 5132, 9474, 930, -15183, 425, 1485, -770, 276, 9300, 8070, 15672, -10295, 6806, 2046, 1725, 1739, 4758, 16845, 13640, 6300, -282, -1581, 17064, -280, -6075, -10929, 4800, -9526, 2676, -150, -360, -657, -10870, -17109, -3410, -6405, 9938, -9498, 2465, 390, -7302, 19800, -8929, -4899, -775, 16101, 2004, 62, 7084, 6975, -4330, 15813, -32280, -7320, -1525, -374, -1446, -7182, -4041, -1125, -1736, -426, -8580, 2269, 3005, -1680, -10930, -735, -8560, -2115, -20235, 30, 1173, 5673, 600, 2201, 2178, 12780, -1930, -125, -11010, -17604, 9504, -924, -2250, -14040, 6180, 761, -14148, 1275, 24706, -8733, -8845, 12618, -7720, 2690, -27900, -3696, 2070, -3255, -4044, -414, 1982, -1074, 2750, 23841, -527, 21300, -6083, -10845, 1506, 1342, 186, -18225, -1380, 6762, 21060, -14211, 8460, -725, 16346, -15348, -6190, -765, -8695, -10230, -16309, -1242, -70, 4500, -4209, -8190, -10736, -22386, 1400, -952, -22335, 186, 16678, -7810, -5100, 25857, 16458, 2325, 3285, -1029, 14762, 2079, -17877, -4575, 2438, -1860, 21180, 396, -1950, -866, 14612, 867, 1370, -345, 18744, 9015, 43380, 23583, -310, -11871, 9798, -705, 1891, 11295, 780, -3976, 9834, 26100, 1870, -960, -8875, 7254, -9606, 2625, 12980, 35508, 644, 2174, -11345, -6750, -25270, 2745, 7590, -525, 14415, -34, -7183, -17652, -150, 27720, -10377, -23160, 22538, 155, -11310, -26136, -26730, -11005, 625, -5898, 18300, 866, 952, -660, 13436, -10248, -15790, -20880, -3805, -3615, -4371, 12420, 198, -8295, 2610, -4140, 20756, 1716, -13450, -3111, 11061, 14091, -45180, -2325, -3410, 1712, 18858, -26085, 5370, 426, 2246, 30690, -6459, 1500, -11332, 5847, 2635, 17892, -6710, -336, -5382, 2202, 18215, 4830, 2100, -4950, -6799, -10929, 3625, -1236, -4350, 9855, 13222, 1785, 28515, -5580, -19785, 122, 6210, 13332, 15830, 38198, -22413, -5850, -29668, -414, 2640, -14823, 4760, 7245, -102, -14736, -26835, -550, -13428, -13485, -3416, 25980, -11625, 20070, -9900, 3765, 11729, 1485, -976, -25515, 2418, 5610, 9300, -4212, 11970, 13590, 31341, -710, -33968, -9414, 710, 1238, 1725, -34035, 10535, -14322, 20917, 16845, -9146, -175, -12070, 4218, 3525, 3510, -19320, 36, 2537, -280, -9455, -47718, -15114, -3255, 4800, 19554, 1704, 1020, -6072, 1875, -28285, 33015, 29340, -6417, -10870, -3190, 18269, -24096, 23400, -6405, -7122, -11415, -38258, -4236, 170, -441, 23343, -7565, -511, 19800, -21030, -274, 8262, 3228, -775, 12621, 6160, 15192, 366, -775, -5940, -3978, 10005, -10011, -4330, 16110, 30371, -156, -39735, -7320, 14454, 2520, -1085, 29976, 18075, 1775, 4290, -20130, -7905, -1125, -2430, -10152, -67380, 1610, -8580, 2883, 23685, -9450, -26166, 10065, -6250, -1518, 25580, 10080, -8560, 6035, 32818, -30873, -1364, 30, -16713, 18630, -12190, -32424, -7500, 49698, 30084, 156, -16516, 12780, 37310, -13423, 2397, 14280, -11010, -3660, -4495, 24966, 17055, 11550, -19628, 15396, -43095, 3158, 6180, 2790, -49918, -51333, 1715, 1275, -4368, 495, 6662, -2310, -610, -5796, 15, 27900, -11077, 2690, -660, 47016, 10488, -6525, 2070, 20418, -1445, 682, 29280, 5175, 9982, -36519, 1202, 14274, 2750, 5615, 17372, 40920, 19785, 21300, 14154, -846, -34846, -527, -18825, 51192, 42174, 5880, -16742, -18225, -40260, -3643, 28365, 14400, 21060, -36498, -12090, 8028, -2139, -50, 12291, -1080, -59180, 13797, -6190, -32610, 31589, -5703, -900, -10230, -9144, -21655, -51120, 29814, 875, -3166, 23628, 7395, -45574, -8190, 29420, -21906, 3069, 6600, 1400, -26787, -3088, -3105, 31290, -2325, -8601, 5367, 43560, 6012, -5100, -1302, -13420, 21252, 9830, 2325, 53361, -12990, 468, 53463, 15950, -96840, -6591, -2440, 34813, -4575, 34800, 7854, 20304, -4338, 21180, -2394, -20700, -12123, 27324, 10825, -28830, -5208, -552, -142, 1370, -25740, -6280, -47649, 5185, 9015, 32943, -560, 20944, -32790, 3875, -2485, -35424, -25680, 72900, -705, -3478, -60705, 40266, -630, 780, 3519, -51150, 1891, 41964, 1800, -22327, 1395, -9745, 6534, -8875, 4260, 50762, -5790, 8120, 2625]}