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lfunc_lfunctions • Show schema
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{'Lhash': '3810971572400460408091718010207', 'a10': 0, 'a2': 0, 'a3': 1, 'a4': -2, 'a5': 3, 'a6': 0, 'a7': -4, 'a8': 0, 'a9': 1, 'accuracy': 100, 'algebraic': True, 'analytic_conductor': 0.40723705030853014, 'analytic_normalization': {'__RealLiteral__': 0, 'data': '0.5', 'prec': 10}, 'bad_lfactors': [[3, [1, -1]], [17, [1, 1]]], 'bad_primes': [3, 17], 'central_character': '51.1', 'coefficient_field': '1.1.1.1', 'conductor': 51, 'conductor_radical': 51, 'degree': 2, 'euler_factors': [[1, 0, 2], [1, -1], [1, -3, 5], [1, 4, 7], [1, 3, 11], [1, 1, 13], [1, 1], [1, 1, 19], [1, -9, 23], [1, -6, 29], [1, -2, 31], [1, 4, 37], [1, 3, 41], [1, 7, 43], [1, 6, 47], [1, 6, 53], [1, -6, 59], [1, -8, 61], [1, 4, 67], [1, -12, 71], [1, -2, 73], [1, 10, 79], [1, 6, 83], [1, 0, 89], [1, 16, 97], [1, 0, 101], [1, -5, 103], [1, -9, 107], [1, -20, 109], [1, 9, 113]], 'euler_factors_factorization': [[[[1, 0, 2], 1]], [[[1, -1], 1]], [[[1, -3, 5], 1]], [[[1, 4, 7], 1]], [[[1, 3, 11], 1]], [[[1, 1, 13], 1]], 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-0.9995774022308896, -0.17992118858497308, 0.3994494400738495, 0.6590979454856031, 0.5884779201789789, 0.24527103905082007, -0.2592862564626609, -0.7860833611394403], 'positive_zeros': ['3.00632648437536992753290178799801', '5.14306131647545403565477997004917', '6.66463575826783700250676263361774', '8.618966621189813626434174913815808', '9.642727340101070443548067847257934', '10.23827402727964604848387184190570', '12.84688978513418512171462235404110', '13.19069815363258335174666499644910', '14.11125275639621900792162393083813', '15.42382989853517912004077694457982'], 'prelabel': '2-51-1.1-c1-0', 'primitive': True, 'rational': True, 'root_analytic_conductor': 0.6381512754108779, 'root_angle': 0.0, 'root_number': '1', 'self_dual': True, 'sign_arg': {'__RealLiteral__': 0, 'data': '0.0', 'prec': 10}, 'spectral_label': 'c1-0', 'trace_hash': 2233362678190722081, 'types': ['CMF'], 'z1': {'__RealLiteral__': 0, 'data': '3.00632648437536992753290178800', 'prec': 103}, 'z2': {'__RealLiteral__': 0, 'data': '5.14306131647545403565477997005', 'prec': 103}, 'z3': {'__RealLiteral__': 0, 'data': '6.66463575826783700250676263362', 'prec': 103}}
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lfunc_search • Show schema
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{'algebraic': True, 'analytic_conductor': 0.40723705030853014, 'bad_primes': [3, 17], 'central_character': '1.1', 'conductor': 51, 'conductor_radical': 51, 'degree': 2, 'dirichlet_coefficients': [1, 0, 1, -2, 3, 0, -4, 0, 1, 0, -3, -2, -1, 0, 3, 4, -1, 0, -1, -6, -4, 0, 9, 0, 4, 0, 1, 8, 6, 0, 2, 0, -3, 0, -12, -2, -4, 0, -1, 0, -3, 0, -7, 6, 3, 0, -6, 4, 9, 0, -1, 2, -6, 0, -9, 0, -1, 0, 6, -6, 8, 0, -4, -8, -3, 0, -4, 2, 9, 0, 12, 0, 2, 0, 4, 2, 12, 0, -10, 12, 1, 0, -6, 8, -3, 0, 6, 0, 0, 0, 4, -18, 2, 0, -3, 0, -16, 0, -3, -8], 'euler11': [1, 3, 11], 'euler13': [1, 1, 13], 'euler17': [1, 1, 0], 'euler19': [1, 1, 19], 'euler2': [1, 0, 2], 'euler23': [1, -9, 23], 'euler29': [1, -6, 29], 'euler3': [1, -1, 0], 'euler31': [1, -2, 31], 'euler37': [1, 4, 37], 'euler41': [1, 3, 41], 'euler43': [1, 7, 43], 'euler47': [1, 6, 47], 'euler5': [1, -3, 5], 'euler53': [1, 6, 53], 'euler59': [1, -6, 59], 'euler61': [1, -8, 61], 'euler67': [1, 4, 67], 'euler7': [1, 4, 7], 'euler71': [1, -12, 71], 'euler73': [1, -2, 73], 'euler79': [1, 10, 79], 'euler83': [1, 6, 83], 'euler89': [1, 0, 89], 'euler97': [1, 16, 97], 'euler_factors': [[1, 0, 2], [1, -1, 0], [1, -3, 5], [1, 4, 7], [1, 3, 11], [1, 1, 13], [1, 1, 0], [1, 1, 19], [1, -9, 23], [1, -6, 29], [1, -2, 31], [1, 4, 37], [1, 3, 41], [1, 7, 43], [1, 6, 47], [1, 6, 53], [1, -6, 59], [1, -8, 61], [1, 4, 67], [1, -12, 71], [1, -2, 73], [1, 10, 79], [1, 6, 83], [1, 0, 89], [1, 16, 97]], 'index': 1, 'instance_types': ['ECQ', 'CMF', 'CMF'], 'instance_urls': ['EllipticCurve/Q/51/a', 'ModularForm/GL2/Q/holomorphic/51/2/a/a/1/1', 'ModularForm/GL2/Q/holomorphic/51/2/a/a'], 'is_instance_Artin': False, 'is_instance_BMF': False, 'is_instance_CMF': True, 'is_instance_DIR': False, 'is_instance_ECNF': False, 'is_instance_ECQ': True, 'is_instance_G2Q': False, 'is_instance_HMF': False, 'is_instance_MaassGL3': False, 'is_instance_MaassGL4': False, 'is_instance_MaassGSp4': False, 'is_instance_NF': False, 'label': '2-51-1.1-c1-0-1', 'motivic_weight': 1, 'mu_imag': [], 'mu_real': [], 'nu_imag': [{'__RealLiteral__': 0, 'data': '0.0', 'prec': 10}], 'nu_real_doubled': [1], 'order_of_vanishing': 0, 'prelabel': '2-51-1.1-c1-0', 'primitive': True, 'rational': True, 'root_analytic_conductor': 0.6381512754108779, 'root_angle': 0.0, 'self_dual': True, 'spectral_label': 'c1-0', 'trace_hash': 2233362678190722081, 'z1': {'__RealLiteral__': 0, 'data': '3.00632648437536992753290178800', 'prec': 103}}
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lfunc_instances • Show schema
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id: 9956568
{'Lhash': '3810971572400460408091718010207', 'Lhash_array': ['3810971572400460408091718010207'], 'label': '2-51-1.1-c1-0-1', 'type': 'CMF', 'url': 'ModularForm/GL2/Q/holomorphic/51/2/a/a'}
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id: 24546143
{'Lhash': '3810971572400460408091718010207', 'Lhash_array': ['3810971572400460408091718010207'], 'label': '2-51-1.1-c1-0-1', 'type': 'CMF', 'url': 'ModularForm/GL2/Q/holomorphic/51/2/a/a/1/1'}
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id: 21816
{'Lhash': '2233362678190722081', 'Lhash_array': ['2233362678190722081'], 'label': '2-51-1.1-c1-0-1', 'type': 'ECQ', 'url': 'EllipticCurve/Q/51/a'}