Formats: - HTML - YAML - JSON - 2026-10-06T15:19:18.844928
  • lfunc_lfunctions • Show schema
    {'Lhash': '391233998882634642551575138089', 'a10': 0, 'a2': 0, 'a3': -617, 'a4': 4096, 'a5': 0, 'a6': 0, 'a7': -231577, 'a8': 0, 'a9': -150752, 'accuracy': 100, 'algebraic': True, 'analytic_conductor': 75.8614868339317, 'analytic_normalization': {'__RealLiteral__': 0, 'data': '6.0', 'prec': 7}, 'bad_lfactors': [[83, [1, -326940373369]]], 'bad_primes': [83], 'central_character': '83.82', 'coefficient_field': '1.1.1.1', 'conductor': 83, 'conductor_radical': 83, 'degree': 2, 'euler_factors': [[1, 0, -4096], [1, 617, 531441], [1, 0, -244140625], [1, 231577, 13841287201], [1, 3198553, 3138428376721], [1, 0, -23298085122481], [1, 30626737, 582622237229761], [1, 0, -2213314919066161], [1, 231011422, 21914624432020321], [1, -991523567, 353814783205469041], [1, 101624713, 787662783788549761], [1, 5012228257, 6582952005840035281], [1, 9474786718, 22563490300366186081], [1, 0, -39959630797262576401], [1, 0, -116191483108948578241], [1, 0, -491258904256726154641], [1, 48671434393, 1779197418239532716881], [1, -86434023647, 2654348974297586158321], [1, 0, -8182718904632857144561], [1, 0, -16409682740640811134241], [1, 0, -22902048046490258711521], [1, 0, -59091511031674153381441], [1, -326940373369], [1, 0, -246990403565262140303521], [1, 0, -693842360995438000295041], [1, 0, -1126825030131969720661201], [1, 0, -1425760886846178945447841], [1, 0, -2252191588960823337718801], [1, -1153705829807, 2812664781782894485727281], [1, 2028122937457, 4334523100191686738306881]], 'euler_factors_factorization': [[[[1, -64], 1], [[1, 64], 1]], [[[1, 617, 531441], 1]], [[[1, -15625], 1], [[1, 15625], 1]], [[[1, 231577, 13841287201], 1]], [[[1, 3198553, 3138428376721], 1]], [[[1, -4826809], 1], [[1, 4826809], 1]], [[[1, 30626737, 582622237229761], 1]], [[[1, -47045881], 1], [[1, 47045881], 1]], [[[1, 231011422, 21914624432020321], 1]], [[[1, -991523567, 353814783205469041], 1]], [[[1, 101624713, 787662783788549761], 1]], [[[1, 5012228257, 6582952005840035281], 1]], [[[1, 9474786718, 22563490300366186081], 1]], [[[1, -6321363049], 1], [[1, 6321363049], 1]], [[[1, -10779215329], 1], [[1, 10779215329], 1]], [[[1, -22164361129], 1], [[1, 22164361129], 1]], [[[1, 48671434393, 1779197418239532716881], 1]], [[[1, -86434023647, 2654348974297586158321], 1]], [[[1, -90458382169], 1], [[1, 90458382169], 1]], [[[1, -128100283921], 1], [[1, 128100283921], 1]], [[[1, -151334226289], 1], [[1, 151334226289], 1]], [[[1, -243087455521], 1], [[1, 243087455521], 1]], [[[1, -326940373369], 1]], [[[1, -496981290961], 1], [[1, 496981290961], 1]], [[[1, -832972004929], 1], [[1, 832972004929], 1]], [[[1, -1061520150601], 1], [[1, 1061520150601], 1]], [[[1, -1194052296529], 1], [[1, 1194052296529], 1]], [[[1, -1500730351849], 1], [[1, 1500730351849], 1]], [[[1, -1153705829807, 2812664781782894485727281], 1]], [[[1, 2028122937457, 4334523100191686738306881], 1]]], 'gamma_factors': [[], [0]], 'group': 'GL2', 'index': 18, 'label': '2-83-83.82-c12-0-18', 'leading_term': '0.404642317704947785477634125690255230928199436708192975000000000000000000000000000000000000', 'load_key': 'CMFs-workshop', 'motivic_weight': 12, 'mu_imag': [], 'mu_real': [], 'nu_imag': [0], 'nu_real_doubled': [12], 'order_of_vanishing': 0, 'origin': 'ModularForm/GL2/Q/holomorphic/83/13/b/a', 'plot_delta': {'__RealLiteral__': 0, 'data': '0.0703125', 'prec': 20}, 'plot_values': [0.4046423177049478, 0.38285996962262614, 0.31803140400947005, 0.2117004235033569, 0.06640149651183655, -0.11439720573068338, -0.3263718910042018, -0.5644407172152756, -0.8228788093350349, -1.095448302750428, -1.3755406628095612, -1.6563282513535538, -1.9309218897465918, -2.1925310093565846, -2.4346228886356425, -2.651077453862407, -2.836334170035742, -2.9855276699139606, -3.094608962041074, -3.1604493207150064, -3.180924288788574, -3.154975613136579, -3.0826493764040412, -2.965109079797278, -2.804622961484276, -2.6045253937963606, -2.3691527790349025, -2.1037549466152874, -1.8143836312139294, -1.5077601697890832, -1.1911250819128356, -0.8720726799625557, -0.5583742809362484, -0.2577939482161564, 0.022099031269922565, 0.2741165408402565, 0.49163241378498523, 0.6687531255077395, 0.8004740428523638, 0.8828171020324416, 0.9129461206399209, 0.8892563870739967, 0.81143570607399, 0.6804947028008071, 0.4987648883643214, 0.2698637525837162, -0.0013730414743440829, -0.3089914506003134, -0.6460490785466149, -1.0047728548675652, -1.3767409912377866, -1.7530851079919239, -2.12470777155181, -2.4825101331375574, -2.8176239298936006, -3.12164181817998, -3.386839869240448, -3.6063860802332663, -3.7745289451834143, -3.886760492870127, -3.939948729368749, -3.9324351143759713, -3.864093540041925, -3.736348251395744, -3.5521492265050245, -3.3159046959018466, -3.0333716944538, -2.711506771630678, -2.358280202672745, -1.9824582068915044, -1.5933587533895142, -1.2005874829225798, -0.8137610635420677, -0.4422258963518165, -0.09478046983761464, 0.2205901941764999, 0.49695966742100445, 0.7286768464936528, 0.9115469464778397, 1.04297029175016, 1.122033549650172, 1.149549372118347, 1.128041905457819, 1.0616772587952426, 0.9561397410767224, 0.8184564329256115, 0.6567743981572406, 0.48009650294118, 0.2979833410821691, 0.12023010593569666, -0.0434716489531019, -0.1838766037042613, -0.29252137476782003, -0.3620419473269565, -0.3864616995354511, -0.36143950358650495, -0.28446838771676747, -0.15501659989061892, 0.02539540372961616, 0.25318655883464625, 0.5227777062638024, 0.826724996246488, 1.155921730862683, 1.4998637594135447, 1.8469711915303173, 2.184957011337476, 2.501231246173928, 2.783327743219669, 3.0193394063617296, 3.1983470018620226, 3.310826398925627, 3.349019397880929, 3.3072541240793276, 3.182202320263096, 2.973062724730543, 2.68166202827364, 2.312467592074865, 1.8725090970962208, 1.371209484031493, 0.820128821256744, 0.2326279882877512, -0.3765378376846841, -0.9916830754062489, -1.5966977535106561, -2.1755784332697026, -2.7129654586155936, -3.194667235343114, -3.6081522275689655, -3.942990072546394, -4.191224651731816, -4.347664085315313, -4.410075379884425, -4.379274768202336, -4.259108525373239, -4.056323094896804, -3.7803275624889765, -3.4428557148165275, -3.0575389487299116, -2.6394049894415197, -2.2043205763308165, -1.7683988399699357, -1.3473939015840821, -0.9561061810266975, -0.6078219374278101, -0.3138096591942655, -0.08289407601277846, 0.07887416723055313, 0.16843867001695084, 0.18583698029104945, 0.13417429549397308, 0.01947069838405535, -0.1496133070109211, -0.3621630390374111, -0.6054928775736663, -0.8657073552642395, -1.1283173013126948, -1.3788854607675076, -1.6036742565562347, -1.7902677317901534, -1.9281402650096378, -2.0091464007010904, -2.027909039759259, -1.982087196318716, -1.872509406094825, -1.7031644799900685, -1.481047410226545, -1.2158646003304352, -0.9196089315224019, -0.6060212154828941, -0.28996004076384807, 0.01329336181101702, 0.28876506032256005, 0.5225207040155654, 0.7023667865938715, 0.8184881123835642, 0.8639907613045535, 0.835323363724356, 0.7325543343178745, 0.5594886931709155, 0.32361498315150095, 0.03588028717693656, -0.2897008672560376, -0.6365902037938521, -0.9864011697510827, -1.3196788122674386, -1.616751218742432, -1.8586155812481853, -2.027819435606851, -2.1092968680842903, -2.091120558614728, -1.965133438342076, -1.7274283944877733, -1.3786506862373415, -0.9241052917228383, -0.37365996906742116, 0.25855599174679356, 0.9546479405534861, 1.6936283900664604, 2.4522387927838656, 3.2058860290294193, 3.9296530772865723, 4.599339129859718, 5.192482228919224, 5.6893174814804235, 6.073626117435412, 6.333435026978311, 6.461532790917593, 6.455776336371999, 6.319171856841837, 6.0597240988929, 5.690059049678588, 5.226835937515057, 4.689974749162314, 4.1017346561386265, 3.4856863545312873, 2.8656269517587134, 2.2644893622276396, 1.7032989919190755, 1.200228710512051, 0.769798767964085, 0.4222615794147954, 0.16320247115413272, -0.0066230428787305345, -0.0912065993103286, -0.09896241733288769, -0.04217664009004623, 0.06374188969071902, 0.20117142442285238, 0.35130101048377127, 0.4952150338802007, 0.6149818169395392, 0.6946886270495037, 0.7213676082473843, 0.6857622326790742, 0.5828916802925409, 0.41238073282440957, 0.17853482126392667, -0.10984680581479307, -0.43991290305150593, -0.7954363580831585, -1.1576910280465027, -1.5064843317567844, -1.821291507798417, -2.082431052536905, -2.272217595468437, -2.3760285896532416, -2.383224753088164, -2.287871069701469, -2.0892150414850943, -1.7918913019207763, -1.4058360333148736, -0.9459101374291202, -0.4312459703323822, 0.11565218809614741, 0.6700097502313492, 1.2060294761864363, 1.6982200919588273], 'positive_zeros': ['0.30862920651179148533908835116404', '2.38483259894561654843019876340265', '3.23404257909401156381187209558034', '5.22344863319334557795943337861497', '6.30861469594715554447479931013451', '6.88177943401389532750221930900371', '8.464549739067672590445757732086388', '10.22750115083136142711227326599932', '10.55618045907570437946752530147949', '11.94992865096724962153336969633117'], 'prelabel': '2-83-83.82-c12-0', 'primitive': True, 'rational': True, 'root_analytic_conductor': 8.709849989174998, 'root_angle': 0.0, 'root_number': '1', 'self_dual': True, 'sign_arg': 0, 'spectral_label': 'c12-0', 'trace_hash': 2213618715861282151, 'types': ['CMF'], 'z1': {'__RealLiteral__': 0, 'data': '0.30862920651179148533908835116', 'prec': 97}, 'z2': {'__RealLiteral__': 0, 'data': '2.38483259894561654843019876340', 'prec': 100}, 'z3': {'__RealLiteral__': 0, 'data': '3.23404257909401156381187209558', 'prec': 100}}
  • lfunc_search • Show schema
    {'algebraic': True, 'analytic_conductor': 75.8614868339317, 'bad_primes': [83], 'central_character': '83.82', 'conductor': 83, 'conductor_radical': 83, 'degree': 2, 'dirichlet_coefficients': [1, 0, -617, 4096, 0, 0, -231577, 0, -150752, 0, -3198553, -2527232, 0, 0, 0, 16777216, -30626737, 0, 0, 0, 142883009, 0, -231011422, 0, 244140625, 0, 420913081, -948539392, 991523567, 0, -101624713, 0, 1973507201, 0, 0, -617480192, -5012228257, 0, 0, 0, -9474786718, 0, 0, -13101273088, 0, 0, 0, -10351542272, 39786619728, 0, 18896696729, 0, 0, 0, 0, 0, 0, 0, -48671434393, 0, 86434023647, 0, 34910695904, 68719476736, 0, 0, 0, -125447114752, 142534047374, 0, 0, 0, 0, 0, -150634765625, 0, 740711308081, 0, 0, 0, -179587577345, 0, 326940373369, 585248804864, 0, 0, -611770040839, 0, 0, 0, 0, -946222784512, 62702447921, 0, 0, 0, 0, 0, 482188261856, 1000000000000], 'euler11': [1, 3198553, 3138428376721], 'euler13': [1, 0, -23298085122481], 'euler17': [1, 30626737, 582622237229761], 'euler19': [1, 0, -2213314919066161], 'euler2': [1, 0, -4096], 'euler23': [1, 231011422, 21914624432020321], 'euler29': [1, -991523567, 353814783205469041], 'euler3': [1, 617, 531441], 'euler31': [1, 101624713, 787662783788549761], 'euler37': [1, 5012228257, 6582952005840035281], 'euler41': [1, 9474786718, 22563490300366186081], 'euler43': [1, 0, -39959630797262576401], 'euler47': [1, 0, -116191483108948578241], 'euler5': [1, 0, -244140625], 'euler53': [1, 0, -491258904256726154641], 'euler59': [1, 48671434393, 1779197418239532716881], 'euler61': [1, -86434023647, 2654348974297586158321], 'euler67': [1, 0, -8182718904632857144561], 'euler7': [1, 231577, 13841287201], 'euler71': [1, 0, -16409682740640811134241], 'euler73': [1, 0, -22902048046490258711521], 'euler79': [1, 0, -59091511031674153381441], 'euler83': [1, -326940373369, 0], 'euler89': [1, 0, -246990403565262140303521], 'euler97': [1, 0, -693842360995438000295041], 'euler_factors': [[1, 0, -4096], [1, 617, 531441], [1, 0, -244140625], [1, 231577, 13841287201], [1, 3198553, 3138428376721], [1, 0, -23298085122481], [1, 30626737, 582622237229761], [1, 0, -2213314919066161], [1, 231011422, 21914624432020321], [1, -991523567, 353814783205469041], [1, 101624713, 787662783788549761], [1, 5012228257, 6582952005840035281], [1, 9474786718, 22563490300366186081], [1, 0, -39959630797262576401], [1, 0, -116191483108948578241], [1, 0, -491258904256726154641], [1, 48671434393, 1779197418239532716881], [1, -86434023647, 2654348974297586158321], [1, 0, -8182718904632857144561], [1, 0, -16409682740640811134241], [1, 0, -22902048046490258711521], [1, 0, -59091511031674153381441], [1, -326940373369, 0], [1, 0, -246990403565262140303521], [1, 0, -693842360995438000295041]], 'index': 18, 'instance_types': ['CMF', 'CMF'], 'instance_urls': ['ModularForm/GL2/Q/holomorphic/83/13/b/a', 'ModularForm/GL2/Q/holomorphic/83/13/b/a/82/1'], 'is_instance_Artin': False, 'is_instance_BMF': False, 'is_instance_CMF': True, 'is_instance_DIR': False, 'is_instance_ECNF': False, 'is_instance_ECQ': False, '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': '2-83-83.82-c12-0-18', 'motivic_weight': 12, 'mu_imag': [], 'mu_real': [], 'nu_imag': [0], 'nu_real_doubled': [12], 'order_of_vanishing': 0, 'prelabel': '2-83-83.82-c12-0', 'primitive': True, 'rational': True, 'root_analytic_conductor': 8.709849989174998, 'root_angle': 0.0, 'self_dual': True, 'spectral_label': 'c12-0', 'trace_hash': 2213618715861282151, 'z1': {'__RealLiteral__': 0, 'data': '0.30862920651179148533908835116', 'prec': 97}}
  • lfunc_instances • Show schema
    • id: 24070964
      {'Lhash': '391233998882634642551575138089', 'Lhash_array': ['391233998882634642551575138089'], 'factors': ['2-83-83.82-c12-0-18'], 'label': '2-83-83.82-c12-0-18', 'type': 'CMF', 'url': 'ModularForm/GL2/Q/holomorphic/83/13/b/a'}
    • id: 24588895
      {'Lhash': '391233998882634642551575138089', 'Lhash_array': ['391233998882634642551575138089'], 'factors': ['2-83-83.82-c12-0-18'], 'label': '2-83-83.82-c12-0-18', 'type': 'CMF', 'url': 'ModularForm/GL2/Q/holomorphic/83/13/b/a/82/1'}