Formats: - HTML - YAML - JSON - 2026-08-13T21:15:55.188719
  • lfunc_lfunctions • Show schema
    {'Lhash': '776883148486425462733061606827,2174034190595267318139208898590', 'a10': 0, 'a2': -24, 'a3': 0, 'a4': 304, 'a5': 0, 'a6': 0, 'a7': 0, 'a8': -2304, 'a9': 1311, 'accuracy': 100, 'algebraic': True, 'analytic_conductor': 127.07264021779676, 'analytic_normalization': {'__RealLiteral__': 0, 'data': '3.0', 'prec': 7}, 'bad_lfactors': [[7, [1]]], 'bad_primes': [7], 'central_character': '49.1', 'coefficient_field': '1.1.1.1', 'conductor': 2401, 'conductor_radical': 7, 'degree': 4, 'euler_factors': [[1, 24, 272, 1536, 4096], [1, 0, -1311, 0, 531441], [1, 0, 1825, 0, 244140625], [1], [1, -2958, 5730563, -5240277438, 3138428376721], [1, 0, -9418418, 0, 23298085122481], [1, 0, -39160991, 0, 582622237229761], [1, 0, -46832879, 0, 2213314919066161], [1, 11826, 331035347, 1750672423314, 21914624432020321], [1, -7956, 1205471126, -4732414341876, 353814783205469041], [1, 0, -1610771759, 0, 787662783788549761], [1, 123154, 8923179747, 315979470173986, 6582952005840035281], [1, 0, 2726428318, 0, 22563490300366186081], [1, 34828, 12945973494, 220160432270572, 39959630797262576401], [1, 0, -20618242031, 0, 116191483108948578241], [1, 121026, 47990545427, 2682463969998354, 491258904256726154641], [1, 0, -37822212479, 0, 1779197418239532716881], [1, 0, -76554740159, 0, 2654348974297586158321], [1, 537554, 253157840067, 48626265168474626, 8182718904632857144561], [1, -203844, 266588661926, -26112474275592324, 16409682740640811134241], [1, 0, -201769475231, 0, 22902048046490258711521], [1, -724462, 617386208403, -176107624201654702, 59091511031674153381441], [1, 0, -606885669938, 0, 106890007738661124410161], [1, 0, 787099021441, 0, 246990403565262140303521], [1, 0, 626523106942, 0, 693842360995438000295041], [1, 0, -368287769999, 0, 1126825030131969720661201], [1, 0, -2384008020431, 0, 1425760886846178945447841], [1, -1321086, 3437777758547, -1982593857602788014, 2252191588960823337718801], [1, 146338, 3359553924243, 245423476020250258, 2812664781782894485727281], [1, -3207012, 6735134997254, -6676844254038094308, 4334523100191686738306881]], 'euler_factors_factorization': [[[[1, 12, 64], 2]], [[[1, 0, -1311, 0, 531441], 1]], [[[1, 0, 1825, 0, 244140625], 1]], [1], [[[1, -1479, 1771561], 2]], [[[1, 0, -9418418, 0, 23298085122481], 1]], [[[1, 0, -39160991, 0, 582622237229761], 1]], [[[1, 0, -46832879, 0, 2213314919066161], 1]], [[[1, 5913, 148035889], 2]], [[[1, -3978, 594823321], 2]], [[[1, 0, -1610771759, 0, 787662783788549761], 1]], [[[1, 61577, 2565726409], 2]], [[[1, 0, 2726428318, 0, 22563490300366186081], 1]], [[[1, 17414, 6321363049], 2]], [[[1, 0, -20618242031, 0, 116191483108948578241], 1]], [[[1, 60513, 22164361129], 2]], [[[1, 0, -37822212479, 0, 1779197418239532716881], 1]], [[[1, 0, -76554740159, 0, 2654348974297586158321], 1]], [[[1, 268777, 90458382169], 2]], [[[1, -101922, 128100283921], 2]], [[[1, 0, -201769475231, 0, 22902048046490258711521], 1]], [[[1, -362231, 243087455521], 2]], [[[1, 0, -606885669938, 0, 106890007738661124410161], 1]], [[[1, 0, 787099021441, 0, 246990403565262140303521], 1]], [[[1, 0, 626523106942, 0, 693842360995438000295041], 1]], [[[1, 0, -368287769999, 0, 1126825030131969720661201], 1]], [[[1, 0, -2384008020431, 0, 1425760886846178945447841], 1]], [[[1, -660543, 1500730351849], 2]], [[[1, 73169, 1677100110841], 2]], [[[1, -1603506, 2081951752609], 2]]], 'gamma_factors': [[], [0, 0]], 'group': 'GL2', 'index': 2, 'label': '4-7e4-1.1-c6e2-0-2', 'load_key': 'CMFs-workshop', 'motivic_weight': 6, 'mu_imag': [], 'mu_real': [], 'nu_imag': [0, 0], 'nu_real_doubled': [6, 6], 'order_of_vanishing': 0, 'origin': 'ModularForm/GL2/Q/holomorphic/49/7/b/a', 'plot_delta': {'__RealLiteral__': 0, 'data': '0.0625', 'prec': 10}, 'plot_values': [0.5830073812519784, 0.574209258383336, 0.5513693595025402, 0.5130169955242903, 0.46060658582511066, 0.396468732688796, 0.32271025978994156, 0.24167342534273234, 0.15614558267284814, 0.0691237656076367, -0.01634775599237461, -0.09724476549003691, -0.1706626571207445, -0.23392973261115244, -0.284744830165547, -0.3212667523028251, -0.3422524062808256, -0.3471235344371686, -0.33609715489788905, -0.3102121939022134, -0.2714325695814125, -0.22262232630019937, -0.16760127097094565, -0.11105973166558751, -0.058546101506052384, -0.01631861180460551, 0.008752702024463805, 0.009391638292228822, -0.021890131188174896, -0.09254546851537655, -0.20970275905355656, -0.3799021587105314, -0.6087281448813829, -0.9005645451364259, -1.2581834634850222, -1.6825599274475636, -2.1724708006021425, -2.7244103475450214, -3.3322688682774446, -3.9873835906178394, -4.678376419911518, -5.391359153175781, -6.110009092893642, -6.815923718194438, -7.48899886064898, -8.10789742922603, -8.650765906169784, -9.095748171127804, -9.42203322473615, -9.610314046873533, -9.644106718794928, -9.510032272400654, -9.199291935643389, -8.707644001448655, -8.03687153969414, -7.194342176928697, -6.194274366909953, -5.056808590500907, -3.808876522663599, -2.4827762388610983, -1.1164826094651354, 0.24819990672036624, 1.5659758427126262, 2.790168718221224, 3.873695468684581, 4.771204583491309, 5.440551644036182, 5.844684818494408, 5.953367351237544, 5.7445284791890465, 5.205885950067325, 4.335534562857895, 3.143079527796791, 1.6493449334147934, -0.11334529470207391, -2.102302056808654, -4.2653053743225575, -6.5423697777713885, -8.867790856437093, -11.17216598081212, -13.385543845378333, -15.439211798705278, -17.26963064985979, -18.819469298066153, -20.04184098474067, -20.90013844468018, -21.37200539372727, -21.447701334679664, -21.133270825227953, -20.447422772888714, -19.423504605602883, -18.105170474109116, -16.547005249936728, -14.809518854815348, -12.958548103535223, -11.060287712100829, -9.179862108537513, -7.377138293287742, -5.705038993119796, -4.206757123838015, -2.9144259208670866, -1.848085133712665, -1.0152139623187246, -0.41143893187482755, -0.02119222964019151, 0.18021777306554354, 0.22487813723715172, 0.14950392133204957, -0.006996153055009345, -0.2061904244841738, -0.4126882205817417, -0.5959904700034859, -0.7325563555214183, -0.8063662751421159, -0.8099973286846373, -0.743807385057358, -0.6158098420102177, -0.43978112294609556, -0.23408025480779615, -0.019279663487653847, 0.1836065343391326, 0.3552567093051058, 0.47976132842681823, 0.5459207103241375, 0.5482914359101195, 0.4873469250062231, 0.36945342630827366, 0.2059890926582358, 0.012209817255446951, -0.19421532932041766, -0.39511449452712033, -0.5731994627304672, -0.7141579963456691, -0.8073295737330536, -0.8473345125304539, -0.8336650666759693, -0.771390260002, -0.6696896754872306, -0.5415318858728633, -0.40157860991307953, -0.2651386234212524, -0.14611783146009016, -0.055844093545434584, -0.0017315392885772593, 0.013367776754363736, -0.008610395846014193, -0.061266256184088014, -0.1345717173474539, -0.21619970922600262, -0.2929270928289241, -0.352550167241374, -0.385097062736308, -0.38460964691553723, -0.349451428341645, -0.28323638954561353, -0.19387573027604002, -0.09326972458779773, 0.004719617516502139, 0.08563611953780348, 0.13683570555468622, 0.1491736392793866, 0.11885489607909923, 0.048396751259126326, -0.052839976558981316, -0.169322217571847, -0.28033553511339065, -0.3626114724432537, -0.39229692332270083, -0.3490114443024954, -0.21763725127974454, 0.007486505237874847, 0.3222821390537362, 0.7102140126702733, 1.1444114782084605, 1.587676535079019, 1.9969574397084726, 2.3263317344481536, 2.532648259661031, 2.5804170887933298, 2.446613480869571, 2.1253116415844238, 1.629559186449508, 0.9932563348420559, 0.26884745471267074, -0.47532815099858944, -1.161215116378517, -1.7083220486275772, -2.0417194364343425, -2.101987182936167, -1.8518011030824648, -1.2849538948739907, -0.42829834337725453, 0.6541545086624366, 1.8657805659493782, 3.080929942531719, 4.155389835353449, 4.937771001237739, 5.282740594276625, 5.06711520926911, 4.201061034400448, 2.6432679702864976, 0.4054450280129155, -2.4397686813359463, -5.763532538844135, -9.384849142201057, -13.083448210944985, -16.616800220525565, -19.73634506958864, -22.213997116841366, -23.854540033431796, -24.524535850762852, -24.15387387144, -22.758787575217845, -20.42850356448833, -17.335962147160068, -13.710399159678696, -9.83386539751946, -6.005642143635763, -2.5281684602478354, 0.3275547342961235, 2.338618854960073, 3.3546550043372805, 3.307590486431723, 2.2202591940119127, 0.20198463068135772, -2.558556758382027, -5.814044419121412, -9.275403462120874, -12.64476698718584, -15.632441680373596, -17.993363306928842, -19.534538843171177, -20.14486051283355, -19.787315431099533, -18.514568516399546, -16.446413945332136, -13.768959306450046, -10.703771345132132, -7.494554574811111, -4.377775474369461, -1.5649415390370809, 0.7761691692185374, 2.5348224246863578, 3.6606694971050295, 4.16678624967374, 4.118828810043694, 3.6270549120877287, 2.8277687653963017, 1.8685494747182312, 0.8906970045942649, 0.014530136762057331, -0.6699916540532574, -1.1117424043669215, -1.2968502322545774, -1.2482593061318092, -1.015517304657795, -0.6671568060840048, -0.27765388427370624, 0.08244552546069543, 0.35617712283704045, 0.5068340800564689, 0.5205628907019864, 0.4065792632388993, 0.19305402192651072, -0.07878743151662597, -0.3617821615983904, -0.6114198568681538, -0.7911720770918891, -0.8793345164275158, -0.869320651313029, -0.771754449632203, -0.6096098238995504, -0.4154857020097982, -0.22437633169846677, -0.06878458064611573, 0.02699086594372266, 0.0502385173183595, 0.0013401375775482607, -0.10683639416381624, -0.25165295298462076, -0.40521042393119483, -0.5387747170859765, -0.6287057645637778, -0.6591835608745297, -0.6262886583647196, -0.5368274779653136, -0.4085589991216201, -0.26544522671975834, -0.13452357480318936, -0.040173527753154865, -0.0003109655467816704, -0.02285374580458665, -0.10438413212615022, -0.23053642983886347, -0.37870883181284865, -0.5214949376730335, -0.6323857032544032, -0.6892848657357715, -0.6801111359068991, -0.6032620093870148, -0.46997197238675475, -0.30053124287876964, -0.12232124461489109, 0.0365125804314083, 0.15149624987067775, 0.20703530753210153, 0.19960945600896757, 0.13856815814159404, 0.04497488430885817, -0.052620057898679316, -0.12386838964333488, -0.1430743554592816, -0.09609720469838802, 0.015657110695646227, 0.17213093783867767, 0.336623932389693, 0.45987371649197345, 0.48780262037684924, 0.37175231958908034, 0.07745890369697395, -0.40430174877102176, -1.0538374263114498, -1.819083462477957, -2.6208365545008654, -3.361273426717094, -3.936636715935699, -4.255956015866311, -4.253652135345104, -3.909923852977485, -3.2535133556904947, -2.3725693061448063, -1.4014281213200808, -0.5147057164329071, 0.09977674427829378, 0.2616067273340783, -0.17201019603522094, -1.2829078354432097, -3.0729407040438077, -5.454454587567501, -8.252291185358125, -11.220169413346351, -14.062295224128574, -16.47547814223707, -18.176576551339032, -18.956080962383144, -18.69135323994998, -17.38942108311879, -15.170390347741355, -12.280728198104036, -9.045297260385476, -5.846877125401288, -3.0621808477489014, -1.0218379494258236, 0.0437733143048563, 0.037367264190816304, -0.9841212614386711, -2.809484192213329, -5.092790210622514, -7.3963699028645, -9.237572765039676, -10.162666088198677, -9.79190632892455, -7.894554316129625, -4.402549950896186, 0.5462125479283239, 6.640112537600849, 13.401090469932686, 20.253419407817145, 26.575974068760488, 31.77818837348075, 35.36954589116427, 37.00818573534386, 36.551619953731, 34.05631592542131, 29.783736612219656, 24.153616102657992, 17.70237997758093, 11.015868029287933, 4.66071095287453, -0.8682484712729919, -5.211320058317645, -8.15541339933643, -9.66549277565994, -9.850375132597051, -8.954572337003285, -7.296784456561197, -5.235554547228291, -3.1104506833989207, -1.2066557575035632, 0.27613473683324474, 1.2383267392570807, 1.6763061532703356, 1.6692850339626193, 1.3518067426687916, 0.8826968500993122, 0.4163907795758082, 0.072741187780398, -0.07405451690442107, -0.00942040939407574, 0.23335210808895596, 0.5785344038180132, 0.9332617259999444, 1.202965806286056, 1.31423608303406, 1.2257802733883831, 0.9370166646284672, 0.48488734427645835, -0.06260879299614669, -0.6228605687882557, -1.1109244909314873, -1.459460366443097, -1.624948293350752, -1.5998437251157156, -1.4065492651735663, -1.096700775266954, -0.7360311212431361, -0.3947166284008191, -0.13250277891964393, 0.01048109129961476, 0.019844761705764, -0.09224953947058065, -0.2914308605873338, -0.5297505550572434, -0.7547959620913194, -0.921937730571355, -0.999801790693204, -0.9782500802674058, -0.8653964936028162, -0.6873025043312606, -0.4785697195237152, -0.27618583907882155, -0.11054823989900876, 0.00011152162075539605, 0.052118025470959985, 0.05629607715802942, 0.033011099208023136, 0.006790474174871729, -0.001060211275103562, 0.021998907423662006, 0.07634648913806721, 0.1499517488573714, 0.22156770435502493, 0.26667199066650304, 0.2639176552050634, 0.20245415922320317, 0.08521531998414524, -0.06912921430870568, -0.22995350683526647, -0.3609591300073481, -0.42861633088325907, -0.41255087093314036, -0.3113065549457475, -0.1482601628738116, 0.03170700232454489, 0.16701032249454373, 0.1914214201650262, 0.047632179774208465, -0.2999282417959548, -0.8528568657654579, -1.575211859648071, -2.3940245857423226, -3.2087671578363564, -3.9083814310194094, -4.386856571365397, -4.56868879777095, -4.4165719965416645, -3.950005255225248, -3.234119908679969, -2.3800228512137096, -1.5162560966323704, -0.7728955482206783, -0.25073641705411265, -0.004789687816688831, -0.030432366101639657, -0.2648040943489184, -0.5990301495872483, -0.9040212653064306, -1.0590564166484162, -0.9887033604518721, -0.6837716796261761, -0.22299839579169184, 0.2355525963889575, 0.4689056139740202, 0.22978131811720034, -0.7056817964138694, -2.4876121202461796, -5.141606146385481, -8.54294469202786, -12.404271144127627, -16.29634962544917, -19.699484705998024, -22.061318398037088, -22.89992359509738, -21.858241017811164, -18.811002480958333, -13.867629167199386, -7.42562124409137, -0.098500883765212, 7.31484419640069, 13.956067229213234, 19.003214453966795, 21.79133112947798, 21.912642905812007, 19.270904241718664, 14.113237067917327, 6.99253585299906, -1.2787407164101579, -9.752947858597045, -17.55976173044085, -23.89097152289793, -27.868548541402923, -29.094489866939448], 'positive_zeros': ['0.61285274376595131798757964621759', '1.59674486255865772407358992621083', '1.71501057957441228679450528033473', '3.80103635563573510036716314497666', '4.62122566597817528276383066700762', '6.50466424085863227894315048369670', '6.74761412627232764767027227554393', '7.44319370427491552486131129137437', '8.003749362503105504736183812476284', '8.940711179047486676003580615923109', '9.047036786990555317737329427626704', '9.809314670971383227725716319706238', '10.15672681276811133704987700446125', '10.62326965612425487298286265024548', '11.45994244414498558376313660711068', '11.96354124776651058735061953976845', '12.57223016689697799374055282622581', '13.67938125732688828552228251765801', '14.00513791113223804225368107721774', '15.04010621949871852735246434323797'], 'prelabel': '4-7e4-1.1-c6e2-0', 'primitive': False, 'rational': True, 'root_analytic_conductor': 3.357476745619052, 'root_angle': 0.0, 'root_number': '1', 'self_dual': True, 'sign_arg': 0, 'spectral_label': 'c6e2-0', 'trace_hash': 853754179688203824, 'types': ['CMF'], 'z1': {'__RealLiteral__': 0, 'data': '0.61285274376595131798757964622', 'prec': 97}, 'z2': {'__RealLiteral__': 0, 'data': '1.59674486255865772407358992621', 'prec': 100}, 'z3': {'__RealLiteral__': 0, 'data': '1.71501057957441228679450528033', 'prec': 100}}
  • lfunc_search • Show schema
    {'algebraic': True, 'analytic_conductor': 127.07264021779676, 'bad_primes': [7], 'central_character': '1.1', 'conductor': 2401, 'conductor_radical': 7, 'degree': 4, 'dirichlet_coefficients': [1, -24, 0, 304, 0, 0, 0, -2304, 1311, 0, 2958, 0, 0, 0, 0, 5376, 0, -31464, 0, 0, 0, -70992, -11826, 0, -1825, 0, 0, 0, 7956, 0, 0, 129024, 0, 0, 0, 398544, -123154, 0, 0, 0, 0, 0, -34828, 899232, 0, 283824, 0, 0, 0, 43800, 0, 0, -121026, 0, 0, 0, 0, -190944, 0, 0, 0, 0, 0, -2265088, 0, 0, -537554, 0, 0, 0, 203844, -3020544, 0, 2955696, 0, 0, 0, 0, 724462, 0, 1187280, 0, 0, 0, 0, 835872, 0, -6815232, 0, 0, 0, -3595104, 0, 0, 0, 0, 0, 0, 3877938, -554800], 'euler11': [1, -2958, 5730563, -5240277438, 3138428376721], 'euler13': [1, 0, -9418418, 0, 23298085122481], 'euler17': [1, 0, -39160991, 0, 582622237229761], 'euler19': [1, 0, -46832879, 0, 2213314919066161], 'euler2': [1, 24, 272, 1536, 4096], 'euler23': [1, 11826, 331035347, 1750672423314, 21914624432020321], 'euler29': [1, -7956, 1205471126, -4732414341876, 353814783205469041], 'euler3': [1, 0, -1311, 0, 531441], 'euler31': [1, 0, -1610771759, 0, 787662783788549761], 'euler37': [1, 123154, 8923179747, 315979470173986, 6582952005840035281], 'euler41': [1, 0, 2726428318, 0, 22563490300366186081], 'euler43': [1, 34828, 12945973494, 220160432270572, 39959630797262576401], 'euler47': [1, 0, -20618242031, 0, 116191483108948578241], 'euler5': [1, 0, 1825, 0, 244140625], 'euler53': [1, 121026, 47990545427, 2682463969998354, 491258904256726154641], 'euler59': [1, 0, -37822212479, 0, 1779197418239532716881], 'euler61': [1, 0, -76554740159, 0, 2654348974297586158321], 'euler67': [1, 537554, 253157840067, 48626265168474626, 8182718904632857144561], 'euler7': [1, 0, 0, 0, 0], 'euler71': [1, -203844, 266588661926, -26112474275592324, 16409682740640811134241], 'euler73': [1, 0, -201769475231, 0, 22902048046490258711521], 'euler79': [1, -724462, 617386208403, -176107624201654702, 59091511031674153381441], 'euler83': [1, 0, -606885669938, 0, 106890007738661124410161], 'euler89': [1, 0, 787099021441, 0, 246990403565262140303521], 'euler97': [1, 0, 626523106942, 0, 693842360995438000295041], 'euler_factors': [[1, 24, 272, 1536, 4096], [1, 0, -1311, 0, 531441], [1, 0, 1825, 0, 244140625], [1, 0, 0, 0, 0], [1, -2958, 5730563, -5240277438, 3138428376721], [1, 0, -9418418, 0, 23298085122481], [1, 0, -39160991, 0, 582622237229761], [1, 0, -46832879, 0, 2213314919066161], [1, 11826, 331035347, 1750672423314, 21914624432020321], [1, -7956, 1205471126, -4732414341876, 353814783205469041], [1, 0, -1610771759, 0, 787662783788549761], [1, 123154, 8923179747, 315979470173986, 6582952005840035281], [1, 0, 2726428318, 0, 22563490300366186081], [1, 34828, 12945973494, 220160432270572, 39959630797262576401], [1, 0, -20618242031, 0, 116191483108948578241], [1, 121026, 47990545427, 2682463969998354, 491258904256726154641], [1, 0, -37822212479, 0, 1779197418239532716881], [1, 0, -76554740159, 0, 2654348974297586158321], [1, 537554, 253157840067, 48626265168474626, 8182718904632857144561], [1, -203844, 266588661926, -26112474275592324, 16409682740640811134241], [1, 0, -201769475231, 0, 22902048046490258711521], [1, -724462, 617386208403, -176107624201654702, 59091511031674153381441], [1, 0, -606885669938, 0, 106890007738661124410161], [1, 0, 787099021441, 0, 246990403565262140303521], [1, 0, 626523106942, 0, 693842360995438000295041]], 'index': 2, 'instance_types': ['CMF'], 'instance_urls': ['ModularForm/GL2/Q/holomorphic/49/7/b/a'], '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': '4-7e4-1.1-c6e2-0-2', 'motivic_weight': 6, 'mu_imag': [], 'mu_real': [], 'nu_imag': [0, 0], 'nu_real_doubled': [6, 6], 'order_of_vanishing': 0, 'prelabel': '4-7e4-1.1-c6e2-0', 'primitive': False, 'rational': True, 'root_analytic_conductor': 3.357476745619052, 'root_angle': 0.0, 'self_dual': True, 'spectral_label': 'c6e2-0', 'trace_hash': 853754179688203824, 'z1': {'__RealLiteral__': 0, 'data': '0.61285274376595131798757964622', 'prec': 97}}
  • lfunc_instances • Show schema
    {'Lhash': '776883148486425462733061606827,2174034190595267318139208898590', 'Lhash_array': ['776883148486425462733061606827', '2174034190595267318139208898590'], 'factors': ['2-7e2-7.6-c6-0-4', '2-7e2-7.6-c6-0-11'], 'label': '4-7e4-1.1-c6e2-0-2', 'type': 'CMF', 'url': 'ModularForm/GL2/Q/holomorphic/49/7/b/a'}