# Stored data for newform 81.8.c.j, downloaded from the LMFDB on 11 May 2026.
{"Nk2": 5184, "analytic_conductor": 25.303187064213862, "analytic_rank": 0, "analytic_rank_proved": true, "char_conductor": 9, "char_degree": 2, "char_is_minimal": false, "char_is_real": false, "char_orbit_index": 3, "char_orbit_label": "c", "char_order": 3, "char_parity": 1, "char_values": [81, 3, [2], [1]], "cm_discs": [], "conrey_index": 28, "dim": 8, "field_disc": 94958855267852766096, "field_disc_factorization": [[2, 4], [3, 4], [223, 2], [1213837, 2]], "field_poly": [40804, 0, 85327, 0, 13863, 0, 229, 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, [17438976, -12026880, 7216992, -868320, 113940, -1890, 483, -15, 1]]], "hecke_orbit": 10, "hecke_orbit_code": 40532534219505745, "hecke_ring_generator_nbound": 5, "hecke_ring_index": 104976, "hecke_ring_index_factorization": [[2, 4], [3, 8]], "hecke_ring_index_proved": true, "inner_twist_count": 2, "inner_twists": [[1, 1, 1, 1, 1, 1, 1], [1, 1, 9, 3, 1, 3, 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": "81.8.c.j", "level": 81, "level_is_powerful": true, "level_is_prime": false, "level_is_prime_power": true, "level_is_prime_square": false, "level_is_square": true, "level_is_squarefree": false, "level_primes": [3], "level_radical": 3, "minimal_twist": "81.8.a.a", "prim_orbit_index": 3, "qexp_display": "q+(4-4\\beta _{1}+\\beta _{2})q^{2}+(-59\\beta _{1}+5\\beta _{3}+\\cdots)q^{4}+\\cdots", "related_objects": [], "relative_dim": 4, "rm_discs": [], "sato_tate_group": "7.2.3.c3", "self_twist_discs": [], "self_twist_type": 0, "space_label": "81.8.c", "trace_display": [15, 0, 192, -800], "trace_hash": 1664673194306869175, "trace_moments": [{"__RealLiteral__": 0, "data": "0.062", "prec": 17}, {"__RealLiteral__": 0, "data": "7.383", "prec": 17}, {"__RealLiteral__": 0, "data": "3.686", "prec": 17}, {"__RealLiteral__": 0, "data": "217.005", "prec": 24}, {"__RealLiteral__": 0, "data": "334.471", "prec": 24}, {"__RealLiteral__": 0, "data": "11758.319", "prec": 30}], "trace_zratio": {"__RealLiteral__": 0, "data": "0.000", "prec": 17}, "traces": [0, 8, 15, 0, -229, 192, 0, -800, -5010, 0, 10938, -5016, 0, 2200, 19452, 0, -40849, 39240, 0, 22480, 96951, 0, 41280, 154560, 0, -164656, 794190, 0, 727720, 118680, 0, -450464, 952065, 0, -664101, 1815216, 0, 1423600, -202740, 0, -1962723, 1027776, 0, -1259000, 7685088, 0, 3943776, 561840, 0, -1370940, 3258420, 0, 303205, 7717680, 0, -851424, 1932180, 0, -1352145, 341472, 0, -2271896, -3142680, 0, 3922946, -517764, 0, 1649560, 1494135, 0, -413508, -13169376, 0, 716200, -5278983, 0, 13960972, -15209160, 0, -1436336, -46652406, 0, -39940980, 1394160, 0, 4525776, -6164088, 0, 36138720, -29871864, 0, -18489616, -479760, 0, 19330980, -38927136, 0, -4456880, -31542030, 0, -20507272, 6680256, 0, -1808240, -14685033, 0, 42087438, -35938080, 0, -20626832, -58484592, 0, -5355860, -20098500, 0, -21279432, -1697862, 0, -6581760, 35001792, 0, -52713868, 89530905, 0, -41533412, 83948352, 0, 125092960, -17273055, 0, 20794389, -21944184, 0, -56352760, 230292840, 0, 49512582, 102245100, 0, -31192880, 198945564, 0, 5953140, 79164480, 0, 262761816, 4850013, 0, 18845605, -51235584, 0, -136338128, 420382680, 0, -197797344, 126185136, 0, -181564880, 296407200, 0, -90207819, 29136048, 0, 257300800, -217140846, 0, 90665988, -65979600, 0, -7447200, -25314174, 0, 81247120, 103315560, 0, -122226464, -119779968, 0, 177870555, -160090848, 0, 105405424, -223632300, 0, -79543488, -312941172, 0, -978840, -204044520, 0, -867951768, 107994912, 0, 230916460, 130302606, 0, 410054115, -538867440, 0, 287636992, -69814356, 0, 305080350, -355328520, 0, 121530408, -488306328, 0, -1146066530, 284945592, 0, -247821800, -440439930, 0, 142424892, -811081344, 0, -326896000, -546974475, 0, 1030680912, -453801240, 0, 150471280, -1016548776, 0, 473087874, 181118760, 0, -4203320, 209834496, 0, -297581055, -226257480, 0, -299440032, -110336640, 0, -393262740, -75430320, 0, -391297868, -727001670, 0, 299783158, 70716600, 0, 193687280, 629626740, 0, 599776701, 135632304, 0, 756757200, 397494624, 0, -206166145, 276348180, 0, -754540504, 2112256878, 0, -635993160, -590880720, 0, -354275856, 51756864, 0, -54291380, 2109849984, 0, 2912128, 733664475, 0, -404319369, 644535168, 0, 831917800, 2829297720, 0, 4506451896, 140810676, 0, -504101840, 405245412, 0, -2137866288, 1915139040, 0, -1152848512, 863067987, 0, -2266669295, 1400940480, 0, -548619600, 2014732626, 0, -1770728190, -1128297000, 0, 1170590888, 136842600, 0, 1538602348, 1465518696, 0, 2624699680, -773393280, 0, -1459804260, 609885984, 0, -1097737100, -2392903770, 0, 3370523200, 129402600, 0, -802485984, -21749685, 0, -1210868400, -872829360, 0, -4569603968, 1923499848, 0, 2849001990, 407351568, 0, 2317273768, -1746592680, 0, 5397957480, -2109276192, 0, 1524647920, -2020189500, 0, -2260602045, 1246024128, 0, -4196845600, 890788920, 0, -1210081197, 696505080, 0, 1204666624, -4495566624, 0, -7635093120, 578007240, 0, 3270837384, -4764723273, 0, 3092851260, -6247420416, 0, 991874904, 2901281250, 0, 1003064660, 742513992, 0, -219680000, -7663248960, 0, -4412149482, -3931156800, 0, 856162840, -3932485632, 0, -26927268, 785834040, 0, -4453490720, -1371627012, 0, 2321558700, 1321258800, 0, 6294977160, -1390752330, 0, 9582425620, -1363700712, 0, -657694368, -3230850015, 0, -3629231181, -8143384080, 0, -7854279680, 668200620, 0, -4102581388, 338252340, 0, -397071760, -142202316, 0, -575637144, -1436461440, 0, 3255036844, -4269902958, 0, 3586301500, 4315299360, 0, 9465288672, -1364643873, 0, -4804843920, 4079407584, 0, 835412464, -233235600, 0, -8173120644, 3101322960, 0, -5003383480, 1950582060, 0, -2992444968, 1140285600, 0, -3362033000, 1363531056, 0, 6996006577, 1013723880, 0, -2359455920, 20740124736, 0, 6164115750, -2339438160, 0, 2410946064, 5908153944, 0, -7130495060, 13859560224, 0, 9866418816, 4308137355, 0, -9264238140, 2633507760, 0, 1704431980, 5357090910, 0, 8246343840, 5647842600, 0, -165238160, 6180799479, 0, 1569379203, 2018816640, 0, 7480482032, -21407436, 0, 18667680, -27132360, 0, -6316220288, 3392689608, 0, -486290040, -6849881184, 0, -2954861380, 9001458165, 0, -9940827217, 10027992624, 0, 261473920, 2185900215, 0, 4584889755, 1864236792, 0, 1992783240, -6300518664, 0, 15432268072, 11455235640, 0, -7863244376, 563952489, 0, -1211850360, -10010496960, 0, -9501513072, -8353671696, 0, -7849895840, -1920092568, 0, -6964488016, 11587963710, 0, -3780666078, -3395209296, 0, 6675383760, 1488300180, 0, 14954154051, 5902026624, 0, 22580492560, -10457339676, 0, 13196182236, -4444149600, 0, 1684724804, 12047301396, 0, -28934710240, 10516600680, 0, -5131867104, -6805569324, 0, 8639224155, -34336905840, 0, -18907179104, 1087626720, 0, 1158252885, -19368446916, 0, -4434844400, -32219464710, 0, -23368245888, -8049150384, 0, -1675651480, -605631966, 0, 15207375604, 15615270000, 0, 14309112224, 5105697180, 0, 12099832755, -16393582320, 0, 5036053920, -13164098712, 0, -7307551800, 9201270804, 0, -3855959744, 10146278640, 0, 11364504648, -31777126080, 0, -24602161880, -13293346440, 0, 4311620319, -12845877360, 0, -12125239920, -43977246006, 0, 2753836482, -3460199880, 0, 7969676192, 1690992288, 0, 14649074545, 5238724680, 0, 9055004976, -23673422301, 0, 5237190240, -8140456608, 0, -3989915540, 7763362080, 0, -3932643152, 27558317496, 0, -9409710560, -6145615980, 0, -1006317867, -29369778144, 0, 4209781840, -14644347612, 0, 20950832256, -8213325900, 0, -9050270624, -15506185368, 0, -24704847600, -15721549920, 0, 6751715156, 19629185961, 0, -21844365575, 12428557776, 0, 16239020128, 31480869840, 0, -3593962677, 7214548344, 0, 15699282720, 21090529440, 0, 15695091030, 18690349020, 0, -11808666320, 20795278848, 0, 4823237052, 11972193120, 0, 12720527328, 731236332, 0, -23601866840, -124514520, 0, -21645185232, 59983700844, 0, -12068596320, -17738009688, 0, 3568900792, 3558589860, 0, -14660833668, 76916287056, 0, 25058979120, 3316063260, 0, 15949441932, 1949626848, 0, 301254940, 13693593252, 0, 31689881112, 40011185400, 0, -37445524720, 24244157601, 0, 10521292800, -41133978720, 0, -7133478432, -12349025256, 0, -7905628760, 18787002624, 0, 643432312, 71968696530, 0, 67437460488, -26006382192, 0, 30594742560, 13810577610, 0, -5614592240, 62797556880, 0, 56540726240, -27882473664, 0, -50100473550, 7381752240, 0, -17482869944, 26268000408, 0, -108058022010, 17910015360, 0, -23181333816, -2282208660, 0, -13740836400, -52588841184, 0, 3490607648, 29644321995, 0, -29007940022, 1256989056, 0, -7270583120, 18380476200, 0, 72807303822, -26314139448, 0, -13445545640, -11134010616, 0, -31817743200, 53022879120, 0, 3471766432, 4828730505, 0, 43301158200, 21368405280, 0, 61042861068, -2008645764, 0, 5849244000, -10562244480, 0, 15376793728, -28518561660, 0, -571285239, -48692950464, 0, -88938824240, -5918037900, 0, 6603835044, 7569291324, 0, -16736091640, -95137916088, 0, -23170590888, 2402683440, 0, 21350113420, -54203235696, 0, 49869831865, -3738090240, 0, 9573468928, 36340276434, 0, 31516791630, 24590440608, 0, 30886990056, -5011938720, 0, 48840497490, 28484152404, 0, 21321295240, -24282304425, 0, -58026447984, -75030606336, 0, -115794711160, -18742429455, 0, 16511209084, -16785147840, 0, 25290520560, -63727089048, 0, -23762353110, -77263625400, 0, 61829987736, -47556099948, 0, 18005075490, -50142349512, 0, 37473428608, 57187482060, 0, -57614501616, 20995547280, 0, 26234411240, -122277872850, 0, -183272898036, -1727856408, 0, -15194359040, -32393585100, 0, 8513927520, 14784650640, 0, -4548302480, 37844737812, 0, 50541563425, 6103377540, 0, 17619529848, -152527211520, 0, 148897838640, -79747150224, 0, 23594880704, -72436152435, 0, -83577324200, 45626274144, 0, -65739553760, -63309293010, 0, 11102684868, 4399715256, 0, 58451334040, -128758811304, 0, 44427724344, 47777549460, 0, 21954393328, -26820213096, 0, 42366235140, -40737107520, 0, -81147690600, 20543859309, 0, -79368507280, -4574042088, 0, -70418128144, 232176061770, 0, -28705528992, -31524289800, 0, 18325099840, 35919504660, 0, 83717104512, 49274328384, 0, 4470468640, -55962715383, 0, -36964790940, 32716955280, 0, -20859471304, 169482793986, 0, 151018712080, -3900313680, 0, -28004014080, 17373969876, 0, 3926612610, -83385481632, 0, -50888815056, 160631765280, 0, -37133223591, 72552173856, 0, 34916226400, -57949681800, 0, -14090850024, -34963678704, 0, 18492248400, -21150732543, 0, -49439645411, 229890873840, 0, 60439508224, 96995890368, 0, -30859226490, 6663990600, 0, -28122872960, 124615381296, 0, 36680071950, -78881897460, 0, -50899370520, 26065403055, 0, -97347898296, 145099573344, 0, 22087531240, -44972034240, 0, 12465441156, 25887725400, 0, -18415975440, -23953121088, 0, 69579620256, 41503317720, 0, -35560912960, 19486372944, 0, -1649097180, 92157017400, 0, 55355214864, 5995947756, 0, 10394577780, 87398939424, 0, 33747637692, 173826518370, 0, 276264932110, 51438711624, 0, 62288619520, 150017855205, 0, -41979346986, -67184492688, 0, -31964212960, -108453338772, 0, -109920876143, -25196957520, 0, -139878344952, 117687007170, 0, -112868655360, -17967506880, 0, -17292671676, -9256748229, 0, -91604889100, -73386231312, 0, 59526189088, 112407668820, 0, -51544112256, -5559763152, 0, 45924069520, -70102587120, 0, 8567140602], "weight": 8, "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, "nf_label": 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], [4, -4, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -59, 0, 5, 1, 0, 0, -1], [0, 50, 0, -7, -1, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [-207, 207, -23, 0, 0, -3, -3, 1], [-653, 0, -87, 87, 11, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1412, 0, 145, -145, -20, -6, 0, 0], [-1277, 1277, -25, 0, 0, -5, -5, 31], [0, 0, 0, 0, 0, 0, 0, 0], [0, 684, 0, -571, 19, 0, -3, -19], [0, 5012, 0, -570, -8, 0, -10, 8], [0, 0, 0, 0, 0, 0, 0, 0], [-10633, 10633, -1635, 0, 0, 30, 30, 39], [4641, 0, -1076, 1076, -20, -20, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2663, 0, -345, 345, 159, 75, 0, 0], [25169, -25169, 3303, 0, 0, 64, 64, -179], [0, 0, 0, 0, 0, 0, 0, 0], [0, 11972, 0, -6298, -176, 0, 42, 176], [0, 39077, 0, -2073, 145, 0, 35, -145], [0, 0, 0, 0, 0, 0, 0, 0], [-41984, 41984, -3908, 0, 0, -84, -84, -356], [99148, 0, -1425, 1425, -436, 50, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [92204, 0, 4476, -4476, -420, -360, 0, 0], [29956, -29956, 1405, 0, 0, -315, -315, -27], [0, 0, 0, 0, 0, 0, 0, 0], [0, -112644, 0, -160, 256, 0, -240, -256], [0, 239649, 0, -5131, -671, 0, -58, 671], [0, 0, 0, 0, 0, 0, 0, 0], [-165132, 165132, 5325, 0, 0, -120, -120, 816], [224513, 0, -5143, 5143, 2289, 165, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [174968, 0, -11703, 11703, 465, 705, 0, 0], [-55684, 55684, -16966, 0, 0, 618, 618, 1824], [0, 0, 0, 0, 0, 0, 0, 0], [0, -501775, 0, 40973, 1369, 0, 666, -1369], [0, 247592, 0, 34450, 1374, 0, 210, -1374], [0, 0, 0, 0, 0, 0, 0, 0], [-310887, 310887, 15757, 0, 0, 1017, 1017, 661], [971936, 0, 39000, -39000, -3680, -1160, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [497996, 0, 17050, -17050, -1448, 150, 0, 0], [133414, -133414, -31734, 0, 0, 338, 338, -1606], [0, 0, 0, 0, 0, 0, 0, 0], [0, -344275, 0, 16730, -4970, 0, -630, 4970], [0, 814816, 0, 2572, -1184, 0, -1048, 1184], [0, 0, 0, 0, 0, 0, 0, 0], [100579, -100579, 93621, 0, 0, -1056, -1056, -3273], [975114, 0, 39816, -39816, 72, 1944, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-130249, 0, -81149, 81149, 5707, -2721, 0, 0], [502516, -502516, 64460, 0, 0, -3560, -3560, -8492], [0, 0, 0, 0, 0, 0, 0, 0], [0, -343492, 0, 15593, 3772, 0, -1314, -3772], [0, 83246, 0, 6170, 374, 0, 1570, -374], [0, 0, 0, 0, 0, 0, 0, 0], [-595812, 595812, -118565, 0, 0, -2925, -2925, -5069], [-428000, 0, -136020, 136020, 3056, 1472, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [528049, 0, 140475, -140475, -3759, 2730, 0, 0], [-140333, 140333, -26498, 0, 0, 4398, 4398, 10734], [0, 0, 0, 0, 0, 0, 0, 0], [0, 449097, 0, -161353, 5137, 0, 4251, -5137], [0, 411465, 0, -152855, -1291, 0, 3712, 1291], [0, 0, 0, 0, 0, 0, 0, 0], [-141052, 141052, -115874, 0, 0, 5238, 5238, 20032], [-1635771, 0, 46765, -46765, -1787, -8735, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [100397, 0, 378, -378, -20790, 1530, 0, 0], [-1294156, 1294156, 137465, 0, 0, 3750, 3750, 19428], [0, 0, 0, 0, 0, 0, 0, 0], [0, 3563300, 0, -264980, -11884, 0, -3480, 11884], [0, -3816250, 0, 64330, 2598, 0, -13686, -2598], [0, 0, 0, 0, 0, 0, 0, 0], [-465723, 465723, -425735, 0, 0, 4845, 4845, 2833], [-5932451, 0, -368641, 368641, 21613, 8266, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-5014648, 0, -7562, 7562, 41224, 1908, 0, 0], [319126, -319126, -140486, 0, 0, -11454, -11454, -17142], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1110438, 0, 38625, 22935, 0, -471, -22935], [0, -1461988, 0, -330870, 4672, 0, 5390, -4672], [0, 0, 0, 0, 0, 0, 0, 0], [9228208, -9228208, 687392, 0, 0, -6624, -6624, -46672], [-3629643, 0, 332240, -332240, -41200, 2720, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2377969, 0, -275065, 275065, -9121, -10245, 0, 0], [21680, -21680, 477800, 0, 0, 2184, 2184, -43248], [0, 0, 0, 0, 0, 0, 0, 0], [0, 4754072, 0, 365480, -24464, 0, -1860, 24464], [0, -9826721, 0, 420769, -32633, 0, 24245, 32633], [0, 0, 0, 0, 0, 0, 0, 0], [-1172628, 1172628, -154082, 0, 0, -20898, -20898, 29326], [-3832980, 0, 429155, -429155, -8680, -7420, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2405052, 0, 556828, -556828, -42764, -8928, 0, 0]]}