Properties

Label 3.23.b.a
Level 3
Weight 23
Character orbit 3.b
Analytic conductor 9.201
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 23 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.20122304526\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{21}\cdot 3^{22} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 14445 - 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -1876448 - 6 \beta_{2} + \beta_{4} ) q^{4} \) \( + ( 5130 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( 48929184 + 46377 \beta_{1} + 44 \beta_{2} + 9 \beta_{3} - 26 \beta_{4} - \beta_{5} ) q^{6} \) \( + ( -574510510 - 126 \beta_{1} - 3906 \beta_{2} + 301 \beta_{4} - 21 \beta_{5} ) q^{7} \) \( + ( -1029272 \beta_{1} + 11868 \beta_{2} - 130 \beta_{3} + 198 \beta_{4} - 198 \beta_{5} ) q^{8} \) \( + ( 9556619193 - 1157814 \beta_{1} + 7398 \beta_{2} - 243 \beta_{3} - 4644 \beta_{4} - 1188 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 14445 - 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -1876448 - 6 \beta_{2} + \beta_{4} ) q^{4} \) \( + ( 5130 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( 48929184 + 46377 \beta_{1} + 44 \beta_{2} + 9 \beta_{3} - 26 \beta_{4} - \beta_{5} ) q^{6} \) \( + ( -574510510 - 126 \beta_{1} - 3906 \beta_{2} + 301 \beta_{4} - 21 \beta_{5} ) q^{7} \) \( + ( -1029272 \beta_{1} + 11868 \beta_{2} - 130 \beta_{3} + 198 \beta_{4} - 198 \beta_{5} ) q^{8} \) \( + ( 9556619193 - 1157814 \beta_{1} + 7398 \beta_{2} - 243 \beta_{3} - 4644 \beta_{4} - 1188 \beta_{5} ) q^{9} \) \( + ( -31144933440 - 30960 \beta_{1} - 608100 \beta_{2} + 15350 \beta_{4} - 5160 \beta_{5} ) q^{10} \) \( + ( -28653498 \beta_{1} + 912192 \beta_{2} + 4124 \beta_{3} + 16731 \beta_{4} - 16731 \beta_{5} ) q^{11} \) \( + ( -220995496800 + 101636008 \beta_{1} - 1594082 \beta_{2} + 2430 \beta_{3} + 132219 \beta_{4} - 41094 \beta_{5} ) q^{12} \) \( + ( 337522082810 - 418104 \beta_{1} - 5813544 \beta_{2} - 192476 \beta_{4} - 69684 \beta_{5} ) q^{13} \) \( + ( -1620354134 \beta_{1} + 3457356 \beta_{2} - 67018 \beta_{3} + 54558 \beta_{4} - 54558 \beta_{5} ) q^{14} \) \( + ( 438005219040 + 3189469050 \beta_{1} + 4712812 \beta_{2} - 5652 \beta_{3} - 353275 \beta_{4} + 134035 \beta_{5} ) q^{15} \) \( + ( -1617244537856 + 3882240 \beta_{1} + 57192384 \beta_{2} + 1251936 \beta_{4} + 647040 \beta_{5} ) q^{16} \) \( + ( -8764344152 \beta_{1} - 91974624 \beta_{2} + 679840 \beta_{3} - 1569564 \beta_{4} + 1569564 \beta_{5} ) q^{17} \) \( + ( 7021191689280 + 26031677049 \beta_{1} + 85189644 \beta_{2} - 102060 \beta_{3} - 6582546 \beta_{4} + 2361636 \beta_{5} ) q^{18} \) \( + ( 16747614778106 + 12436830 \beta_{1} + 227197554 \beta_{2} - 3319509 \beta_{4} + 2072805 \beta_{5} ) q^{19} \) \( + ( -70431700880 \beta_{1} + 128772456 \beta_{2} - 4653676 \beta_{3} + 1800900 \beta_{4} - 1800900 \beta_{5} ) q^{20} \) \( + ( -131513198881302 + 30427229294 \beta_{1} - 458890138 \beta_{2} + 1093743 \beta_{3} + 44920764 \beta_{4} - 9485532 \beta_{5} ) q^{21} \) \( + ( 174315052612800 - 139725360 \beta_{1} - 2113379220 \beta_{2} - 35896130 \beta_{4} - 23287560 \beta_{5} ) q^{22} \) \( + ( 229342967284 \beta_{1} + 1650549264 \beta_{2} + 21904400 \beta_{3} + 31820994 \beta_{4} - 31820994 \beta_{5} ) q^{23} \) \( + ( -412077872212992 - 438049918584 \beta_{1} - 498617780 \beta_{2} - 4103658 \beta_{3} - 25835746 \beta_{4} - 36373118 \beta_{5} ) q^{24} \) \( + ( 232761543955465 + 24807960 \beta_{1} - 1808206200 \beta_{2} + 370278700 \beta_{4} + 4134660 \beta_{5} ) q^{25} \) \( + ( 854480769754 \beta_{1} - 2665616016 \beta_{2} - 67518472 \beta_{3} - 54834408 \beta_{4} + 54834408 \beta_{5} ) q^{26} \) \( + ( -1788609593150235 - 1328092775910 \beta_{1} + 8188414497 \beta_{2} - 5117580 \beta_{3} - 710908407 \beta_{4} + 205448967 \beta_{5} ) q^{27} \) \( + ( 7428451349748800 + 1507113216 \beta_{1} + 32341640436 \beta_{2} - 1203847806 \beta_{4} + 251185536 \beta_{5} ) q^{28} \) \( + ( 1845671669086 \beta_{1} - 24377637474 \beta_{2} + 102204507 \beta_{3} - 424364472 \beta_{4} + 424364472 \beta_{5} ) q^{29} \) \( + ( -19361520572800320 + 1519183372320 \beta_{1} - 19284827580 \beta_{2} + 126325980 \beta_{3} + 3288317850 \beta_{4} + 77788620 \beta_{5} ) q^{30} \) \( + ( 19018277066858882 + 374943690 \beta_{1} + 2540689110 \beta_{2} + 618062065 \beta_{4} + 62490615 \beta_{5} ) q^{31} \) \( + ( -8942846509312 \beta_{1} + 68176579200 \beta_{2} + 151257920 \beta_{3} + 1233645120 \beta_{4} - 1233645120 \beta_{5} ) q^{32} \) \( + ( -29141670462268320 + 10284532969734 \beta_{1} + 28228350002 \beta_{2} - 604857825 \beta_{3} - 7021833728 \beta_{4} - 1029204472 \beta_{5} ) q^{33} \) \( + ( 53169644748066048 - 19917760320 \beta_{1} - 348864513456 \beta_{2} + 2816973576 \beta_{4} - 3319626720 \beta_{5} ) q^{34} \) \( + ( -30569891520980 \beta_{1} + 48693536136 \beta_{2} - 1283764356 \beta_{3} + 731981250 \beta_{4} - 731981250 \beta_{5} ) q^{35} \) \( + ( -117931661043775200 + 22412841845904 \beta_{1} - 130588760574 \beta_{2} + 1253033388 \beta_{3} + 8199023445 \beta_{4} - 3528570492 \beta_{5} ) q^{36} \) \( + ( 113963812346776970 + 30780634104 \beta_{1} + 450865290984 \beta_{2} + 10357546236 \beta_{4} + 5130105684 \beta_{5} ) q^{37} \) \( + ( 31644572163938 \beta_{1} - 28053543852 \beta_{2} + 3184221210 \beta_{3} - 159789582 \beta_{4} + 159789582 \beta_{5} ) q^{38} \) \( + ( -173007534854498958 - 20179000337944 \beta_{1} + 384806543018 \beta_{2} + 734541372 \beta_{3} - 8449920864 \beta_{4} + 17478754992 \beta_{5} ) q^{39} \) \( + ( 296991620083169280 + 12925509120 \beta_{1} + 568980912000 \beta_{2} - 58925960000 \beta_{4} + 2154251520 \beta_{5} ) q^{40} \) \( + ( 141748686524828 \beta_{1} - 1552244904612 \beta_{2} - 2313832314 \beta_{3} - 27966569616 \beta_{4} + 27966569616 \beta_{5} ) q^{41} \) \( + ( -184785733917570240 - 274776720907878 \beta_{1} - 336192768044 \beta_{2} - 11544734880 \beta_{3} + 34496477666 \beta_{4} - 13027247576 \beta_{5} ) q^{42} \) \( + ( 45545555028294890 + 139953853566 \beta_{1} + 2316820862226 \beta_{2} + 2623893979 \beta_{4} + 23325642261 \beta_{5} ) q^{43} \) \( + ( 131746014957488 \beta_{1} + 3272565755208 \beta_{2} - 8962073084 \beta_{3} + 57478452084 \beta_{4} - 57478452084 \beta_{5} ) q^{44} \) \( + ( 538103789363240640 + 105921048556530 \beta_{1} + 330306662106 \beta_{2} + 27674284149 \beta_{3} - 108452973600 \beta_{4} + 5106026160 \beta_{5} ) q^{45} \) \( + ( -1391614486174737792 - 701071339680 \beta_{1} - 13840144493976 \beta_{2} + 359270360996 \beta_{4} - 116845223280 \beta_{5} ) q^{46} \) \( + ( -223395939914968 \beta_{1} - 602051441712 \beta_{2} + 32361621880 \beta_{3} - 7283601972 \beta_{4} + 7283601972 \beta_{5} ) q^{47} \) \( + ( 1732127992604421120 + 128867403546112 \beta_{1} - 2207439158720 \beta_{2} - 8121176640 \beta_{3} + 7687553184 \beta_{4} - 140299884864 \beta_{5} ) q^{48} \) \( + ( -1382467921536112845 + 519121227240 \beta_{1} + 11494856392824 \beta_{2} - 473805989804 \beta_{4} + 86520204540 \beta_{5} ) q^{49} \) \( + ( -999056389893975 \beta_{1} + 4417092471120 \beta_{2} - 42645402520 \beta_{3} + 74307501000 \beta_{4} - 74307501000 \beta_{5} ) q^{50} \) \( + ( 2372949865379217024 + 1652544752821416 \beta_{1} + 1634413907928 \beta_{2} - 101608188588 \beta_{3} + 721925884176 \beta_{4} + 256453854672 \beta_{5} ) q^{51} \) \( + ( -3772765993050447040 + 376197387264 \beta_{1} + 9155894644644 \beta_{2} - 480989698374 \beta_{4} + 62699564544 \beta_{5} ) q^{52} \) \( + ( -1942448441217582 \beta_{1} - 23302946997054 \beta_{2} - 21681642315 \beta_{3} - 418447086624 \beta_{4} + 418447086624 \beta_{5} ) q^{53} \) \( + ( 8063967659322937824 + 434566393371153 \beta_{1} + 3684200429520 \beta_{2} + 215380577961 \beta_{3} - 1173331763568 \beta_{4} + 134511463431 \beta_{5} ) q^{54} \) \( + ( -6812431193356240320 + 2061126124920 \beta_{1} + 32427642378600 \beta_{2} + 320743283900 \beta_{4} + 343521020820 \beta_{5} ) q^{55} \) \( + ( 5120435694901968 \beta_{1} + 1588423591608 \beta_{2} + 208980741116 \beta_{3} + 50755500684 \beta_{4} - 50755500684 \beta_{5} ) q^{56} \) \( + ( 7286898641799519090 - 397565266613614 \beta_{1} + 12787141226582 \beta_{2} - 43828511535 \beta_{3} - 944552576196 \beta_{4} - 148229825364 \beta_{5} ) q^{57} \) \( + ( -11214326357688726720 - 2858709116880 \beta_{1} - 72502705859820 \beta_{2} + 4142925651970 \beta_{4} - 476451519480 \beta_{5} ) q^{58} \) \( + ( -1633393408632738 \beta_{1} + 55429198222152 \beta_{2} - 378936008816 \beta_{3} + 949206824451 \beta_{4} - 949206824451 \beta_{5} ) q^{59} \) \( + ( -7385262750622218240 - 17062142048110800 \beta_{1} - 22798129515512 \beta_{2} - 407066678748 \beta_{3} + 2116146492500 \beta_{4} - 752838896660 \beta_{5} ) q^{60} \) \( + ( 20274875064690915482 - 9243547773480 \beta_{1} - 115478118105720 \beta_{2} - 6430168575380 \beta_{4} - 1540591295580 \beta_{5} ) q^{61} \) \( + ( 17063688080943802 \beta_{1} + 7677109232700 \beta_{2} + 2639468270 \beta_{3} + 137374036470 \beta_{4} - 137374036470 \beta_{5} ) q^{62} \) \( + ( -30749150589284280510 + 8961672357123822 \beta_{1} - 103872256020666 \beta_{2} + 359545404960 \beta_{3} + 3504810772827 \beta_{4} - 1337512172787 \beta_{5} ) q^{63} \) \( + ( 47533933649079009280 + 10712125071360 \beta_{1} + 213261682587648 \beta_{2} - 5787710788608 \beta_{4} + 1785354178560 \beta_{5} ) q^{64} \) \( + ( -17491962233392220 \beta_{1} + 8123791190004 \beta_{2} + 1292471349666 \beta_{3} + 283546773000 \beta_{4} - 283546773000 \beta_{5} ) q^{65} \) \( + ( -62440131935737005120 - 4773257665089120 \beta_{1} + 198143786203380 \beta_{2} + 314552676780 \beta_{3} + 934733411730 \beta_{4} + 4673009880540 \beta_{5} ) q^{66} \) \( + ( 12616928514362543450 + 3833494278534 \beta_{1} - 110260641720726 \beta_{2} + 29025368838271 \beta_{4} + 638915713089 \beta_{5} ) q^{67} \) \( + ( 916999386755392 \beta_{1} - 370502688353568 \beta_{2} - 1923215217680 \beta_{3} - 6822178208208 \beta_{4} + 6822178208208 \beta_{5} ) q^{68} \) \( + ( -38907670286930486976 + 74225457266626548 \beta_{1} + 129269458472140 \beta_{2} + 675502671546 \beta_{3} - 23826701863288 \beta_{4} - 284518686104 \beta_{5} ) q^{69} \) \( + ( 185601169149500039040 + 39218317961760 \beta_{1} + 928743264031800 \beta_{2} - 45850771889300 \beta_{4} + 6536386326960 \beta_{5} ) q^{70} \) \( + ( -98794912910399396 \beta_{1} + 250365650754144 \beta_{2} - 1208646032312 \beta_{3} + 4341317402862 \beta_{4} - 4341317402862 \beta_{5} ) q^{71} \) \( + ( -106643477288524953600 - 33765661439396568 \beta_{1} - 423812300590980 \beta_{2} - 3669059660850 \beta_{3} + 3385713867030 \beta_{4} + 2384236685610 \beta_{5} ) q^{72} \) \( + ( -9529746884922331150 - 65328327784224 \beta_{1} - 1290658485813984 \beta_{2} + 33642170457264 \beta_{4} - 10888054630704 \beta_{5} ) q^{73} \) \( + ( 88666307248824746 \beta_{1} + 150995297031696 \beta_{2} + 5466299337672 \beta_{3} + 3282019518888 \beta_{4} - 3282019518888 \beta_{5} ) q^{74} \) \( + ( -55662271019893056555 + 30260892562523680 \beta_{1} + 331285315634905 \beta_{2} + 828441952020 \beta_{3} + 47949249801600 \beta_{4} - 15784012678320 \beta_{5} ) q^{75} \) \( + ( -121871633778707327680 - 46304594346240 \beta_{1} - 1076159130500796 \beta_{2} + 50735981899466 \beta_{4} - 7717432391040 \beta_{5} ) q^{76} \) \( + ( 1889598851216132 \beta_{1} + 803588590532196 \beta_{2} - 451502297190 \beta_{3} + 14301421013376 \beta_{4} - 14301421013376 \beta_{5} ) q^{77} \) \( + ( 122625075661512736320 - 157824758051864982 \beta_{1} - 301364656240856 \beta_{2} + 11626025243490 \beta_{3} + 15113459017364 \beta_{4} - 4902689113394 \beta_{5} ) q^{78} \) \( + ( 64089565981887956066 - 3010854737430 \beta_{1} + 1131550243681974 \beta_{2} - 196955192662079 \beta_{4} - 501809122905 \beta_{5} ) q^{79} \) \( + ( 202747290122657280 \beta_{1} - 147434401671936 \beta_{2} - 8997711042944 \beta_{3} - 3596797641600 \beta_{4} + 3596797641600 \beta_{5} ) q^{80} \) \( + ( 518294964539348526321 - 110474773267881492 \beta_{1} - 2258954359182972 \beta_{2} - 14123243333934 \beta_{3} - 27627454270836 \beta_{4} + 21535099677396 \beta_{5} ) q^{81} \) \( + ( -861143245406166844800 + 91772178564960 \beta_{1} + 325584506722920 \beta_{2} + 200658633782180 \beta_{4} + 15295363094160 \beta_{5} ) q^{82} \) \( + ( 34003207646456242 \beta_{1} - 2675712056966352 \beta_{2} - 6367120771380 \beta_{3} - 48462763957047 \beta_{4} + 48462763957047 \beta_{5} ) q^{83} \) \( + ( 1116444314971178125632 - 160805973660209584 \beta_{1} + 6722175072501308 \beta_{2} - 7259054221188 \beta_{3} - 220447008641034 \beta_{4} + 14977621565172 \beta_{5} ) q^{84} \) \( + ( -1128763308901974001920 + 357092487781920 \beta_{1} + 5320937387272800 \beta_{2} + 105100679293200 \beta_{4} + 59515414630320 \beta_{5} ) q^{85} \) \( + ( 79369104360680914 \beta_{1} + 158778288194964 \beta_{2} + 30635346705338 \beta_{3} + 6117685150482 \beta_{4} - 6117685150482 \beta_{5} ) q^{86} \) \( + ( 832567709417874706080 + 401432750057724462 \beta_{1} + 261204269819156 \beta_{2} + 14694666079500 \beta_{3} + 103534499888551 \beta_{4} + 53765885501729 \beta_{5} ) q^{87} \) \( + ( -67361337553443563520 - 349969339169280 \beta_{1} - 4152330054549120 \beta_{2} - 280082044156480 \beta_{4} - 58328223194880 \beta_{5} ) q^{88} \) \( + ( -399911605035813772 \beta_{1} - 1072084412469972 \beta_{2} + 22064794832606 \beta_{3} - 16780279347756 \beta_{4} + 16780279347756 \beta_{5} ) q^{89} \) \( + ( -643130603516748893760 + 888345761915997360 \beta_{1} - 17868780042472980 \beta_{2} + 14894976493080 \beta_{3} + 380442779538750 \beta_{4} - 121886095623840 \beta_{5} ) q^{90} \) \( + ( -467790416916030651884 - 73876957677180 \beta_{1} - 418077682187844 \beta_{2} - 135534157627526 \beta_{4} - 12312826279530 \beta_{5} ) q^{91} \) \( + ( -1845630926541296224 \beta_{1} + 10547348962704624 \beta_{2} - 110001890907720 \beta_{3} + 176559600308184 \beta_{4} - 176559600308184 \beta_{5} ) q^{92} \) \( + ( 347870046752875342170 - 97884416544336346 \beta_{1} + 19101655092795782 \beta_{2} + 423740237355 \beta_{3} + 66475275096060 \beta_{4} - 33979978732140 \beta_{5} ) q^{93} \) \( + ( 1355954196740911573248 - 996671619984960 \beta_{1} - 17384239838735856 \beta_{2} + 128841028719976 \beta_{4} - 166111936664160 \beta_{5} ) q^{94} \) \( + ( 1207709126043265740 \beta_{1} - 1465268686532688 \beta_{2} + 13461535776848 \beta_{3} - 24723204854850 \beta_{4} + 24723204854850 \beta_{5} ) q^{95} \) \( + ( -2511503598815115706368 - 9500162146055424 \beta_{1} + 1095143052412032 \beta_{2} - 98150089894848 \beta_{3} - 306561088598208 \beta_{4} - 90324753607488 \beta_{5} ) q^{96} \) \( + ( 260871948084498324290 + 542468629927944 \beta_{1} + 4596139597346904 \beta_{2} + 740834039130916 \beta_{4} + 90411438321324 \beta_{5} ) q^{97} \) \( + ( 361457102632425107 \beta_{1} - 5149690927750992 \beta_{2} + 176493610303640 \beta_{3} - 73048736891592 \beta_{4} + 73048736891592 \beta_{5} ) q^{98} \) \( + ( -1015894616616020496960 - 1752363909710236242 \beta_{1} - 32456255795788752 \beta_{2} + 110053665984456 \beta_{3} - 1226209469975901 \beta_{4} + 233297081440461 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 86670q^{3} \) \(\mathstrut -\mathstrut 11258688q^{4} \) \(\mathstrut +\mathstrut 293575104q^{6} \) \(\mathstrut -\mathstrut 3447063060q^{7} \) \(\mathstrut +\mathstrut 57339715158q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 86670q^{3} \) \(\mathstrut -\mathstrut 11258688q^{4} \) \(\mathstrut +\mathstrut 293575104q^{6} \) \(\mathstrut -\mathstrut 3447063060q^{7} \) \(\mathstrut +\mathstrut 57339715158q^{9} \) \(\mathstrut -\mathstrut 186869600640q^{10} \) \(\mathstrut -\mathstrut 1325972980800q^{12} \) \(\mathstrut +\mathstrut 2025132496860q^{13} \) \(\mathstrut +\mathstrut 2628031314240q^{15} \) \(\mathstrut -\mathstrut 9703467227136q^{16} \) \(\mathstrut +\mathstrut 42127150135680q^{18} \) \(\mathstrut +\mathstrut 100485688668636q^{19} \) \(\mathstrut -\mathstrut 789079193287812q^{21} \) \(\mathstrut +\mathstrut 1045890315676800q^{22} \) \(\mathstrut -\mathstrut 2472467233277952q^{24} \) \(\mathstrut +\mathstrut 1396569263732790q^{25} \) \(\mathstrut -\mathstrut 10731657558901410q^{27} \) \(\mathstrut +\mathstrut 44570708098492800q^{28} \) \(\mathstrut -\mathstrut 116169123436801920q^{30} \) \(\mathstrut +\mathstrut 114109662401153292q^{31} \) \(\mathstrut -\mathstrut 174850022773609920q^{33} \) \(\mathstrut +\mathstrut 319017868488396288q^{34} \) \(\mathstrut -\mathstrut 707589966262651200q^{36} \) \(\mathstrut +\mathstrut 683782874080661820q^{37} \) \(\mathstrut -\mathstrut 1038045209126993748q^{39} \) \(\mathstrut +\mathstrut 1781949720499015680q^{40} \) \(\mathstrut -\mathstrut 1108714403505421440q^{42} \) \(\mathstrut +\mathstrut 273273330169769340q^{43} \) \(\mathstrut +\mathstrut 3228622736179443840q^{45} \) \(\mathstrut -\mathstrut 8349686917048426752q^{46} \) \(\mathstrut +\mathstrut 10392767955626526720q^{48} \) \(\mathstrut -\mathstrut 8294807529216677070q^{49} \) \(\mathstrut +\mathstrut 14237699192275302144q^{51} \) \(\mathstrut -\mathstrut 22636595958302682240q^{52} \) \(\mathstrut +\mathstrut 48383805955937626944q^{54} \) \(\mathstrut -\mathstrut 40874587160137441920q^{55} \) \(\mathstrut +\mathstrut 43721391850797114540q^{57} \) \(\mathstrut -\mathstrut 67285958146132360320q^{58} \) \(\mathstrut -\mathstrut 44311576503733309440q^{60} \) \(\mathstrut +\mathstrut 121649250388145492892q^{61} \) \(\mathstrut -\mathstrut 184494903535705683060q^{63} \) \(\mathstrut +\mathstrut 285203601894474055680q^{64} \) \(\mathstrut -\mathstrut 374640791614422030720q^{66} \) \(\mathstrut +\mathstrut 75701571086175260700q^{67} \) \(\mathstrut -\mathstrut 233446021721582921856q^{69} \) \(\mathstrut +\mathstrut 1113607014897000234240q^{70} \) \(\mathstrut -\mathstrut 639860863731149721600q^{72} \) \(\mathstrut -\mathstrut 57178481309533986900q^{73} \) \(\mathstrut -\mathstrut 333973626119358339330q^{75} \) \(\mathstrut -\mathstrut 731229802672243966080q^{76} \) \(\mathstrut +\mathstrut 735750453969076417920q^{78} \) \(\mathstrut +\mathstrut 384537395891327736396q^{79} \) \(\mathstrut +\mathstrut 3109769787236091157926q^{81} \) \(\mathstrut -\mathstrut 5166859472437001068800q^{82} \) \(\mathstrut +\mathstrut 6698665889827068753792q^{84} \) \(\mathstrut -\mathstrut 6772579853411844011520q^{85} \) \(\mathstrut +\mathstrut 4995406256507248236480q^{87} \) \(\mathstrut -\mathstrut 404168025320661381120q^{88} \) \(\mathstrut -\mathstrut 3858783621100493362560q^{90} \) \(\mathstrut -\mathstrut 2806742501496183911304q^{91} \) \(\mathstrut +\mathstrut 2087220280517252053020q^{93} \) \(\mathstrut +\mathstrut 8135725180445469439488q^{94} \) \(\mathstrut -\mathstrut 15069021592890694238208q^{96} \) \(\mathstrut +\mathstrut 1565231688506989945740q^{97} \) \(\mathstrut -\mathstrut 6095367699696122981760q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(126474\) \(x^{4}\mathstrut +\mathstrut \) \(3861674040\) \(x^{2}\mathstrut +\mathstrut \) \(9831214131200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 65 \nu^{5} + 12256 \nu^{4} + 8098250 \nu^{3} + 909321664 \nu^{2} + 227790065720 \nu + 4539979294720 \)\()/61954080\)
\(\beta_{3}\)\(=\)\((\)\( 1369 \nu^{5} + 6128 \nu^{4} + 101935546 \nu^{3} + 454660832 \nu^{2} + 309897772600 \nu + 2269989647360 \)\()/5162840\)
\(\beta_{4}\)\(=\)\((\)\( 65 \nu^{5} + 12256 \nu^{4} + 8098250 \nu^{3} + 2396219584 \nu^{2} + 227790065720 \nu + 67224621806080 \)\()/10325680\)
\(\beta_{5}\)\(=\)\((\)\( -1625 \nu^{5} + 189968 \nu^{4} - 202456250 \nu^{3} + 16324832672 \nu^{2} - 5695866816440 \nu + 164396642835200 \)\()/15488520\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut -\mathstrut \) \(6070752\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(99\) \(\beta_{5}\mathstrut +\mathstrut \) \(99\) \(\beta_{4}\mathstrut -\mathstrut \) \(65\) \(\beta_{3}\mathstrut +\mathstrut \) \(5934\) \(\beta_{2}\mathstrut -\mathstrut \) \(4708940\) \(\beta_{1}\)\()/864\)
\(\nu^{4}\)\(=\)\((\)\(6740\) \(\beta_{5}\mathstrut -\mathstrut \) \(118031\) \(\beta_{4}\mathstrut +\mathstrut \) \(1382186\) \(\beta_{2}\mathstrut +\mathstrut \) \(40440\) \(\beta_{1}\mathstrut +\mathstrut \) \(595607370912\)\()/216\)
\(\nu^{5}\)\(=\)\((\)\(3625423\) \(\beta_{5}\mathstrut -\mathstrut \) \(3625423\) \(\beta_{4}\mathstrut +\mathstrut \) \(4049125\) \(\beta_{3}\mathstrut -\mathstrut \) \(227318438\) \(\beta_{2}\mathstrut +\mathstrut \) \(167163893020\) \(\beta_{1}\)\()/432\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
281.771i
210.435i
52.8797i
52.8797i
210.435i
281.771i
3381.25i 169408. + 51788.6i −7.23853e6 6.36810e7i 1.75110e8 5.72810e8i −2.60308e9 1.02933e10i 2.60170e10 + 1.75468e10i −2.15321e11
2.2 2525.22i −174303. + 31617.4i −2.18245e6 4.89124e7i 7.98410e7 + 4.40153e8i −1.80726e7 5.08037e9i 2.93817e10 1.10220e10i 1.23515e11
2.3 634.557i 48229.8 170455.i 3.79164e6 2.56634e6i −1.08163e8 3.06045e7i 8.97619e8 5.06753e9i −2.67288e10 1.64420e10i −1.62849e9
2.4 634.557i 48229.8 + 170455.i 3.79164e6 2.56634e6i −1.08163e8 + 3.06045e7i 8.97619e8 5.06753e9i −2.67288e10 + 1.64420e10i −1.62849e9
2.5 2525.22i −174303. 31617.4i −2.18245e6 4.89124e7i 7.98410e7 4.40153e8i −1.80726e7 5.08037e9i 2.93817e10 + 1.10220e10i 1.23515e11
2.6 3381.25i 169408. 51788.6i −7.23853e6 6.36810e7i 1.75110e8 + 5.72810e8i −2.60308e9 1.02933e10i 2.60170e10 1.75468e10i −2.15321e11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{23}^{\mathrm{new}}(3, [\chi])\).