Properties

Label 3.27.b.a
Level 3
Weight 27
Character orbit 3.b
Analytic conductor 12.849
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 27 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(12.8487876219\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{33}\cdot 3^{38}\cdot 5^{3} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -157635 - 5 \beta_{1} - \beta_{2} ) q^{3} + ( -31106084 + 7 \beta_{1} - 16 \beta_{2} + \beta_{3} ) q^{4} + ( 24403 \beta_{1} - 41 \beta_{2} - \beta_{4} ) q^{5} + ( 533532636 - 568030 \beta_{1} + 87 \beta_{2} - 11 \beta_{3} + \beta_{5} ) q^{6} + ( -10465408030 + 3912 \beta_{1} - 9014 \beta_{2} + 181 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{7} + ( -39967501 \beta_{1} - 14488 \beta_{2} - 185 \beta_{3} + 372 \beta_{4} + 42 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{8} + ( 441904690665 + 27817404 \beta_{1} + 125217 \beta_{2} - 15459 \beta_{3} - 1725 \beta_{4} - 12 \beta_{5} + 15 \beta_{6} + 9 \beta_{7} ) q^{9} +O(q^{10})\) \( q +\beta_{1} q^{2} +(-157635 - 5 \beta_{1} - \beta_{2}) q^{3} +(-31106084 + 7 \beta_{1} - 16 \beta_{2} + \beta_{3}) q^{4} +(24403 \beta_{1} - 41 \beta_{2} - \beta_{4}) q^{5} +(533532636 - 568030 \beta_{1} + 87 \beta_{2} - 11 \beta_{3} + \beta_{5}) q^{6} +(-10465408030 + 3912 \beta_{1} - 9014 \beta_{2} + 181 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{7}) q^{7} +(-39967501 \beta_{1} - 14488 \beta_{2} - 185 \beta_{3} + 372 \beta_{4} + 42 \beta_{5} + \beta_{6} + 3 \beta_{7}) q^{8} +(441904690665 + 27817404 \beta_{1} + 125217 \beta_{2} - 15459 \beta_{3} - 1725 \beta_{4} - 12 \beta_{5} + 15 \beta_{6} + 9 \beta_{7}) q^{9} +(-2395047072600 + 100075 \beta_{1} - 214950 \beta_{2} + 87095 \beta_{3} - 100 \beta_{4} + 140 \beta_{5} + 135 \beta_{6} - 35 \beta_{7}) q^{10} +(984347523 \beta_{1} - 3449530 \beta_{2} - 21112 \beta_{3} + 4745 \beta_{4} - 1680 \beta_{5} + 851 \beta_{6} - 120 \beta_{7}) q^{11} +(45206518388940 + 940613470 \beta_{1} + 28370552 \beta_{2} - 1397196 \beta_{3} + 13716 \beta_{4} - 486 \beta_{5} + 4185 \beta_{6} - 405 \beta_{7}) q^{12} +(-49739419328950 - 35220562 \beta_{1} + 81850644 \beta_{2} + 2092930 \beta_{3} - 17084 \beta_{4} - 2024 \beta_{5} + 16578 \beta_{6} + 506 \beta_{7}) q^{13} +(-26509833779 \beta_{1} - 327692800 \beta_{2} - 1852993 \beta_{3} - 358780 \beta_{4} + 29946 \beta_{5} + 54497 \beta_{6} + 2139 \beta_{7}) q^{14} +(-89986725878400 - 9419299826 \beta_{1} + 7721832 \beta_{2} + 2027405 \beta_{3} + 183627 \beta_{4} + 16460 \beta_{5} + 149040 \beta_{6} + 8505 \beta_{7}) q^{15} +(1838542346916928 - 1382747888 \beta_{1} + 3189063424 \beta_{2} - 46819728 \beta_{3} - 335360 \beta_{4} + 13312 \beta_{5} + 338688 \beta_{6} - 3328 \beta_{7}) q^{16} +(-280274780452 \beta_{1} - 3028996616 \beta_{2} - 18589368 \beta_{3} + 3271484 \beta_{4} - 301392 \beta_{5} + 615228 \beta_{6} - 21528 \beta_{7}) q^{17} +(-2737611720104040 + 1595891767896 \beta_{1} - 703543914 \beta_{2} + 100942587 \beta_{3} - 9140004 \beta_{4} - 241200 \beta_{5} + 805239 \beta_{6} - 109971 \beta_{7}) q^{18} +(2040050194420250 - 1422872413 \beta_{1} + 3281416004 \beta_{2} - 49159362 \beta_{3} - 347395 \beta_{4} + 3944 \beta_{5} + 348381 \beta_{6} - 986 \beta_{7}) q^{19} +(-6895319537598 \beta_{1} + 8052355376 \beta_{2} + 51492090 \beta_{3} - 18099464 \beta_{4} + 1679580 \beta_{5} - 1800330 \beta_{6} + 119970 \beta_{7}) q^{20} +(24322605045748506 + 8528472775973 \beta_{1} + 7218730633 \beta_{2} - 946387908 \beta_{3} + 83797605 \beta_{4} + 2161728 \beta_{5} - 7114176 \beta_{6} + 970380 \beta_{7}) q^{21} +(-96531085060221240 + 69091141295 \beta_{1} - 159293374350 \beta_{2} + 2620562923 \beta_{3} + 16054060 \beta_{4} - 806564 \beta_{5} - 16255701 \beta_{6} + 201641 \beta_{7}) q^{22} +(-46088621734102 \beta_{1} + 152719605596 \beta_{2} + 898371904 \beta_{3} + 12164286 \beta_{4} - 2102016 \beta_{5} - 27834830 \beta_{6} - 150144 \beta_{7}) q^{23} +(-57792893538115008 + 115519201794607 \beta_{1} - 20847905016 \beta_{2} + 1833378995 \beta_{3} - 330355260 \beta_{4} - 12844270 \beta_{5} - 34171227 \beta_{6} - 6091281 \beta_{7}) q^{24} +(246423606756631225 + 18070884750 \beta_{1} - 43065671900 \beta_{2} - 6739474550 \beta_{3} + 24232700 \beta_{4} + 6702200 \beta_{5} - 22557150 \beta_{6} - 1675550 \beta_{7}) q^{25} +(-160974787822218 \beta_{1} - 110298954880 \beta_{2} - 826255620 \beta_{3} + 771243280 \beta_{4} - 47705112 \beta_{5} + 31251204 \beta_{6} - 3407508 \beta_{7}) q^{26} +(-986831126995349115 + 83441291496282 \beta_{1} - 401923968867 \beta_{2} + 20122324182 \beta_{3} + 616209147 \beta_{4} + 50142456 \beta_{5} + 140474547 \beta_{6} + 27215838 \beta_{7}) q^{27} +(1915218941491163000 - 1215140720322 \beta_{1} + 2803940683104 \beta_{2} - 33523857966 \beta_{3} - 327746304 \beta_{4} - 22277376 \beta_{5} + 322176960 \beta_{6} + 5569344 \beta_{7}) q^{28} +(-226769765330007 \beta_{1} - 3238860617715 \beta_{2} - 17927338160 \beta_{3} - 5300607475 \beta_{4} + 458176992 \beta_{5} + 508070776 \beta_{6} + 32726928 \beta_{7}) q^{29} +(924810074974960200 - 231190617217995 \beta_{1} + 2409619122450 \beta_{2} - 89759754855 \beta_{3} - 64284300 \beta_{4} - 112538160 \beta_{5} + 644793885 \beta_{6} - 81523665 \beta_{7}) q^{30} +(-1159064869156260238 - 34208156790 \beta_{1} + 114283466950 \beta_{2} + 186070877179 \beta_{3} - 451728445 \beta_{4} - 40938172 \beta_{5} + 441493902 \beta_{6} + 10234543 \beta_{7}) q^{31} +(3720462463360336 \beta_{1} + 2514403153280 \beta_{2} + 12141057552 \beta_{3} + 13220935360 \beta_{4} - 2223670176 \beta_{5} - 126267024 \beta_{6} - 158833584 \beta_{7}) q^{32} +(-8146075022366561280 - 4235202928889980 \beta_{1} + 1402945774683 \beta_{2} - 204725219453 \beta_{3} - 637174107 \beta_{4} + 53637052 \beta_{5} - 1603167795 \beta_{6} + 121775319 \beta_{7}) q^{33} +(27655782815188880352 + 8279364499172 \beta_{1} - 19182901107976 \beta_{2} - 186175153548 \beta_{3} + 3473666000 \beta_{4} + 800532752 \beta_{5} - 3273532812 \beta_{6} - 200133188 \beta_{7}) q^{34} +(3907607256380582 \beta_{1} + 28502020417776 \beta_{2} + 167826488360 \beta_{3} - 4572630014 \beta_{4} + 6149201520 \beta_{5} - 5944598470 \beta_{6} + 439228680 \beta_{7}) q^{35} +(-\)\(12\!\cdots\!88\)\( - 7984963477370739 \beta_{1} - 35986685813568 \beta_{2} + 1829557117407 \beta_{3} - 34208379960 \beta_{4} + 267946116 \beta_{5} - 6502069254 \beta_{6} + 179375022 \beta_{7}) q^{36} +(\)\(10\!\cdots\!30\)\( + 22800753996362 \beta_{1} - 52794273294244 \beta_{2} - 325874118378 \beta_{3} + 6269725244 \beta_{4} - 3986530552 \beta_{5} - 7266357882 \beta_{6} + 996632638 \beta_{7}) q^{37} +(6805120900904695 \beta_{1} + 2808669604032 \beta_{2} + 21009333289 \beta_{3} - 13490077348 \beta_{4} - 5174076810 \beta_{5} - 67528457 \beta_{6} - 369576915 \beta_{7}) q^{38} +(-\)\(19\!\cdots\!66\)\( + 3671137776619256 \beta_{1} + 61842404964862 \beta_{2} - 2334900501666 \beta_{3} + 264753550620 \beta_{4} + 1168579224 \beta_{5} + 5364453474 \beta_{6} - 1513297242 \beta_{7}) q^{39} +(\)\(51\!\cdots\!00\)\( - 102446361533600 \beta_{1} + 236600964032000 \beta_{2} - 1751051108960 \beta_{3} - 24818150400 \beta_{4} + 9435463680 \beta_{5} + 27177016320 \beta_{6} - 2358865920 \beta_{7}) q^{40} +(-24467108750108078 \beta_{1} - 167571735701670 \beta_{2} - 937596552128 \beta_{3} - 180080035270 \beta_{4} - 31993610880 \beta_{5} + 32942022064 \beta_{6} - 2285257920 \beta_{7}) q^{41} +(-\)\(83\!\cdots\!20\)\( + 91480136459532955 \beta_{1} + 33009592948014 \beta_{2} - 2901396744091 \beta_{3} - 739545830316 \beta_{4} - 12227890108 \beta_{5} + 65916444789 \beta_{6} + 3682631223 \beta_{7}) q^{42} +(\)\(84\!\cdots\!70\)\( - 12233926233017 \beta_{1} + 31360273005164 \beta_{2} + 16219530131554 \beta_{3} - 40355504979 \beta_{4} + 776060184 \beta_{5} + 40549520025 \beta_{6} - 194015046 \beta_{7}) q^{43} +(-277994660714740694 \beta_{1} - 417874392089360 \beta_{2} - 2740738259902 \beta_{3} + 1003122689880 \beta_{4} + 145884240012 \beta_{5} + 69040712942 \beta_{6} + 10420302858 \beta_{7}) q^{44} +(-\)\(21\!\cdots\!00\)\( + 143201642901143517 \beta_{1} + 192127585723911 \beta_{2} - 12193242539130 \beta_{3} + 261157141971 \beta_{4} + 30629325240 \beta_{5} - 46317437190 \beta_{6} - 2516283090 \beta_{7}) q^{45} +(\)\(45\!\cdots\!52\)\( - 176332234228638 \beta_{1} + 399953455766684 \beta_{2} - 41451088682518 \beta_{3} - 1254122840 \beta_{4} - 82106293688 \beta_{5} - 19272450582 \beta_{6} + 20526573422 \beta_{7}) q^{46} +(162977558759932116 \beta_{1} + 1583192560718992 \beta_{2} + 9841922195472 \beta_{3} - 1854116405668 \beta_{4} - 240676488864 \beta_{5} - 280161629028 \beta_{6} - 17191177776 \beta_{7}) q^{47} +(-\)\(83\!\cdots\!40\)\( - 123488133005337056 \beta_{1} - 1308633229974400 \beta_{2} + 86933394970560 \beta_{3} + 3069036133056 \beta_{4} + 36665714016 \beta_{5} - 157718556432 \beta_{6} - 8899978032 \beta_{7}) q^{48} +(\)\(57\!\cdots\!39\)\( + 1841045531589922 \beta_{1} - 4251495129826756 \beta_{2} + 33205492912102 \beta_{3} + 640266302340 \beta_{4} + 257534375112 \beta_{5} - 575882708562 \beta_{6} - 64383593778 \beta_{7}) q^{49} +(695695039874444925 \beta_{1} + 527324565142400 \beta_{2} + 3640171656500 \beta_{3} - 2076520603600 \beta_{4} - 120594957000 \beta_{5} - 100026920500 \beta_{6} - 8613925500 \beta_{7}) q^{50} +(-\)\(77\!\cdots\!24\)\( - 446775478055660616 \beta_{1} + 578589062211972 \beta_{2} - 85586977863300 \beta_{3} - 5878778071860 \beta_{4} - 428883478464 \beta_{5} - 671736559788 \beta_{6} + 24641989452 \beta_{7}) q^{51} +(\)\(12\!\cdots\!40\)\( - 1695839416945978 \beta_{1} + 3919179750283616 \beta_{2} - 14693459299446 \beta_{3} - 621582294016 \beta_{4} - 228170564608 \beta_{5} + 564539652864 \beta_{6} + 57042641152 \beta_{7}) q^{52} +(72883336565719471 \beta_{1} + 1703661519239139 \beta_{2} + 5275247295104 \beta_{3} + 18505039236619 \beta_{4} + 1310122313472 \beta_{5} - 313994866336 \beta_{6} + 93580165248 \beta_{7}) q^{53} +(-\)\(81\!\cdots\!64\)\( - 2560590476517809745 \beta_{1} + 3819384732219597 \beta_{2} - 152431016064498 \beta_{3} + 563397024420 \beta_{4} + 1029799603269 \beta_{5} + 2073927643785 \beta_{6} - 16830615213 \beta_{7}) q^{54} +(\)\(39\!\cdots\!00\)\( + 1581714411956250 \beta_{1} - 3609640131727300 \beta_{2} + 257031636321350 \beta_{3} - 436486480100 \beta_{4} - 788838294200 \beta_{5} + 239276906550 \beta_{6} + 197209573550 \beta_{7}) q^{55} +(3621039901968845718 \beta_{1} - 21673477286956720 \beta_{2} - 118924244604034 \beta_{3} - 34873526374360 \beta_{4} - 2129050790412 \beta_{5} + 3958752265106 \beta_{6} - 152075056458 \beta_{7}) q^{56} +(-\)\(85\!\cdots\!70\)\( - 108239450394568942 \beta_{1} - 1498946983708007 \beta_{2} + 89127906323391 \beta_{3} + 3341342546631 \beta_{4} + 43683963228 \beta_{5} - 185601204819 \beta_{6} - 6206099229 \beta_{7}) q^{57} +(\)\(22\!\cdots\!60\)\( - 20576624734271615 \beta_{1} + 47494443363678350 \beta_{2} - 498168261510299 \beta_{3} - 4216465878060 \beta_{4} + 2835565961892 \beta_{5} + 4925357368533 \beta_{6} - 708891490473 \beta_{7}) q^{58} +(-3500267332020309089 \beta_{1} + 18721056905223010 \beta_{2} + 109513220618880 \beta_{3} + 3945024648365 \beta_{4} - 170824441536 \beta_{5} - 3402841611933 \beta_{6} - 12201745824 \beta_{7}) q^{59} +(\)\(16\!\cdots\!00\)\( + 7519315702014535786 \beta_{1} - 2622568904481552 \beta_{2} + 399568538419010 \beta_{3} + 14010498208728 \beta_{4} - 6069153827380 \beta_{5} + 3641363718030 \beta_{6} - 56148758550 \beta_{7}) q^{60} +(-\)\(39\!\cdots\!38\)\( + 27759391795992130 \beta_{1} - 64361263538967700 \beta_{2} - 851237200152194 \beta_{3} + 9693903193260 \beta_{4} - 2632427787288 \beta_{5} - 10352010140082 \beta_{6} + 658106946822 \beta_{7}) q^{61} +(-14015440072744484711 \beta_{1} - 3772155021838720 \beta_{2} - 40219105795957 \beta_{3} + 70404234108500 \beta_{4} + 4131138510786 \beta_{5} + 786561198869 \beta_{6} + 295081322199 \beta_{7}) q^{62} +(\)\(82\!\cdots\!70\)\( + 9735196663869880188 \beta_{1} - 31779635048715294 \beta_{2} + 1169943350536491 \beta_{3} - 6616330235607 \beta_{4} + 13586567685060 \beta_{5} - 18973653788328 \beta_{6} + 140673284703 \beta_{7}) q^{63} +(-\)\(24\!\cdots\!72\)\( + 19017716470179584 \beta_{1} - 43522712046825472 \beta_{2} + 2438252574531840 \beta_{3} - 1514246840320 \beta_{4} - 4623269576704 \beta_{5} + 358429446144 \beta_{6} + 1155817394176 \beta_{7}) q^{64} +(-13470878652680851122 \beta_{1} + 125407735843548654 \beta_{2} + 765525661912240 \beta_{3} - 105543764192506 \beta_{4} - 1177043236320 \beta_{5} - 23788683173480 \beta_{6} - 84074516880 \beta_{7}) q^{65} +(\)\(41\!\cdots\!80\)\( + 6610818157850606337 \beta_{1} + 69960755553490890 \beta_{2} - 4723239174127947 \beta_{3} - 138772295055900 \beta_{4} - 1846998221904 \beta_{5} + 8802647573049 \beta_{6} + 319678476579 \beta_{7}) q^{66} +(-\)\(37\!\cdots\!10\)\( + 56143477099460937 \beta_{1} - 129601856009662424 \beta_{2} + 1266837235506568 \beta_{3} + 21700361153489 \beta_{4} + 13782210779360 \beta_{5} - 18254808458649 \beta_{6} - 3445552694840 \beta_{7}) q^{67} +(13788701180783452696 \beta_{1} - 152205196817527232 \beta_{2} - 944024430641480 \beta_{3} + 194438904469408 \beta_{4} - 3507159360432 \beta_{5} + 29900015974024 \beta_{6} - 250511382888 \beta_{7}) q^{68} +(\)\(35\!\cdots\!96\)\( + 17537909434056278296 \beta_{1} - 26150279682516558 \beta_{2} + 3269517942102074 \beta_{3} + 330443745688710 \beta_{4} - 47149736242072 \beta_{5} - 1470964286034 \beta_{6} - 1361299707630 \beta_{7}) q^{69} +(-\)\(38\!\cdots\!00\)\( - 192750549939442250 \beta_{1} + 445287157171810100 \beta_{2} - 2591630218584050 \beta_{3} - 59632774777800 \beta_{4} - 6694456141800 \beta_{5} + 57959160742350 \beta_{6} + 1673614035450 \beta_{7}) q^{70} +(52987082925886589086 \beta_{1} - 91681945454361620 \beta_{2} - 463351889841584 \beta_{3} - 296348311528070 \beta_{4} - 25331200920480 \beta_{5} + 17363432376622 \beta_{6} - 1809371494320 \beta_{7}) q^{71} +(\)\(60\!\cdots\!40\)\( - \)\(16\!\cdots\!61\)\( \beta_{1} + 54910552206162600 \beta_{2} - 120623552006625 \beta_{3} - 77410166620140 \beta_{4} + 85350827878650 \beta_{5} + 74561083644105 \beta_{6} + 725561257563 \beta_{7}) q^{72} +(-\)\(21\!\cdots\!70\)\( - 125633529111646752 \beta_{1} + 288464778141194784 \beta_{2} - 11044612523886192 \beta_{3} - 21433091537904 \beta_{4} - 12476859655488 \beta_{5} + 18313876624032 \beta_{6} + 3119214913872 \beta_{7}) q^{73} +(\)\(11\!\cdots\!94\)\( \beta_{1} - 356296028804392320 \beta_{2} - 1964189308832156 \beta_{3} - 637196819109520 \beta_{4} + 72976510241880 \beta_{5} + 53073322101148 \beta_{6} + 5212607874420 \beta_{7}) q^{74} +(\)\(68\!\cdots\!25\)\( - 17170361733833171675 \beta_{1} - 249529981088707225 \beta_{2} + 9298996714576350 \beta_{3} - 493882220562300 \beta_{4} - 1919250887400 \beta_{5} - 15816231013950 \beta_{6} + 2605058605350 \beta_{7}) q^{75} +(-\)\(53\!\cdots\!56\)\( + 148054372041395302 \beta_{1} - 340408825075642656 \beta_{2} + 10584488024100458 \beta_{3} + 21276538131200 \beta_{4} - 5747980178176 \beta_{5} - 22713533175744 \beta_{6} + 1436995044544 \beta_{7}) q^{76} +(-\)\(12\!\cdots\!30\)\( \beta_{1} + 531565774697489334 \beta_{2} + 2539430851943744 \beta_{3} + 2453021406707014 \beta_{4} + 12393685299072 \beta_{5} - 80768102405056 \beta_{6} + 885263235648 \beta_{7}) q^{77} +(-\)\(36\!\cdots\!00\)\( - 14789796384578905040 \beta_{1} + 25662234551252646 \beta_{2} - 5936859552925762 \beta_{3} - 216474538425264 \beta_{4} - 182360615202478 \beta_{5} - 140746801608108 \beta_{6} - 1597383387684 \beta_{7}) q^{78} +(\)\(73\!\cdots\!70\)\( + 419651169126067562 \beta_{1} - 969395305126829466 \beta_{2} + 5984907610064203 \beta_{3} + 144003073111315 \beta_{4} + 46786659912964 \beta_{5} - 132306408133074 \beta_{6} - 11696664978241 \beta_{7}) q^{79} +(\)\(27\!\cdots\!48\)\( \beta_{1} + 1471090605995256064 \beta_{2} + 8861015033523040 \beta_{3} - 519432372634496 \beta_{4} - 254378659523520 \beta_{5} - 247948434896480 \beta_{6} - 18169904251680 \beta_{7}) q^{80} +(-\)\(92\!\cdots\!19\)\( + \)\(40\!\cdots\!22\)\( \beta_{1} + 968791587245231310 \beta_{2} - 43633981478269188 \beta_{3} + 954420535018530 \beta_{4} + 194772517017120 \beta_{5} - 92484268932816 \beta_{6} - 6306685856820 \beta_{7}) q^{81} +(\)\(24\!\cdots\!40\)\( + 954744831529132130 \beta_{1} - 2203900911169134500 \beta_{2} + 21921161524039082 \beta_{3} + 291405428806440 \beta_{4} + 68960354881224 \beta_{5} - 274165340086134 \beta_{6} - 17240088720306 \beta_{7}) q^{82} +(-\)\(82\!\cdots\!59\)\( \beta_{1} - 1023831408209364918 \beta_{2} - 4460472524887768 \beta_{3} - 6711358313643653 \beta_{4} + 194666294683056 \beta_{5} + 117229094463377 \beta_{6} + 13904735334504 \beta_{7}) q^{83} +(-\)\(73\!\cdots\!76\)\( - 36854418053699886628 \beta_{1} - 1465305466204855568 \beta_{2} + 61249747228956648 \beta_{3} + 3780775590105960 \beta_{4} + 49177055085012 \beta_{5} - 172698440583294 \beta_{6} - 3809837443770 \beta_{7}) q^{84} +(\)\(45\!\cdots\!00\)\( - 914882147286621800 \beta_{1} + 2127574981693507600 \beta_{2} + 62311225759234920 \beta_{3} - 599286179318000 \beta_{4} - 348890324598560 \beta_{5} + 512063598168360 \beta_{6} + 87222581149640 \beta_{7}) q^{85} +(-\)\(26\!\cdots\!81\)\( \beta_{1} + 34787838481088960 \beta_{2} - 1469597039773847 \beta_{3} + 6625801517813340 \beta_{4} + 317535346875894 \beta_{5} + 9776910415543 \beta_{6} + 22681096205421 \beta_{7}) q^{86} +(-\)\(82\!\cdots\!00\)\( + \)\(12\!\cdots\!70\)\( \beta_{1} + 251811891614601024 \beta_{2} - 12758837245400153 \beta_{3} - 7603351913381391 \beta_{4} - 6147317050172 \beta_{5} + 996885674006328 \beta_{6} + 38275047163179 \beta_{7}) q^{87} +(\)\(20\!\cdots\!60\)\( - 2946908023875761440 \beta_{1} + 6749695382651584000 \beta_{2} - 347675592295650784 \beta_{3} - 66549547637760 \beta_{4} + 214584810012672 \beta_{5} + 120195750140928 \beta_{6} - 53646202503168 \beta_{7}) q^{88} +(-\)\(14\!\cdots\!66\)\( \beta_{1} - 6819894052555258150 \beta_{2} - 42924963032263160 \beta_{3} + 11187057421791870 \beta_{4} - 37277734209744 \beta_{5} + 1345648765393708 \beta_{6} - 2662695300696 \beta_{7}) q^{89} +(-\)\(14\!\cdots\!00\)\( - \)\(11\!\cdots\!25\)\( \beta_{1} + 202011063113068650 \beta_{2} + 120067561593003195 \beta_{3} - 3658160596389300 \beta_{4} - 493751516610060 \beta_{5} - 95922044615565 \beta_{6} - 11495337679935 \beta_{7}) q^{90} +(\)\(72\!\cdots\!32\)\( - 3315680563705370572 \beta_{1} + 7698405101329274076 \beta_{2} + 158803129671937438 \beta_{3} - 1142714027833250 \beta_{4} + 626061276199528 \beta_{5} + 1299229346883132 \beta_{6} - 156515319049882 \beta_{7}) q^{91} +(\)\(44\!\cdots\!36\)\( \beta_{1} + 4078555226552088096 \beta_{2} + 30049020817448828 \beta_{3} - 23592231898173104 \beta_{4} - 1252571416936728 \beta_{5} - 796440065135068 \beta_{6} - 89469386924052 \beta_{7}) q^{92} +(-\)\(90\!\cdots\!30\)\( + \)\(17\!\cdots\!91\)\( \beta_{1} + 1133769556434907513 \beta_{2} - 242035225765149906 \beta_{3} + 8574853651070625 \beta_{4} - 39512271600792 \beta_{5} + 713819008998702 \beta_{6} - 96468600635658 \beta_{7}) q^{93} +(-\)\(16\!\cdots\!68\)\( + 10818822860915458452 \beta_{1} - 24908120803651041896 \beta_{2} + 597705834828226276 \beta_{3} + 1740935683817360 \beta_{4} - 876966647928880 \beta_{5} - 1960177345799580 \beta_{6} + 219241661982220 \beta_{7}) q^{94} +(\)\(30\!\cdots\!14\)\( \beta_{1} + 1565743753228638212 \beta_{2} + 9449453666464640 \beta_{3} - 648942852390718 \beta_{4} - 249091666696320 \beta_{5} - 266939009662930 \beta_{6} - 17792261906880 \beta_{7}) q^{95} +(\)\(83\!\cdots\!48\)\( - \)\(71\!\cdots\!40\)\( \beta_{1} + 998807487292550016 \beta_{2} - 182625528933752496 \beta_{3} + 12289489770064320 \beta_{4} + 2546026789128672 \beta_{5} - 3092953882559184 \beta_{6} + 38840855627664 \beta_{7}) q^{96} +(\)\(10\!\cdots\!30\)\( - 1380270968723766398 \beta_{1} + 3054588790824803996 \beta_{2} - 727414828290484586 \beta_{3} + 1045271629086324 \beta_{4} - 336790210716408 \beta_{5} - 1129469181765426 \beta_{6} + 84197552679102 \beta_{7}) q^{97} +(\)\(16\!\cdots\!31\)\( \beta_{1} + 17512467485501190272 \beta_{2} + 94273479192647692 \beta_{3} + 32551674924860112 \beta_{4} + 5059516246209096 \beta_{5} - 3522017940298508 \beta_{6} + 361394017586364 \beta_{7}) q^{98} +(\)\(15\!\cdots\!40\)\( + \)\(39\!\cdots\!49\)\( \beta_{1} + 6408458057206318950 \beta_{2} + 367335309314599650 \beta_{3} + 1818706016128365 \beta_{4} - 3736686634768584 \beta_{5} - 1253107622351727 \beta_{6} + 187451750304954 \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 1261080q^{3} - 248848672q^{4} + 4268261088q^{6} - 83723264240q^{7} + 3535237525320q^{9} + O(q^{10}) \) \( 8q - 1261080q^{3} - 248848672q^{4} + 4268261088q^{6} - 83723264240q^{7} + 3535237525320q^{9} - 19160376580800q^{10} + 361652147111520q^{12} - 397915354631600q^{13} - 719893807027200q^{15} + 14708338775335424q^{16} - 21900893760832320q^{18} + 16320401555362000q^{19} + 194580840365988048q^{21} - 772248680481769920q^{22} - 462343148304920064q^{24} + 1971388854053049800q^{25} - 7894649015962792920q^{27} + 15321751531929304000q^{28} + 7398480599799681600q^{30} - 9272518953250081904q^{31} - 65168600178932490240q^{33} + \)\(22\!\cdots\!16\)\(q^{34} - \)\(10\!\cdots\!04\)\(q^{36} + \)\(87\!\cdots\!40\)\(q^{37} - \)\(15\!\cdots\!28\)\(q^{39} + \)\(41\!\cdots\!00\)\(q^{40} - \)\(67\!\cdots\!60\)\(q^{42} + \)\(67\!\cdots\!60\)\(q^{43} - \)\(16\!\cdots\!00\)\(q^{45} + \)\(36\!\cdots\!16\)\(q^{46} - \)\(66\!\cdots\!20\)\(q^{48} + \)\(45\!\cdots\!12\)\(q^{49} - \)\(62\!\cdots\!92\)\(q^{51} + \)\(99\!\cdots\!20\)\(q^{52} - \)\(65\!\cdots\!12\)\(q^{54} + \)\(31\!\cdots\!00\)\(q^{55} - \)\(68\!\cdots\!60\)\(q^{57} + \)\(17\!\cdots\!80\)\(q^{58} + \)\(13\!\cdots\!00\)\(q^{60} - \)\(31\!\cdots\!04\)\(q^{61} + \)\(65\!\cdots\!60\)\(q^{63} - \)\(19\!\cdots\!76\)\(q^{64} + \)\(33\!\cdots\!40\)\(q^{66} - \)\(30\!\cdots\!80\)\(q^{67} + \)\(28\!\cdots\!68\)\(q^{69} - \)\(30\!\cdots\!00\)\(q^{70} + \)\(48\!\cdots\!20\)\(q^{72} - \)\(17\!\cdots\!60\)\(q^{73} + \)\(55\!\cdots\!00\)\(q^{75} - \)\(42\!\cdots\!48\)\(q^{76} - \)\(29\!\cdots\!00\)\(q^{78} + \)\(58\!\cdots\!60\)\(q^{79} - \)\(73\!\cdots\!52\)\(q^{81} + \)\(19\!\cdots\!20\)\(q^{82} - \)\(58\!\cdots\!08\)\(q^{84} + \)\(36\!\cdots\!00\)\(q^{85} - \)\(66\!\cdots\!00\)\(q^{87} + \)\(16\!\cdots\!80\)\(q^{88} - \)\(11\!\cdots\!00\)\(q^{90} + \)\(57\!\cdots\!56\)\(q^{91} - \)\(72\!\cdots\!40\)\(q^{93} - \)\(12\!\cdots\!44\)\(q^{94} + \)\(66\!\cdots\!84\)\(q^{96} + \)\(87\!\cdots\!40\)\(q^{97} + \)\(12\!\cdots\!20\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 2728193 x^{6} + 2071419806976 x^{4} + 503906274711956480 x^{2} + 26754716118873014272000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-471665 \nu^{7} + 20280832 \nu^{6} - 1161523912305 \nu^{5} + 96171313767936 \nu^{4} - 686320481127321600 \nu^{3} + 104640782566978301952 \nu^{2} - 90815125969473991429120 \nu + 16685784668286991737159680\)\()/ 10515390452241776640 \)
\(\beta_{3}\)\(=\)\((\)\(-471665 \nu^{7} + 20280832 \nu^{6} - 1161523912305 \nu^{5} + 96171313767936 \nu^{4} - 686320481127321600 \nu^{3} + 199279296637154291712 \nu^{2} - 90870331769348260756480 \nu + 81233817572450902744985600\)\()/ 657211903265111040 \)
\(\beta_{4}\)\(=\)\((\)\(33155731 \nu^{7} - 831514112 \nu^{6} + 161318375210643 \nu^{5} - 3943023864485376 \nu^{4} + 226050560840163311616 \nu^{3} - 4290272085246110380032 \nu^{2} + 79431919774169303133578240 \nu - 684117171399766661223546880\)\()/ 10515390452241776640 \)
\(\beta_{5}\)\(=\)\((\)\(-285348169 \nu^{7} - 1501425874432 \nu^{6} - 1257431393410377 \nu^{5} - 3479789658634814976 \nu^{4} - 1316771913624071245824 \nu^{3} - 1743919598111293576630272 \nu^{2} - 156536114475731790394649600 \nu - 144195973498442279917312409600\)\()/ 10515390452241776640 \)
\(\beta_{6}\)\(=\)\((\)\(1134243497 \nu^{7} + 40135766528 \nu^{6} + 2819748401572521 \nu^{5} + 190323029946745344 \nu^{4} + 1709708026004582940672 \nu^{3} + 223235748434693428482048 \nu^{2} + 242868321397791732026567680 \nu + 44037365474183930793174630400\)\()/ 3505130150747258880 \)
\(\beta_{7}\)\(=\)\((\)\(-798776683 \nu^{7} + 4240177508864 \nu^{6} - 2394922098782571 \nu^{5} + 9914999902202233344 \nu^{4} - 1847989159438615044096 \nu^{3} + 5085119964463877422436352 \nu^{2} - 220530647460590778399785984 \nu + 444053741770755584702392238080\)\()/ 2103078090448355328 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 16 \beta_{2} + 7 \beta_{1} - 98214948\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} + \beta_{6} + 42 \beta_{5} + 372 \beta_{4} - 185 \beta_{3} - 14488 \beta_{2} - 174185229 \beta_{1}\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-208 \beta_{7} + 21168 \beta_{6} + 832 \beta_{5} - 20960 \beta_{4} - 15509145 \beta_{3} + 400643056 \beta_{2} - 174502127 \beta_{1} + 1069263967740228\)\()/1296\)
\(\nu^{5}\)\(=\)\((\)\(-60258747 \beta_{7} - 24668905 \beta_{6} - 843622458 \beta_{5} - 5414815892 \beta_{4} + 3862601057 \beta_{3} + 400218502488 \beta_{2} + 2310447184770517 \beta_{1}\)\()/15552\)
\(\nu^{6}\)\(=\)\((\)\(328777328 \beta_{7} - 16389815568 \beta_{6} - 1315109312 \beta_{5} + 16061038240 \beta_{4} + 8489690829747 \beta_{3} - 254935355386448 \beta_{2} + 110925592130341 \beta_{1} - 525318624435425550444\)\()/432\)
\(\nu^{7}\)\(=\)\((\)\(36368554359273 \beta_{7} + 24692188805027 \beta_{6} + 509159761029822 \beta_{5} + 2812147381080796 \beta_{4} - 2644946314083787 \beta_{3} - 313424214724866504 \beta_{2} - 1219407371621474544455 \beta_{1}\)\()/5184\)

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
1286.17i
818.442i
575.513i
269.997i
269.997i
575.513i
818.442i
1286.17i
15434.0i −1.57937e6 + 217880.i −1.71101e8 7.56364e8i 3.36276e9 + 2.43760e10i −6.43856e10 1.60501e12i 2.44692e12 6.88223e11i −1.16738e13
2.2 9821.30i 1.39737e6 767608.i −2.93491e7 1.01800e9i −7.53891e9 1.37240e10i 1.32967e11 3.70850e11i 1.36342e12 2.14527e12i −9.99808e12
2.3 6906.16i 618315. + 1.46954e6i 1.94139e7 1.04255e9i 1.01489e10 4.27018e9i −1.82788e11 5.97540e11i −1.77724e12 + 1.81728e12i 7.19999e12
2.4 3239.96i −1.06686e6 1.18477e6i 5.66115e7 1.50979e9i −3.83861e9 + 3.45659e9i 7.23448e10 4.00850e11i −2.65486e11 + 2.52796e12i 4.89166e12
2.5 3239.96i −1.06686e6 + 1.18477e6i 5.66115e7 1.50979e9i −3.83861e9 3.45659e9i 7.23448e10 4.00850e11i −2.65486e11 2.52796e12i 4.89166e12
2.6 6906.16i 618315. 1.46954e6i 1.94139e7 1.04255e9i 1.01489e10 + 4.27018e9i −1.82788e11 5.97540e11i −1.77724e12 1.81728e12i 7.19999e12
2.7 9821.30i 1.39737e6 + 767608.i −2.93491e7 1.01800e9i −7.53891e9 + 1.37240e10i 1.32967e11 3.70850e11i 1.36342e12 + 2.14527e12i −9.99808e12
2.8 15434.0i −1.57937e6 217880.i −1.71101e8 7.56364e8i 3.36276e9 2.43760e10i −6.43856e10 1.60501e12i 2.44692e12 + 6.88223e11i −1.16738e13
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
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 \(S_{27}^{\mathrm{new}}(3, [\chi])\).