Properties

Label 3.43.b.b
Level 3
Weight 43
Character orbit 3.b
Analytic conductor 33.518
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 43 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(33.5183121516\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{73}\cdot 3^{104}\cdot 5^{6}\cdot 7^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1415600433 + 546 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -2916625296824 + 22 \beta_{1} + 78 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 17753465 \beta_{1} - 319 \beta_{2} - \beta_{4} ) q^{5} \) \( + ( -3991721810167224 + 288634312 \beta_{1} + 42450 \beta_{2} + 915 \beta_{3} + 15 \beta_{4} - \beta_{7} ) q^{6} \) \( + ( -3484551832576486 + 2088211 \beta_{1} + 7269609 \beta_{2} - 43776 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{7} \) \( + ( -2097629597340 \beta_{1} - 89312098 \beta_{2} + 29109 \beta_{3} + 2655 \beta_{4} - \beta_{5} - \beta_{6} + 42 \beta_{7} - 22 \beta_{8} + \beta_{9} ) q^{8} \) \( + ( -873291365178333255 - 5328568034312 \beta_{1} + 1688583242 \beta_{2} - 8490197 \beta_{3} + 39173 \beta_{4} - 116 \beta_{5} - 54 \beta_{6} - 414 \beta_{7} - 843 \beta_{8} - 4 \beta_{10} + \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(1415600433 + 546 \beta_{1} + \beta_{2}) q^{3}\) \(+(-2916625296824 + 22 \beta_{1} + 78 \beta_{2} + \beta_{3}) q^{4}\) \(+(17753465 \beta_{1} - 319 \beta_{2} - \beta_{4}) q^{5}\) \(+(-3991721810167224 + 288634312 \beta_{1} + 42450 \beta_{2} + 915 \beta_{3} + 15 \beta_{4} - \beta_{7}) q^{6}\) \(+(-3484551832576486 + 2088211 \beta_{1} + 7269609 \beta_{2} - 43776 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8}) q^{7}\) \(+(-2097629597340 \beta_{1} - 89312098 \beta_{2} + 29109 \beta_{3} + 2655 \beta_{4} - \beta_{5} - \beta_{6} + 42 \beta_{7} - 22 \beta_{8} + \beta_{9}) q^{8}\) \(+(-873291365178333255 - 5328568034312 \beta_{1} + 1688583242 \beta_{2} - 8490197 \beta_{3} + 39173 \beta_{4} - 116 \beta_{5} - 54 \beta_{6} - 414 \beta_{7} - 843 \beta_{8} - 4 \beta_{10} + \beta_{11}) q^{9}\) \(+(-\)\(12\!\cdots\!20\)\( + 14608593575 \beta_{1} + 51188616772 \beta_{2} + 37551180 \beta_{3} - 1860 \beta_{4} + 4975 \beta_{5} + 5579 \beta_{6} - 2839 \beta_{7} - 12690 \beta_{8} + 124 \beta_{10}) q^{10}\) \(+(49145482890366 \beta_{1} - 36128316342 \beta_{2} + 11832000 \beta_{3} + 1777442 \beta_{4} + 718 \beta_{5} + 1028 \beta_{6} - 34864 \beta_{7} - 5251 \beta_{8} - 160 \beta_{9} - 118 \beta_{10} + 54 \beta_{11}) q^{11}\) \(+(\)\(41\!\cdots\!32\)\( - 4830418052192023 \beta_{1} - 2905704983004 \beta_{2} + 3055972278 \beta_{3} + 14445120 \beta_{4} - 100092 \beta_{5} - 109872 \beta_{6} + 9576 \beta_{7} + 562415 \beta_{8} + 1701 \beta_{9} - 3219 \beta_{10} + 240 \beta_{11}) q^{12}\) \(+(\)\(86\!\cdots\!66\)\( - 2526457929944 \beta_{1} - 8823211090070 \beta_{2} + 24046553596 \beta_{3} - 46410 \beta_{4} + 606762 \beta_{5} + 108628 \beta_{6} + 350636 \beta_{7} + 2210260 \beta_{8} + 3094 \beta_{10}) q^{13}\) \(+(141145674671305672 \beta_{1} - 3112073983104 \beta_{2} + 1036904046 \beta_{3} + 362622467 \beta_{4} + 332590 \beta_{5} + 458430 \beta_{6} - 15897773 \beta_{7} + 419175 \beta_{8} - 90040 \beta_{9} - 46607 \beta_{10} + 9504 \beta_{11}) q^{14}\) \(+(-\)\(37\!\cdots\!80\)\( - 711494493088674775 \beta_{1} + 27421953522771 \beta_{2} + 206979473730 \beta_{3} + 1278463602 \beta_{4} - 697680 \beta_{5} + 17006031 \beta_{6} - 1645736 \beta_{7} - 26898840 \beta_{8} + 349920 \beta_{9} + 9666 \beta_{10} - 270 \beta_{11}) q^{15}\) \(+(\)\(25\!\cdots\!68\)\( + 80301707457474 \beta_{1} + 278369136082832 \beta_{2} - 2906624515408 \beta_{3} + 29453550 \beta_{4} + 25494686 \beta_{5} - 35721242 \beta_{6} + 254478412 \beta_{7} - 116096802 \beta_{8} - 1963570 \beta_{10}) q^{16}\) \(+(2999828807829131742 \beta_{1} - 92808351668512 \beta_{2} + 33989956902 \beta_{3} + 44034003830 \beta_{4} + 33207366 \beta_{5} + 49411316 \beta_{6} - 1645733072 \beta_{7} + 85523798 \beta_{8} + 392064 \beta_{9} - 5810510 \beta_{10} - 608850 \beta_{11}) q^{17}\) \(+(\)\(38\!\cdots\!80\)\( + 26874861070845791232 \beta_{1} - 5692943629404324 \beta_{2} - 3701134197936 \beta_{3} - 49165477740 \beta_{4} - 575354205 \beta_{5} + 45945711 \beta_{6} - 1481093019 \beta_{7} + 1289549538 \beta_{8} - 9815256 \beta_{9} - 37233648 \beta_{10} - 470880 \beta_{11}) q^{18}\) \(+(\)\(14\!\cdots\!10\)\( - 2280928514406054 \beta_{1} - 8002416954343434 \beta_{2} - 16239260875162 \beta_{3} + 1204695000 \beta_{4} + 1754614052 \beta_{5} + 1507894226 \beta_{6} + 17058291504 \beta_{7} - 591566159 \beta_{8} - 80313000 \beta_{10}) q^{19}\) \(+(-\)\(24\!\cdots\!20\)\( \beta_{1} - 44256449096397116 \beta_{2} + 14738733093510 \beta_{3} + 4662929167306 \beta_{4} + 1589429490 \beta_{5} + 2421302290 \beta_{6} - 79575203500 \beta_{7} - 4320653420 \beta_{8} + 74561790 \beta_{9} - 303211240 \beta_{10} + 15910560 \beta_{11}) q^{20}\) \(+(\)\(75\!\cdots\!06\)\( + 32250409945400429501 \beta_{1} - 14957234553433661 \beta_{2} - 285937516645782 \beta_{3} - 4437776049117 \beta_{4} + 22040320362 \beta_{5} + 8715456648 \beta_{6} - 40124081520 \beta_{7} - 21172313194 \beta_{8} + 16951680 \beta_{9} - 375671706 \beta_{10} + 17658906 \beta_{11}) q^{21}\) \(+(-\)\(35\!\cdots\!80\)\( - 91913186467338464 \beta_{1} - 320888981262294868 \beta_{2} + 979959195343980 \beta_{3} + 14555209695 \beta_{4} - 40443157858 \beta_{5} + 784835326 \beta_{6} + 109589651683 \beta_{7} + 78750644769 \beta_{8} - 970347313 \beta_{10}) q^{22}\) \(+(\)\(41\!\cdots\!80\)\( \beta_{1} + 554393358432307124 \beta_{2} - 175129490524740 \beta_{3} + 35829571861200 \beta_{4} - 6217967392 \beta_{5} - 10389795392 \beta_{6} + 325215631744 \beta_{7} + 92437992418 \beta_{8} - 1923045248 \beta_{9} + 1537040800 \beta_{10} - 237630240 \beta_{11}) q^{23}\) \(+(\)\(17\!\cdots\!72\)\( - \)\(20\!\cdots\!86\)\( \beta_{1} - 159909710332630938 \beta_{2} - 9630226823186535 \beta_{3} - 162490343688231 \beta_{4} - 201098916567 \beta_{5} - 204852429135 \beta_{6} + 740516422462 \beta_{7} + 119584371744 \beta_{8} + 2734307685 \beta_{9} + 5170159422 \beta_{10} - 338798592 \beta_{11}) q^{24}\) \(+(-\)\(12\!\cdots\!55\)\( + 2791816737682726780 \beta_{1} + 9796480686096222418 \beta_{2} + 21594431356302140 \beta_{3} - 447873153990 \beta_{4} + 340659941950 \beta_{5} - 58222990144 \beta_{6} - 4344092209036 \beta_{7} - 1400375586840 \beta_{8} + 29858210266 \beta_{10}) q^{25}\) \(+(-\)\(65\!\cdots\!46\)\( \beta_{1} + 3392647578755941792 \beta_{2} - 1078984853057592 \beta_{3} + 114887943257932 \beta_{4} - 277204291416 \beta_{5} - 402152553816 \beta_{6} + 13574551343964 \beta_{7} - 204381565908 \beta_{8} + 21414453840 \beta_{9} + 45070389780 \beta_{10} + 2144211264 \beta_{11}) q^{26}\) \(+(\)\(38\!\cdots\!17\)\( + \)\(88\!\cdots\!70\)\( \beta_{1} - 9480791645327101995 \beta_{2} - 83653348628991564 \beta_{3} - 236834423499330 \beta_{4} - 145679788818 \beta_{5} + 1876064393286 \beta_{6} - 1842114015216 \beta_{7} + 3527439353757 \beta_{8} - 51555603168 \beta_{9} + 87807729510 \beta_{10} + 4125190410 \beta_{11}) q^{27}\) \(+(-\)\(10\!\cdots\!24\)\( - 22264979322167538906 \beta_{1} - 77520194937092061652 \beta_{2} + 456442104441368146 \beta_{3} + 1273542668790 \beta_{4} - 1273024251482 \beta_{5} - 3578327902770 \beta_{6} + 4560455164636 \beta_{7} + 25579098502822 \beta_{8} - 84902844586 \beta_{10}) q^{28}\) \(+(\)\(46\!\cdots\!91\)\( \beta_{1} + 68468336538188979883 \beta_{2} - 22442604097895268 \beta_{3} - 3107019848106607 \beta_{4} + 1781225699996 \beta_{5} + 2606047287496 \beta_{6} - 87558370409504 \beta_{7} + 20218037266588 \beta_{8} - 98644874240 \beta_{9} - 298004510060 \beta_{10} - 9535827924 \beta_{11}) q^{29}\) \(+(\)\(52\!\cdots\!40\)\( - \)\(79\!\cdots\!80\)\( \beta_{1} - \)\(17\!\cdots\!64\)\( \beta_{2} - 2449025933307036240 \beta_{3} + 14687377480165605 \beta_{4} + 7926570304770 \beta_{5} - 12822703015518 \beta_{6} - 7945768351647 \beta_{7} + 26361574781335 \beta_{8} + 456210056520 \beta_{9} - 1180116129063 \beta_{10} - 33352746720 \beta_{11}) q^{30}\) \(+(-\)\(68\!\cdots\!58\)\( + 60871895192575971253 \beta_{1} + \)\(21\!\cdots\!67\)\( \beta_{2} + 1054436753268964644 \beta_{3} + 17575875056520 \beta_{4} - 560039824666 \beta_{5} + 62826850794809 \beta_{6} + 304367587341456 \beta_{7} - 77985939892568 \beta_{8} - 1171725003768 \beta_{10}) q^{31}\) \(+(\)\(31\!\cdots\!88\)\( \beta_{1} + \)\(12\!\cdots\!84\)\( \beta_{2} - 403172216917834896 \beta_{3} - 17507276207019440 \beta_{4} + 11594715317328 \beta_{5} + 17115028098128 \beta_{6} - 572238083668256 \beta_{7} + 297053644497632 \beta_{8} - 379949605968 \beta_{9} - 1998199485440 \beta_{10} - 27973416960 \beta_{11}) q^{32}\) \(+(\)\(36\!\cdots\!00\)\( + \)\(46\!\cdots\!50\)\( \beta_{1} - \)\(22\!\cdots\!46\)\( \beta_{2} - 1689459054693231153 \beta_{3} + 19462929129440025 \beta_{4} + 26758541839374 \beta_{5} + 83368777685274 \beta_{6} - 172041290229010 \beta_{7} + 697408560905517 \beta_{8} - 1858644309888 \beta_{9} + 1446970507074 \beta_{10} + 164370387105 \beta_{11}) q^{33}\) \(+(-\)\(21\!\cdots\!88\)\( - \)\(35\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!80\)\( \beta_{2} + 5708245931542005712 \beta_{3} - 41974148407740 \beta_{4} - 2691146645144 \beta_{5} - 429275277331096 \beta_{6} - 1289378015066284 \beta_{7} + 3070090861069308 \beta_{8} + 2798276560516 \beta_{10}) q^{34}\) \(+(-\)\(14\!\cdots\!10\)\( \beta_{1} + \)\(15\!\cdots\!24\)\( \beta_{2} - 4919569077808324110 \beta_{3} + 91618641767467426 \beta_{4} - 127464087577490 \beta_{5} - 185644514196540 \beta_{6} + 6252805275893200 \beta_{7} + 2752430377097820 \beta_{8} + 8612459531360 \beta_{9} + 21006834960490 \beta_{10} + 802269013590 \beta_{11}) q^{35}\) \(+(-\)\(20\!\cdots\!88\)\( + \)\(34\!\cdots\!10\)\( \beta_{1} + \)\(56\!\cdots\!98\)\( \beta_{2} + 47961970676163974707 \beta_{3} - 611404142043693334 \beta_{4} - 746534163295070 \beta_{5} - 237212762126094 \beta_{6} + 2006575814259252 \beta_{7} + 6793149734962320 \beta_{8} - 3384463703310 \beta_{9} + 36910918353740 \beta_{10} - 184865736608 \beta_{11}) q^{36}\) \(+(-\)\(84\!\cdots\!46\)\( - \)\(43\!\cdots\!32\)\( \beta_{1} - \)\(15\!\cdots\!18\)\( \beta_{2} - \)\(15\!\cdots\!56\)\( \beta_{3} - 1147224466397430 \beta_{4} + 709751122075526 \beta_{5} + 1495376796337844 \beta_{6} - 8001184842502572 \beta_{7} + 716982118977796 \beta_{8} + 76481631093162 \beta_{10}) q^{37}\) \(+(\)\(21\!\cdots\!20\)\( \beta_{1} + \)\(11\!\cdots\!20\)\( \beta_{2} - 35720019454590976758 \beta_{3} + 655992249758586245 \beta_{4} + 18815454459770 \beta_{5} - 3333002533430 \beta_{6} - 457502869511555 \beta_{7} + 23044732513447177 \beta_{8} - 55425842246480 \beta_{9} + 8693350952335 \beta_{10} - 6379538806080 \beta_{11}) q^{38}\) \(+(-\)\(91\!\cdots\!66\)\( + \)\(13\!\cdots\!92\)\( \beta_{1} + \)\(20\!\cdots\!44\)\( \beta_{2} + 74493381445152508254 \beta_{3} - 138279185353359726 \beta_{4} + 2808888635366970 \beta_{5} - 2225764643587758 \beta_{6} + 23735362655160 \beta_{7} + 29720431147222400 \beta_{8} + 92051893795680 \beta_{9} - 154830277502862 \beta_{10} - 4635952773582 \beta_{11}) q^{39}\) \(+(\)\(11\!\cdots\!40\)\( - \)\(15\!\cdots\!80\)\( \beta_{1} - \)\(55\!\cdots\!24\)\( \beta_{2} - \)\(23\!\cdots\!80\)\( \beta_{3} + 8177009552764020 \beta_{4} - 9133363299643500 \beta_{5} - 4360382447321148 \beta_{6} + 65551369243907208 \beta_{7} + 111962799804010260 \beta_{8} - 545133970184268 \beta_{10}) q^{40}\) \(+(-\)\(13\!\cdots\!46\)\( \beta_{1} + \)\(31\!\cdots\!42\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} + 476003503180218458 \beta_{4} + 3332201172163952 \beta_{5} + 5057904007928992 \beta_{6} - 166555980273214976 \beta_{7} + 76656839077038016 \beta_{8} + 129094866532480 \beta_{9} - 628248907297712 \beta_{10} + 25743641595696 \beta_{11}) q^{41}\) \(+(-\)\(23\!\cdots\!20\)\( + \)\(17\!\cdots\!51\)\( \beta_{1} - \)\(64\!\cdots\!20\)\( \beta_{2} + \)\(26\!\cdots\!16\)\( \beta_{3} + 13144016868568046400 \beta_{4} + 13816671975758169 \beta_{5} + 26966015547089085 \beta_{6} - 84698075923889069 \beta_{7} + 123912361933517430 \beta_{8} - 567108152514144 \beta_{9} - 283300016232384 \beta_{10} + 47152845256320 \beta_{11}) q^{42}\) \(+(-\)\(59\!\cdots\!74\)\( + \)\(60\!\cdots\!94\)\( \beta_{1} + \)\(21\!\cdots\!90\)\( \beta_{2} - \)\(23\!\cdots\!14\)\( \beta_{3} - 12193538678218920 \beta_{4} + 53354696834508748 \beta_{5} + 29749695595994814 \beta_{6} - 11520006491334608 \beta_{7} - 63224307731705243 \beta_{8} + 812902578547928 \beta_{10}) q^{43}\) \(+(-\)\(32\!\cdots\!88\)\( \beta_{1} + \)\(64\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!38\)\( \beta_{3} - 29567179455041808750 \beta_{4} - 8827127314312166 \beta_{5} - 12897065471201606 \beta_{6} + 433558333876235172 \beta_{7} + 108127354968439012 \beta_{8} + 595102322771990 \beta_{9} + 1476126757369272 \beta_{10} - 7432084769760 \beta_{11}) q^{44}\) \(+(\)\(14\!\cdots\!20\)\( + \)\(28\!\cdots\!75\)\( \beta_{1} - \)\(33\!\cdots\!33\)\( \beta_{2} - \)\(67\!\cdots\!30\)\( \beta_{3} - 5689115433474608349 \beta_{4} - 169679754214655580 \beta_{5} - 128622873454392204 \beta_{6} + 444246798965078844 \beta_{7} + 119783329023618450 \beta_{8} + 1709201398677120 \beta_{9} + 1760905462774716 \beta_{10} - 261691869188070 \beta_{11}) q^{45}\) \(+(-\)\(30\!\cdots\!68\)\( + \)\(23\!\cdots\!84\)\( \beta_{1} + \)\(82\!\cdots\!48\)\( \beta_{2} + \)\(11\!\cdots\!12\)\( \beta_{3} - 56062444523463030 \beta_{4} - 130000631173932724 \beta_{5} - 180788853269335700 \beta_{6} - 1063415271851927030 \beta_{7} + 182978200270120518 \beta_{8} + 3737496301564202 \beta_{10}) q^{46}\) \(+(-\)\(81\!\cdots\!56\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2} + \)\(55\!\cdots\!28\)\( \beta_{3} - 8679787591094778220 \beta_{4} - 36035964299863188 \beta_{5} - 57395269793864088 \beta_{6} + 1842347824281923616 \beta_{7} - 477030004569362160 \beta_{8} - 6410669871719232 \beta_{9} + 7807744971718020 \beta_{10} - 623968609369860 \beta_{11}) q^{47}\) \(+(\)\(32\!\cdots\!68\)\( + \)\(30\!\cdots\!14\)\( \beta_{1} + \)\(10\!\cdots\!64\)\( \beta_{2} - \)\(31\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!30\)\( \beta_{4} + 673799503480140582 \beta_{5} + 357396143585259150 \beta_{6} - 612253356951803940 \beta_{7} - 2348878615077689994 \beta_{8} - 1313665274101968 \beta_{9} + 11670882528203334 \beta_{10} + 951006028032000 \beta_{11}) q^{48}\) \(+(\)\(41\!\cdots\!19\)\( + \)\(40\!\cdots\!84\)\( \beta_{1} + \)\(14\!\cdots\!82\)\( \beta_{2} + \)\(14\!\cdots\!92\)\( \beta_{3} + 110076481076468550 \beta_{4} - 216227351991727454 \beta_{5} + 495545759869914768 \beta_{6} + 1897644274728081292 \beta_{7} - 3138772688537094392 \beta_{8} - 7338432071764570 \beta_{10}) q^{49}\) \(+(-\)\(20\!\cdots\!75\)\( \beta_{1} - \)\(33\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + \)\(74\!\cdots\!20\)\( \beta_{4} + 180009721152002200 \beta_{5} + 284670755155986200 \beta_{6} - 9169642344018215500 \beta_{7} - 6318782614976091100 \beta_{8} + 26114659347405200 \beta_{9} - 38408513000529700 \beta_{10} + 4501326471220800 \beta_{11}) q^{50}\) \(+(-\)\(22\!\cdots\!84\)\( + \)\(18\!\cdots\!06\)\( \beta_{1} - \)\(69\!\cdots\!84\)\( \beta_{2} + \)\(22\!\cdots\!42\)\( \beta_{3} - 94002296260392111402 \beta_{4} - 934031550832143102 \beta_{5} - 903938823629348976 \beta_{6} - 1467526056154210752 \beta_{7} - 5432192599246997886 \beta_{8} - 5497337127276000 \beta_{9} - 83960277410664738 \beta_{10} - 2129028921168534 \beta_{11}) q^{51}\) \(+(\)\(51\!\cdots\!84\)\( + \)\(10\!\cdots\!12\)\( \beta_{1} + \)\(38\!\cdots\!12\)\( \beta_{2} - \)\(11\!\cdots\!86\)\( \beta_{3} + 690657464104372440 \beta_{4} + 2442328404944616344 \beta_{5} + 222789436399712312 \beta_{6} + 9932613411608639856 \beta_{7} - 11179486927752459368 \beta_{8} - 46043830940291496 \beta_{10}) q^{52}\) \(+(-\)\(29\!\cdots\!03\)\( \beta_{1} - \)\(43\!\cdots\!83\)\( \beta_{2} + \)\(13\!\cdots\!36\)\( \beta_{3} - \)\(13\!\cdots\!65\)\( \beta_{4} + 174573546472649824 \beta_{5} + 223340934823913024 \beta_{6} - 8110215537000890368 \beta_{7} - 8416008518164080640 \beta_{8} - 53104198053048064 \beta_{9} - 15846292228276960 \beta_{10} - 17586106070937120 \beta_{11}) q^{53}\) \(+(-\)\(64\!\cdots\!44\)\( + \)\(69\!\cdots\!72\)\( \beta_{1} + \)\(22\!\cdots\!22\)\( \beta_{2} + \)\(34\!\cdots\!83\)\( \beta_{3} + \)\(15\!\cdots\!04\)\( \beta_{4} - 2647603414831269798 \beta_{5} + 2967248678245760826 \beta_{6} + 810046758653354160 \beta_{7} - 11531052288052183989 \beta_{8} - 21203161983090000 \beta_{9} + 95906698780215141 \beta_{10} + 1540745827017408 \beta_{11}) q^{54}\) \(+(\)\(61\!\cdots\!40\)\( + \)\(14\!\cdots\!10\)\( \beta_{1} + \)\(51\!\cdots\!86\)\( \beta_{2} - \)\(33\!\cdots\!20\)\( \beta_{3} - 723865535185059480 \beta_{4} - 6883128641028711100 \beta_{5} - 5185122038541225638 \beta_{6} - 24636801176998769072 \beta_{7} - 12375628883899455630 \beta_{8} + 48257702345670632 \beta_{10}) q^{55}\) \(+(-\)\(19\!\cdots\!04\)\( \beta_{1} + \)\(16\!\cdots\!68\)\( \beta_{2} - \)\(58\!\cdots\!42\)\( \beta_{3} - \)\(46\!\cdots\!14\)\( \beta_{4} - 2096646022971549650 \beta_{5} - 3091459333845989330 \beta_{6} + \)\(10\!\cdots\!76\)\( \beta_{7} - 3020592466976388620 \beta_{8} + 33515631591269330 \beta_{9} + 356697773832344064 \beta_{10} + 37651693052547072 \beta_{11}) q^{56}\) \(+(-\)\(63\!\cdots\!66\)\( + \)\(17\!\cdots\!86\)\( \beta_{1} + \)\(94\!\cdots\!12\)\( \beta_{2} - \)\(19\!\cdots\!23\)\( \beta_{3} + \)\(25\!\cdots\!05\)\( \beta_{4} + 17014592672035675416 \beta_{5} - 3878791801572915738 \beta_{6} + 27972959210241791094 \beta_{7} - 5835744305419952613 \beta_{8} + 345261779769783168 \beta_{9} + 523200949162308432 \beta_{10} + 6731154274077855 \beta_{11}) q^{57}\) \(+(-\)\(34\!\cdots\!20\)\( - \)\(97\!\cdots\!43\)\( \beta_{1} - \)\(33\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} - 14498216830807542180 \beta_{4} + 10345516018604200629 \beta_{5} + 11020572723402843945 \beta_{6} - \)\(11\!\cdots\!17\)\( \beta_{7} + \)\(10\!\cdots\!38\)\( \beta_{8} + 966547788720502812 \beta_{10}) q^{58}\) \(+(-\)\(42\!\cdots\!68\)\( \beta_{1} + \)\(58\!\cdots\!86\)\( \beta_{2} - \)\(18\!\cdots\!46\)\( \beta_{3} + \)\(74\!\cdots\!76\)\( \beta_{4} + 1521338410390347432 \beta_{5} + 2234006006409130032 \beta_{6} - 74899591903765044288 \beta_{7} + \)\(12\!\cdots\!91\)\( \beta_{8} - 76773419213316480 \beta_{9} - 258228043134309000 \beta_{10} - 1094020055997048 \beta_{11}) q^{59}\) \(+(\)\(56\!\cdots\!80\)\( + \)\(10\!\cdots\!60\)\( \beta_{1} - \)\(48\!\cdots\!16\)\( \beta_{2} - \)\(23\!\cdots\!90\)\( \beta_{3} - \)\(19\!\cdots\!02\)\( \beta_{4} - 50196959085934867770 \beta_{5} - 31444731812531974986 \beta_{6} - 47423081658940507364 \beta_{7} + \)\(35\!\cdots\!80\)\( \beta_{8} - 1507333772303192970 \beta_{9} - 1226775123750886236 \beta_{10} - 20416178277427680 \beta_{11}) q^{60}\) \(+(-\)\(71\!\cdots\!18\)\( - \)\(35\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} + \)\(51\!\cdots\!56\)\( \beta_{3} + 48437058855522550770 \beta_{4} - 31992110279623562754 \beta_{5} + 13600507913042141940 \beta_{6} + \)\(48\!\cdots\!80\)\( \beta_{7} + \)\(23\!\cdots\!28\)\( \beta_{8} - 3229137257034836718 \beta_{10}) q^{61}\) \(+(-\)\(10\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!96\)\( \beta_{2} - \)\(58\!\cdots\!34\)\( \beta_{3} + \)\(37\!\cdots\!15\)\( \beta_{4} + 6192613349849377942 \beta_{5} + 9720693910919749542 \beta_{6} - \)\(31\!\cdots\!49\)\( \beta_{7} + \)\(39\!\cdots\!23\)\( \beta_{8} + 1325434448373679928 \beta_{9} - 1248793343544819755 \beta_{10} - 289304254494623520 \beta_{11}) q^{62}\) \(+(\)\(84\!\cdots\!66\)\( + \)\(93\!\cdots\!17\)\( \beta_{1} + \)\(32\!\cdots\!31\)\( \beta_{2} - \)\(47\!\cdots\!36\)\( \beta_{3} + \)\(23\!\cdots\!00\)\( \beta_{4} + \)\(12\!\cdots\!30\)\( \beta_{5} + \)\(16\!\cdots\!61\)\( \beta_{6} - \)\(21\!\cdots\!12\)\( \beta_{7} + 30519113735268589002 \beta_{8} + 2891223297474617088 \beta_{9} - 2752567189169301872 \beta_{10} + 1729197910514360 \beta_{11}) q^{63}\) \(+(-\)\(11\!\cdots\!52\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(82\!\cdots\!84\)\( \beta_{2} - \)\(45\!\cdots\!60\)\( \beta_{3} + 93077006070560160 \beta_{4} + \)\(13\!\cdots\!48\)\( \beta_{5} - \)\(13\!\cdots\!60\)\( \beta_{6} - \)\(12\!\cdots\!40\)\( \beta_{7} + \)\(15\!\cdots\!44\)\( \beta_{8} - 6205133738037344 \beta_{10}) q^{64}\) \(+(-\)\(65\!\cdots\!90\)\( \beta_{1} + \)\(67\!\cdots\!86\)\( \beta_{2} - \)\(21\!\cdots\!40\)\( \beta_{3} + \)\(42\!\cdots\!14\)\( \beta_{4} + 32877313081927186340 \beta_{5} + 46913312895438083640 \beta_{6} - \)\(15\!\cdots\!00\)\( \beta_{7} + \)\(24\!\cdots\!80\)\( \beta_{8} - 5719219015811749760 \beta_{9} - 5190050380481904340 \beta_{10} + 979204301608359060 \beta_{11}) q^{65}\) \(+(-\)\(34\!\cdots\!60\)\( + \)\(97\!\cdots\!89\)\( \beta_{1} - \)\(28\!\cdots\!76\)\( \beta_{2} + \)\(14\!\cdots\!16\)\( \beta_{3} + \)\(29\!\cdots\!16\)\( \beta_{4} - \)\(32\!\cdots\!19\)\( \beta_{5} - \)\(16\!\cdots\!35\)\( \beta_{6} + \)\(80\!\cdots\!75\)\( \beta_{7} + \)\(49\!\cdots\!58\)\( \beta_{8} + 821059237229834520 \beta_{9} + 11668991388370109436 \beta_{10} + 46799062789718112 \beta_{11}) q^{66}\) \(+(\)\(53\!\cdots\!94\)\( - \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(39\!\cdots\!00\)\( \beta_{2} + \)\(14\!\cdots\!94\)\( \beta_{3} - \)\(14\!\cdots\!60\)\( \beta_{4} - \)\(18\!\cdots\!00\)\( \beta_{5} + \)\(41\!\cdots\!24\)\( \beta_{6} - \)\(84\!\cdots\!64\)\( \beta_{7} + \)\(10\!\cdots\!99\)\( \beta_{8} + 9702076195460143504 \beta_{10}) q^{67}\) \(+(-\)\(14\!\cdots\!00\)\( \beta_{1} - \)\(44\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!96\)\( \beta_{3} - \)\(45\!\cdots\!80\)\( \beta_{4} - \)\(21\!\cdots\!84\)\( \beta_{5} - \)\(32\!\cdots\!84\)\( \beta_{6} + \)\(10\!\cdots\!88\)\( \beta_{7} - \)\(16\!\cdots\!68\)\( \beta_{8} + 7441072734339762664 \beta_{9} + 38058471425796110880 \beta_{10} - 1276991255609930880 \beta_{11}) q^{68}\) \(+(-\)\(75\!\cdots\!64\)\( - \)\(60\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2} + \)\(21\!\cdots\!74\)\( \beta_{3} + \)\(28\!\cdots\!62\)\( \beta_{4} + \)\(57\!\cdots\!48\)\( \beta_{5} - \)\(82\!\cdots\!32\)\( \beta_{6} - \)\(77\!\cdots\!12\)\( \beta_{7} - \)\(21\!\cdots\!66\)\( \beta_{8} - 16881953865506881920 \beta_{9} - 12139846431010517448 \beta_{10} + 342439535327626674 \beta_{11}) q^{69}\) \(+(\)\(10\!\cdots\!20\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} + \)\(40\!\cdots\!08\)\( \beta_{2} - \)\(72\!\cdots\!60\)\( \beta_{3} - \)\(48\!\cdots\!90\)\( \beta_{4} - \)\(58\!\cdots\!00\)\( \beta_{5} - \)\(82\!\cdots\!64\)\( \beta_{6} - \)\(71\!\cdots\!66\)\( \beta_{7} - \)\(89\!\cdots\!90\)\( \beta_{8} + 32149905924380290746 \beta_{10}) q^{70}\) \(+(\)\(49\!\cdots\!52\)\( \beta_{1} - \)\(37\!\cdots\!04\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(30\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!76\)\( \beta_{5} + \)\(45\!\cdots\!96\)\( \beta_{6} - \)\(14\!\cdots\!88\)\( \beta_{7} - \)\(68\!\cdots\!62\)\( \beta_{8} + 22908718978296143680 \beta_{9} - 56896572652534862236 \beta_{10} - 1193379332316330852 \beta_{11}) q^{71}\) \(+(-\)\(83\!\cdots\!20\)\( - \)\(25\!\cdots\!44\)\( \beta_{1} + \)\(29\!\cdots\!74\)\( \beta_{2} + \)\(43\!\cdots\!61\)\( \beta_{3} - \)\(67\!\cdots\!65\)\( \beta_{4} - \)\(14\!\cdots\!25\)\( \beta_{5} + \)\(26\!\cdots\!91\)\( \beta_{6} + \)\(21\!\cdots\!18\)\( \beta_{7} - \)\(15\!\cdots\!46\)\( \beta_{8} + 31278030056265181833 \beta_{9} + 65268515579800556268 \beta_{10} - 2287163759923814400 \beta_{11}) q^{72}\) \(+(\)\(39\!\cdots\!26\)\( + \)\(32\!\cdots\!72\)\( \beta_{1} + \)\(11\!\cdots\!64\)\( \beta_{2} + \)\(30\!\cdots\!76\)\( \beta_{3} + \)\(27\!\cdots\!00\)\( \beta_{4} + \)\(14\!\cdots\!84\)\( \beta_{5} + \)\(89\!\cdots\!28\)\( \beta_{6} + \)\(31\!\cdots\!20\)\( \beta_{7} - \)\(53\!\cdots\!76\)\( \beta_{8} - \)\(18\!\cdots\!60\)\( \beta_{10}) q^{73}\) \(+(\)\(54\!\cdots\!98\)\( \beta_{1} - \)\(35\!\cdots\!16\)\( \beta_{2} + \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(17\!\cdots\!76\)\( \beta_{4} + \)\(12\!\cdots\!64\)\( \beta_{5} + \)\(13\!\cdots\!44\)\( \beta_{6} - \)\(54\!\cdots\!32\)\( \beta_{7} - \)\(71\!\cdots\!88\)\( \beta_{8} - \)\(11\!\cdots\!80\)\( \beta_{9} - 5121486512253713204 \beta_{10} + 6704739121317838272 \beta_{11}) q^{74}\) \(+(\)\(85\!\cdots\!85\)\( - \)\(58\!\cdots\!10\)\( \beta_{1} - \)\(11\!\cdots\!41\)\( \beta_{2} - \)\(14\!\cdots\!30\)\( \beta_{3} + \)\(21\!\cdots\!10\)\( \beta_{4} - \)\(19\!\cdots\!50\)\( \beta_{5} - \)\(10\!\cdots\!62\)\( \beta_{6} - \)\(14\!\cdots\!28\)\( \beta_{7} + \)\(11\!\cdots\!80\)\( \beta_{8} - 4397310536661727200 \beta_{9} - \)\(36\!\cdots\!82\)\( \beta_{10} + 4355976054920352450 \beta_{11}) q^{75}\) \(+(-\)\(90\!\cdots\!16\)\( - \)\(33\!\cdots\!54\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(46\!\cdots\!58\)\( \beta_{3} - \)\(31\!\cdots\!30\)\( \beta_{4} + \)\(27\!\cdots\!82\)\( \beta_{5} - \)\(47\!\cdots\!42\)\( \beta_{6} - \)\(30\!\cdots\!68\)\( \beta_{7} + \)\(12\!\cdots\!26\)\( \beta_{8} + \)\(21\!\cdots\!22\)\( \beta_{10}) q^{76}\) \(+(\)\(82\!\cdots\!86\)\( \beta_{1} + \)\(28\!\cdots\!82\)\( \beta_{2} - \)\(90\!\cdots\!24\)\( \beta_{3} + \)\(93\!\cdots\!10\)\( \beta_{4} + \)\(10\!\cdots\!04\)\( \beta_{5} + \)\(15\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!28\)\( \beta_{7} + \)\(90\!\cdots\!80\)\( \beta_{8} + \)\(14\!\cdots\!56\)\( \beta_{9} - \)\(20\!\cdots\!60\)\( \beta_{10} - 5694961424158829520 \beta_{11}) q^{77}\) \(+(-\)\(96\!\cdots\!60\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} + \)\(46\!\cdots\!72\)\( \beta_{2} - \)\(38\!\cdots\!26\)\( \beta_{3} - \)\(34\!\cdots\!10\)\( \beta_{4} + \)\(18\!\cdots\!64\)\( \beta_{5} - \)\(83\!\cdots\!08\)\( \beta_{6} + \)\(29\!\cdots\!18\)\( \beta_{7} + \)\(22\!\cdots\!76\)\( \beta_{8} + 4473099909512558064 \beta_{9} + \)\(60\!\cdots\!00\)\( \beta_{10} + 5276676518851907520 \beta_{11}) q^{78}\) \(+(-\)\(13\!\cdots\!30\)\( + \)\(65\!\cdots\!61\)\( \beta_{1} + \)\(22\!\cdots\!39\)\( \beta_{2} - \)\(18\!\cdots\!72\)\( \beta_{3} + \)\(29\!\cdots\!60\)\( \beta_{4} - \)\(82\!\cdots\!98\)\( \beta_{5} + \)\(54\!\cdots\!77\)\( \beta_{6} + \)\(56\!\cdots\!28\)\( \beta_{7} - \)\(10\!\cdots\!84\)\( \beta_{8} - 19894644432837095624 \beta_{10}) q^{79}\) \(+(\)\(15\!\cdots\!60\)\( \beta_{1} + \)\(45\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!40\)\( \beta_{3} - \)\(94\!\cdots\!76\)\( \beta_{4} - \)\(67\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!60\)\( \beta_{6} + \)\(33\!\cdots\!00\)\( \beta_{7} + \)\(72\!\cdots\!80\)\( \beta_{8} + \)\(10\!\cdots\!40\)\( \beta_{9} + \)\(11\!\cdots\!60\)\( \beta_{10} - 8219886641823191040 \beta_{11}) q^{80}\) \(+(-\)\(24\!\cdots\!99\)\( - \)\(58\!\cdots\!56\)\( \beta_{1} + \)\(46\!\cdots\!18\)\( \beta_{2} - \)\(49\!\cdots\!30\)\( \beta_{3} + \)\(12\!\cdots\!68\)\( \beta_{4} + \)\(12\!\cdots\!82\)\( \beta_{5} + \)\(20\!\cdots\!84\)\( \beta_{6} - \)\(12\!\cdots\!88\)\( \beta_{7} + \)\(11\!\cdots\!26\)\( \beta_{8} - \)\(32\!\cdots\!00\)\( \beta_{9} + \)\(29\!\cdots\!90\)\( \beta_{10} - 42026205399678549234 \beta_{11}) q^{81}\) \(+(\)\(10\!\cdots\!80\)\( - \)\(27\!\cdots\!26\)\( \beta_{1} - \)\(97\!\cdots\!92\)\( \beta_{2} + \)\(75\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} - \)\(26\!\cdots\!42\)\( \beta_{5} - \)\(24\!\cdots\!86\)\( \beta_{6} - \)\(20\!\cdots\!78\)\( \beta_{7} + \)\(23\!\cdots\!96\)\( \beta_{8} + \)\(82\!\cdots\!88\)\( \beta_{10}) q^{82}\) \(+(\)\(28\!\cdots\!14\)\( \beta_{1} + \)\(56\!\cdots\!26\)\( \beta_{2} - \)\(18\!\cdots\!32\)\( \beta_{3} - \)\(13\!\cdots\!50\)\( \beta_{4} + \)\(74\!\cdots\!22\)\( \beta_{5} + \)\(10\!\cdots\!72\)\( \beta_{6} - \)\(36\!\cdots\!24\)\( \beta_{7} + \)\(14\!\cdots\!75\)\( \beta_{8} - \)\(40\!\cdots\!92\)\( \beta_{9} - \)\(12\!\cdots\!10\)\( \beta_{10} - 20623419216894564450 \beta_{11}) q^{83}\) \(+(-\)\(96\!\cdots\!56\)\( - \)\(95\!\cdots\!76\)\( \beta_{1} - \)\(92\!\cdots\!88\)\( \beta_{2} + \)\(36\!\cdots\!92\)\( \beta_{3} - \)\(67\!\cdots\!78\)\( \beta_{4} - \)\(30\!\cdots\!02\)\( \beta_{5} - \)\(41\!\cdots\!46\)\( \beta_{6} + \)\(27\!\cdots\!28\)\( \beta_{7} + \)\(35\!\cdots\!24\)\( \beta_{8} + \)\(72\!\cdots\!30\)\( \beta_{9} - \)\(83\!\cdots\!32\)\( \beta_{10} + 66855885574104352224 \beta_{11}) q^{84}\) \(+(\)\(14\!\cdots\!40\)\( - \)\(21\!\cdots\!60\)\( \beta_{1} - \)\(74\!\cdots\!24\)\( \beta_{2} - \)\(41\!\cdots\!00\)\( \beta_{3} + \)\(42\!\cdots\!20\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5} + \)\(14\!\cdots\!72\)\( \beta_{6} + \)\(56\!\cdots\!28\)\( \beta_{7} + \)\(36\!\cdots\!40\)\( \beta_{8} - \)\(28\!\cdots\!28\)\( \beta_{10}) q^{85}\) \(+(-\)\(53\!\cdots\!52\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(67\!\cdots\!94\)\( \beta_{3} + \)\(59\!\cdots\!53\)\( \beta_{4} + \)\(89\!\cdots\!70\)\( \beta_{5} + \)\(13\!\cdots\!30\)\( \beta_{6} - \)\(44\!\cdots\!87\)\( \beta_{7} - \)\(14\!\cdots\!35\)\( \beta_{8} - \)\(35\!\cdots\!40\)\( \beta_{9} - \)\(16\!\cdots\!13\)\( \beta_{10} + \)\(19\!\cdots\!16\)\( \beta_{11}) q^{86}\) \(+(-\)\(90\!\cdots\!40\)\( + \)\(55\!\cdots\!47\)\( \beta_{1} - \)\(88\!\cdots\!67\)\( \beta_{2} + \)\(15\!\cdots\!26\)\( \beta_{3} + \)\(22\!\cdots\!30\)\( \beta_{4} - \)\(33\!\cdots\!84\)\( \beta_{5} + \)\(88\!\cdots\!53\)\( \beta_{6} - \)\(20\!\cdots\!68\)\( \beta_{7} - \)\(23\!\cdots\!36\)\( \beta_{8} + \)\(89\!\cdots\!56\)\( \beta_{9} - \)\(38\!\cdots\!30\)\( \beta_{10} + 58742554106544044490 \beta_{11}) q^{87}\) \(+(\)\(21\!\cdots\!80\)\( + \)\(80\!\cdots\!12\)\( \beta_{1} + \)\(28\!\cdots\!88\)\( \beta_{2} - \)\(25\!\cdots\!60\)\( \beta_{3} - \)\(23\!\cdots\!80\)\( \beta_{4} - \)\(87\!\cdots\!56\)\( \beta_{5} + \)\(15\!\cdots\!80\)\( \beta_{6} + \)\(60\!\cdots\!48\)\( \beta_{7} - \)\(72\!\cdots\!92\)\( \beta_{8} + \)\(15\!\cdots\!52\)\( \beta_{10}) q^{88}\) \(+(-\)\(21\!\cdots\!72\)\( \beta_{1} - \)\(25\!\cdots\!86\)\( \beta_{2} + \)\(80\!\cdots\!26\)\( \beta_{3} - \)\(88\!\cdots\!56\)\( \beta_{4} + \)\(30\!\cdots\!58\)\( \beta_{5} + \)\(87\!\cdots\!08\)\( \beta_{6} - \)\(22\!\cdots\!32\)\( \beta_{7} - \)\(51\!\cdots\!86\)\( \beta_{8} + \)\(12\!\cdots\!00\)\( \beta_{9} - \)\(17\!\cdots\!30\)\( \beta_{10} - \)\(28\!\cdots\!82\)\( \beta_{11}) q^{89}\) \(+(-\)\(20\!\cdots\!80\)\( + \)\(38\!\cdots\!35\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2} - \)\(80\!\cdots\!40\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4} + \)\(23\!\cdots\!75\)\( \beta_{5} - \)\(31\!\cdots\!09\)\( \beta_{6} + \)\(28\!\cdots\!89\)\( \beta_{7} - \)\(14\!\cdots\!70\)\( \beta_{8} - \)\(59\!\cdots\!00\)\( \beta_{9} + \)\(61\!\cdots\!56\)\( \beta_{10} - \)\(32\!\cdots\!00\)\( \beta_{11}) q^{90}\) \(+(\)\(71\!\cdots\!52\)\( + \)\(65\!\cdots\!26\)\( \beta_{1} + \)\(23\!\cdots\!50\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} + \)\(13\!\cdots\!40\)\( \beta_{4} - \)\(42\!\cdots\!96\)\( \beta_{5} - \)\(45\!\cdots\!34\)\( \beta_{6} - \)\(11\!\cdots\!76\)\( \beta_{7} + \)\(90\!\cdots\!92\)\( \beta_{8} - \)\(93\!\cdots\!96\)\( \beta_{10}) q^{91}\) \(+(-\)\(54\!\cdots\!88\)\( \beta_{1} - \)\(52\!\cdots\!48\)\( \beta_{2} + \)\(17\!\cdots\!68\)\( \beta_{3} + \)\(45\!\cdots\!60\)\( \beta_{4} - \)\(56\!\cdots\!76\)\( \beta_{5} - \)\(83\!\cdots\!76\)\( \beta_{6} + \)\(27\!\cdots\!12\)\( \beta_{7} - \)\(12\!\cdots\!28\)\( \beta_{8} + \)\(45\!\cdots\!76\)\( \beta_{9} + \)\(95\!\cdots\!80\)\( \beta_{10} - \)\(62\!\cdots\!60\)\( \beta_{11}) q^{92}\) \(+(\)\(12\!\cdots\!86\)\( + \)\(20\!\cdots\!67\)\( \beta_{1} - \)\(80\!\cdots\!11\)\( \beta_{2} - \)\(58\!\cdots\!92\)\( \beta_{3} + \)\(19\!\cdots\!55\)\( \beta_{4} - \)\(22\!\cdots\!34\)\( \beta_{5} - \)\(33\!\cdots\!76\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7} + \)\(18\!\cdots\!64\)\( \beta_{8} + \)\(72\!\cdots\!64\)\( \beta_{9} + \)\(12\!\cdots\!98\)\( \beta_{10} + 82453065139714535160 \beta_{11}) q^{93}\) \(+(\)\(59\!\cdots\!52\)\( + \)\(31\!\cdots\!72\)\( \beta_{1} + \)\(10\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!96\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4} + \)\(25\!\cdots\!60\)\( \beta_{5} + \)\(30\!\cdots\!96\)\( \beta_{6} - \)\(69\!\cdots\!96\)\( \beta_{7} - \)\(21\!\cdots\!20\)\( \beta_{8} + \)\(10\!\cdots\!16\)\( \beta_{10}) q^{94}\) \(+(-\)\(28\!\cdots\!60\)\( \beta_{1} - \)\(53\!\cdots\!72\)\( \beta_{2} + \)\(16\!\cdots\!60\)\( \beta_{3} - \)\(51\!\cdots\!68\)\( \beta_{4} - \)\(44\!\cdots\!60\)\( \beta_{5} - \)\(74\!\cdots\!60\)\( \beta_{6} + \)\(23\!\cdots\!00\)\( \beta_{7} - \)\(12\!\cdots\!70\)\( \beta_{8} - \)\(18\!\cdots\!60\)\( \beta_{9} + \)\(10\!\cdots\!60\)\( \beta_{10} + \)\(28\!\cdots\!60\)\( \beta_{11}) q^{95}\) \(+(-\)\(14\!\cdots\!92\)\( + \)\(33\!\cdots\!20\)\( \beta_{1} + \)\(64\!\cdots\!72\)\( \beta_{2} - \)\(20\!\cdots\!48\)\( \beta_{3} + \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(36\!\cdots\!28\)\( \beta_{5} + \)\(49\!\cdots\!24\)\( \beta_{6} - \)\(14\!\cdots\!92\)\( \beta_{7} - \)\(17\!\cdots\!04\)\( \beta_{8} + \)\(53\!\cdots\!00\)\( \beta_{9} - \)\(25\!\cdots\!12\)\( \beta_{10} + \)\(12\!\cdots\!20\)\( \beta_{11}) q^{96}\) \(+(\)\(90\!\cdots\!14\)\( + \)\(27\!\cdots\!92\)\( \beta_{1} + \)\(96\!\cdots\!66\)\( \beta_{2} + \)\(35\!\cdots\!84\)\( \beta_{3} - \)\(51\!\cdots\!30\)\( \beta_{4} + \)\(17\!\cdots\!34\)\( \beta_{5} + \)\(11\!\cdots\!20\)\( \beta_{6} - \)\(10\!\cdots\!32\)\( \beta_{7} - \)\(19\!\cdots\!48\)\( \beta_{8} + \)\(34\!\cdots\!02\)\( \beta_{10}) q^{97}\) \(+(-\)\(25\!\cdots\!21\)\( \beta_{1} + \)\(63\!\cdots\!96\)\( \beta_{2} - \)\(20\!\cdots\!12\)\( \beta_{3} + \)\(60\!\cdots\!60\)\( \beta_{4} + \)\(34\!\cdots\!12\)\( \beta_{5} + \)\(51\!\cdots\!12\)\( \beta_{6} - \)\(17\!\cdots\!64\)\( \beta_{7} + \)\(23\!\cdots\!40\)\( \beta_{8} + \)\(16\!\cdots\!68\)\( \beta_{9} - \)\(63\!\cdots\!00\)\( \beta_{10} - \)\(28\!\cdots\!20\)\( \beta_{11}) q^{98}\) \(+(-\)\(38\!\cdots\!80\)\( + \)\(27\!\cdots\!78\)\( \beta_{1} + \)\(78\!\cdots\!20\)\( \beta_{2} - \)\(19\!\cdots\!34\)\( \beta_{3} + \)\(45\!\cdots\!64\)\( \beta_{4} + \)\(34\!\cdots\!48\)\( \beta_{5} + \)\(13\!\cdots\!10\)\( \beta_{6} - \)\(96\!\cdots\!84\)\( \beta_{7} + \)\(27\!\cdots\!09\)\( \beta_{8} - \)\(13\!\cdots\!00\)\( \beta_{9} - \)\(22\!\cdots\!48\)\( \beta_{10} - \)\(76\!\cdots\!32\)\( \beta_{11}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 16987205196q^{3} \) \(\mathstrut -\mathstrut 34999503561888q^{4} \) \(\mathstrut -\mathstrut 47900661722006688q^{6} \) \(\mathstrut -\mathstrut 41814621990917832q^{7} \) \(\mathstrut -\mathstrut 10479496382139999060q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 16987205196q^{3} \) \(\mathstrut -\mathstrut 34999503561888q^{4} \) \(\mathstrut -\mathstrut 47900661722006688q^{6} \) \(\mathstrut -\mathstrut 41814621990917832q^{7} \) \(\mathstrut -\mathstrut 10479496382139999060q^{9} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!40\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!84\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!92\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!60\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!16\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!60\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!72\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!60\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!64\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!60\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!04\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!88\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!80\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!96\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!56\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!56\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!52\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!40\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!88\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!40\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!16\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!16\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!28\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!08\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!08\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!28\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!80\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!92\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!40\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!60\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!16\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!92\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!24\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!20\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!28\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!68\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!40\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!12\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!20\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!88\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!60\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!72\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!60\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!24\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!32\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!24\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!04\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!68\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!60\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut +\mathstrut \) \(33864221333\) \(x^{10}\mathstrut +\mathstrut \) \(431067677694832840128\) \(x^{8}\mathstrut +\mathstrut \) \(2571749156180178266268142478336\) \(x^{6}\mathstrut +\mathstrut \) \(7347696986799808047021630608244043939840\) \(x^{4}\mathstrut +\mathstrut \) \(8856020548466312086094523118699705898743155916800\) \(x^{2}\mathstrut +\mathstrut \) \(2517175751433570113519145367463712046356978756810178560000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 36 \nu \)
\(\beta_{2}\)\(=\)\((\)\(71\!\cdots\!25\) \(\nu^{11}\mathstrut -\mathstrut \) \(41\!\cdots\!56\) \(\nu^{10}\mathstrut +\mathstrut \) \(21\!\cdots\!25\) \(\nu^{9}\mathstrut -\mathstrut \) \(11\!\cdots\!28\) \(\nu^{8}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(10\!\cdots\!68\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(70\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(83\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(19\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(93\!\cdots\!25\) \(\nu^{11}\mathstrut +\mathstrut \) \(54\!\cdots\!28\) \(\nu^{10}\mathstrut -\mathstrut \) \(27\!\cdots\!25\) \(\nu^{9}\mathstrut +\mathstrut \) \(14\!\cdots\!64\) \(\nu^{8}\mathstrut -\mathstrut \) \(29\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(57\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(24\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(96\!\cdots\!00\)\()/\)\(97\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(20\!\cdots\!83\) \(\nu^{11}\mathstrut +\mathstrut \) \(13\!\cdots\!64\) \(\nu^{10}\mathstrut +\mathstrut \) \(74\!\cdots\!99\) \(\nu^{9}\mathstrut +\mathstrut \) \(35\!\cdots\!32\) \(\nu^{8}\mathstrut +\mathstrut \) \(91\!\cdots\!04\) \(\nu^{7}\mathstrut +\mathstrut \) \(34\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(43\!\cdots\!68\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!44\) \(\nu^{4}\mathstrut +\mathstrut \) \(70\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(22\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(95\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(61\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(17\!\cdots\!27\) \(\nu^{11}\mathstrut +\mathstrut \) \(38\!\cdots\!24\) \(\nu^{10}\mathstrut -\mathstrut \) \(47\!\cdots\!51\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!72\) \(\nu^{8}\mathstrut -\mathstrut \) \(47\!\cdots\!56\) \(\nu^{7}\mathstrut +\mathstrut \) \(20\!\cdots\!92\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!92\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!64\) \(\nu^{4}\mathstrut -\mathstrut \) \(30\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(37\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(91\!\cdots\!00\)\()/\)\(73\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(54\!\cdots\!37\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!96\) \(\nu^{10}\mathstrut -\mathstrut \) \(14\!\cdots\!81\) \(\nu^{9}\mathstrut +\mathstrut \) \(66\!\cdots\!88\) \(\nu^{8}\mathstrut -\mathstrut \) \(13\!\cdots\!36\) \(\nu^{7}\mathstrut +\mathstrut \) \(75\!\cdots\!68\) \(\nu^{6}\mathstrut -\mathstrut \) \(56\!\cdots\!52\) \(\nu^{5}\mathstrut +\mathstrut \) \(34\!\cdots\!56\) \(\nu^{4}\mathstrut -\mathstrut \) \(72\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(52\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(18\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(59\!\cdots\!87\) \(\nu^{11}\mathstrut +\mathstrut \) \(33\!\cdots\!60\) \(\nu^{10}\mathstrut +\mathstrut \) \(16\!\cdots\!31\) \(\nu^{9}\mathstrut +\mathstrut \) \(95\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(17\!\cdots\!36\) \(\nu^{7}\mathstrut +\mathstrut \) \(97\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(74\!\cdots\!52\) \(\nu^{5}\mathstrut +\mathstrut \) \(41\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(11\!\cdots\!60\) \(\nu^{3}\mathstrut +\mathstrut \) \(66\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(26\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(22\!\cdots\!00\)\()/\)\(19\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(82\!\cdots\!25\) \(\nu^{11}\mathstrut +\mathstrut \) \(51\!\cdots\!68\) \(\nu^{10}\mathstrut +\mathstrut \) \(24\!\cdots\!25\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!84\) \(\nu^{8}\mathstrut +\mathstrut \) \(26\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!04\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(54\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(87\!\cdots\!40\) \(\nu^{2}\mathstrut +\mathstrut \) \(96\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(25\!\cdots\!00\)\()/\)\(14\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(33\!\cdots\!45\) \(\nu^{11}\mathstrut -\mathstrut \) \(13\!\cdots\!44\) \(\nu^{10}\mathstrut -\mathstrut \) \(17\!\cdots\!15\) \(\nu^{9}\mathstrut -\mathstrut \) \(34\!\cdots\!12\) \(\nu^{8}\mathstrut -\mathstrut \) \(11\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(32\!\cdots\!12\) \(\nu^{6}\mathstrut -\mathstrut \) \(57\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!64\) \(\nu^{4}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(19\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(16\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(71\!\cdots\!00\)\()/\)\(29\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(11\!\cdots\!89\) \(\nu^{11}\mathstrut -\mathstrut \) \(85\!\cdots\!84\) \(\nu^{10}\mathstrut +\mathstrut \) \(32\!\cdots\!57\) \(\nu^{9}\mathstrut -\mathstrut \) \(22\!\cdots\!12\) \(\nu^{8}\mathstrut +\mathstrut \) \(33\!\cdots\!92\) \(\nu^{7}\mathstrut -\mathstrut \) \(21\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!44\) \(\nu^{5}\mathstrut -\mathstrut \) \(83\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(21\!\cdots\!20\) \(\nu^{3}\mathstrut -\mathstrut \) \(12\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(36\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(49\!\cdots\!00\)\()/\)\(29\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(35\!\cdots\!95\) \(\nu^{11}\mathstrut -\mathstrut \) \(54\!\cdots\!64\) \(\nu^{10}\mathstrut +\mathstrut \) \(21\!\cdots\!55\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!52\) \(\nu^{8}\mathstrut +\mathstrut \) \(38\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(13\!\cdots\!32\) \(\nu^{6}\mathstrut +\mathstrut \) \(27\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(53\!\cdots\!64\) \(\nu^{4}\mathstrut +\mathstrut \) \(77\!\cdots\!80\) \(\nu^{3}\mathstrut -\mathstrut \) \(82\!\cdots\!20\) \(\nu^{2}\mathstrut +\mathstrut \) \(62\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(30\!\cdots\!00\)\()/\)\(58\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/36\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(78\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\) \(\beta_{1}\mathstrut -\mathstrut \) \(7314671807928\)\()/1296\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(42\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2655\) \(\beta_{4}\mathstrut +\mathstrut \) \(29109\) \(\beta_{3}\mathstrut -\mathstrut \) \(89312098\) \(\beta_{2}\mathstrut -\mathstrut \) \(10893722619548\) \(\beta_{1}\)\()/46656\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(981785\) \(\beta_{10}\mathstrut -\mathstrut \) \(58048401\) \(\beta_{8}\mathstrut +\mathstrut \) \(127239206\) \(\beta_{7}\mathstrut -\mathstrut \) \(17860621\) \(\beta_{6}\mathstrut +\mathstrut \) \(12747343\) \(\beta_{5}\mathstrut +\mathstrut \) \(14726775\) \(\beta_{4}\mathstrut -\mathstrut \) \(8050382024360\) \(\beta_{3}\mathstrut -\mathstrut \) \(375386873757752\) \(\beta_{2}\mathstrut -\mathstrut \) \(104984681137695\) \(\beta_{1}\mathstrut +\mathstrut \) \(39841909549909302436887744\)\()/839808\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(1748338560\) \(\beta_{11}\mathstrut -\mathstrut \) \(124887467840\) \(\beta_{10}\mathstrut -\mathstrut \) \(1123258478149\) \(\beta_{9}\mathstrut +\mathstrut \) \(42755108592174\) \(\beta_{8}\mathstrut -\mathstrut \) \(81944368595858\) \(\beta_{7}\mathstrut +\mathstrut \) \(2169200883909\) \(\beta_{6}\mathstrut +\mathstrut \) \(1824181335109\) \(\beta_{5}\mathstrut -\mathstrut \) \(4013408134683995\) \(\beta_{4}\mathstrut -\mathstrut \) \(57203947530296265\) \(\beta_{3}\mathstrut +\mathstrut \) \(175897071470702953322\) \(\beta_{2}\mathstrut +\mathstrut \) \(8546285920837473035941388\) \(\beta_{1}\)\()/3779136\)
\(\nu^{6}\)\(=\)\((\)\(449720372971920511\) \(\beta_{10}\mathstrut +\mathstrut \) \(28235950194240100679\) \(\beta_{8}\mathstrut -\mathstrut \) \(59587386496284945130\) \(\beta_{7}\mathstrut +\mathstrut \) \(6803985349595592555\) \(\beta_{6}\mathstrut -\mathstrut \) \(4388140130772292057\) \(\beta_{5}\mathstrut -\mathstrut \) \(6745805594578807665\) \(\beta_{4}\mathstrut +\mathstrut \) \(2432103200369640023459864\) \(\beta_{3}\mathstrut +\mathstrut \) \(69133990020928838819651848\) \(\beta_{2}\mathstrut +\mathstrut \) \(19072672217765669540866953\) \(\beta_{1}\mathstrut -\mathstrut \) \(10418818187960863539442663221913566528\)\()/22674816\)
\(\nu^{7}\)\(=\)\((\)\(1873917644796389911680\) \(\beta_{11}\mathstrut +\mathstrut \) \(61064203052821794733760\) \(\beta_{10}\mathstrut +\mathstrut \) \(352860993536814481184019\) \(\beta_{9}\mathstrut -\mathstrut \) \(16421154765224898123681122\) \(\beta_{8}\mathstrut +\mathstrut \) \(31906044454266556073801438\) \(\beta_{7}\mathstrut -\mathstrut \) \(866480624957163266012819\) \(\beta_{6}\mathstrut -\mathstrut \) \(697489168936506478841619\) \(\beta_{5}\mathstrut +\mathstrut \) \(1359943658168172381970928205\) \(\beta_{4}\mathstrut +\mathstrut \) \(21990323972292899272993188207\) \(\beta_{3}\mathstrut -\mathstrut \) \(67676219627709295532816245333062\) \(\beta_{2}\mathstrut -\mathstrut \) \(2373460107374169274305608634734797716\) \(\beta_{1}\)\()/\)\(102036672\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(155849888114969881446557399049\) \(\beta_{10}\mathstrut -\mathstrut \) \(10228384833489753266555105669377\) \(\beta_{8}\mathstrut +\mathstrut \) \(21013361465646158684138082736326\) \(\beta_{7}\mathstrut -\mathstrut \) \(2064102773183042893105141362621\) \(\beta_{6}\mathstrut +\mathstrut \) \(1296534130421852609025083407071\) \(\beta_{5}\mathstrut +\mathstrut \) \(2337748321724548221698360985735\) \(\beta_{4}\mathstrut -\mathstrut \) \(717399361561916356144472691870845736\) \(\beta_{3}\mathstrut -\mathstrut \) \(12754627553518062666210031417555407544\) \(\beta_{2}\mathstrut -\mathstrut \) \(3444579366517530083824162293964929967\) \(\beta_{1}\mathstrut +\mathstrut \) \(2893490052701920529423576434629460453336989087936\)\()/\)\(612220032\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(93493096762857075630129628402560\) \(\beta_{11}\mathstrut -\mathstrut \) \(2471623097608888718250487704740160\) \(\beta_{10}\mathstrut -\mathstrut \) \(11752066367156359414949633752576365\) \(\beta_{9}\mathstrut +\mathstrut \) \(602196965354523777368872796458246558\) \(\beta_{8}\mathstrut -\mathstrut \) \(1186643832913876391099814222796438050\) \(\beta_{7}\mathstrut +\mathstrut \) \(32579366034460616192746397474197485\) \(\beta_{6}\mathstrut +\mathstrut \) \(25734229587964010224115704487442285\) \(\beta_{5}\mathstrut -\mathstrut \) \(45605992939688330559354995568540251955\) \(\beta_{4}\mathstrut -\mathstrut \) \(805708903871292559345548241764857163537\) \(\beta_{3}\mathstrut +\mathstrut \) \(2480935241155707258378247272188345743335610\) \(\beta_{2}\mathstrut +\mathstrut \) \(75172958155956288221625286901431700628183585388\) \(\beta_{1}\)\()/\)\(306110016\)
\(\nu^{10}\)\(=\)\((\)\(1814243159679611481239750736028756174461\) \(\beta_{10}\mathstrut +\mathstrut \) \(123886106305968197840977564657468841448661\) \(\beta_{8}\mathstrut -\mathstrut \) \(247906869022120526964055484168165078185662\) \(\beta_{7}\mathstrut +\mathstrut \) \(21917227852344016407845513530018217397057\) \(\beta_{6}\mathstrut -\mathstrut \) \(14162121295828003150064648615801818287243\) \(\beta_{5}\mathstrut -\mathstrut \) \(27213647395194172218596261040431342616915\) \(\beta_{4}\mathstrut +\mathstrut \) \(7781371599124703746521014419132931874816417864\) \(\beta_{3}\mathstrut +\mathstrut \) \(86671722881098765678227362163778681216873339672\) \(\beta_{2}\mathstrut +\mathstrut \) \(22604785250624792951096825415578773179471358363\) \(\beta_{1}\mathstrut -\mathstrut \) \(30547796305282421163531488735705569176383922298069547284416\)\()/\)\(612220032\)
\(\nu^{11}\)\(=\)\((\)\(12\!\cdots\!20\) \(\beta_{11}\mathstrut +\mathstrut \) \(30\!\cdots\!80\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\!\cdots\!93\) \(\beta_{9}\mathstrut -\mathstrut \) \(69\!\cdots\!70\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\!\cdots\!06\) \(\beta_{7}\mathstrut -\mathstrut \) \(38\!\cdots\!13\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\!\cdots\!13\) \(\beta_{5}\mathstrut +\mathstrut \) \(48\!\cdots\!15\) \(\beta_{4}\mathstrut +\mathstrut \) \(92\!\cdots\!13\) \(\beta_{3}\mathstrut -\mathstrut \) \(28\!\cdots\!54\) \(\beta_{2}\mathstrut -\mathstrut \) \(80\!\cdots\!48\) \(\beta_{1}\)\()/\)\(306110016\)

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
104257.i
103520.i
74991.2i
59345.7i
52282.8i
19978.7i
19978.7i
52282.8i
59345.7i
74991.2i
103520.i
104257.i
3.75326e6i 1.02269e10 2.19746e9i −9.68891e12 5.14242e14i −8.24764e15 3.83843e16i 7.24611e17 1.98580e19i 9.97613e19 4.49466e19i 1.93008e21
2.2 3.72673e6i −8.44421e9 6.17368e9i −9.49045e12 9.25532e14i −2.30076e16 + 3.14693e16i 7.62151e16 1.89780e19i 3.31903e19 + 1.04264e20i −3.44920e21
2.3 2.69968e6i −4.62216e9 + 9.38374e9i −2.89024e12 1.56653e14i 2.53331e16 + 1.24784e16i 1.13686e17 4.07061e18i −6.66902e19 8.67463e19i 4.22913e20
2.4 2.13644e6i −3.48827e9 9.86159e9i −1.66350e11 7.27116e14i −2.10687e16 + 7.45250e15i −7.17356e17 9.04079e18i −8.50829e19 + 6.87998e19i 1.55344e21
2.5 1.88218e6i 9.19591e9 + 4.98540e9i 8.55436e11 4.92433e14i 9.38344e15 1.73084e16i −8.30626e17 9.88801e18i 5.97105e19 + 9.16906e19i −9.26849e20
2.6 719233.i 5.62540e9 8.81895e9i 3.88075e12 4.30398e14i −6.34288e15 4.04597e15i 6.12563e17 5.95439e18i −4.61287e19 9.92202e19i −3.09557e20
2.7 719233.i 5.62540e9 + 8.81895e9i 3.88075e12 4.30398e14i −6.34288e15 + 4.04597e15i 6.12563e17 5.95439e18i −4.61287e19 + 9.92202e19i −3.09557e20
2.8 1.88218e6i 9.19591e9 4.98540e9i 8.55436e11 4.92433e14i 9.38344e15 + 1.73084e16i −8.30626e17 9.88801e18i 5.97105e19 9.16906e19i −9.26849e20
2.9 2.13644e6i −3.48827e9 + 9.86159e9i −1.66350e11 7.27116e14i −2.10687e16 7.45250e15i −7.17356e17 9.04079e18i −8.50829e19 6.87998e19i 1.55344e21
2.10 2.69968e6i −4.62216e9 9.38374e9i −2.89024e12 1.56653e14i 2.53331e16 1.24784e16i 1.13686e17 4.07061e18i −6.66902e19 + 8.67463e19i 4.22913e20
2.11 3.72673e6i −8.44421e9 + 6.17368e9i −9.49045e12 9.25532e14i −2.30076e16 3.14693e16i 7.62151e16 1.89780e19i 3.31903e19 1.04264e20i −3.44920e21
2.12 3.75326e6i 1.02269e10 + 2.19746e9i −9.68891e12 5.14242e14i −8.24764e15 + 3.83843e16i 7.24611e17 1.98580e19i 9.97613e19 + 4.49466e19i 1.93008e21
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.12
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

This newform can be constructed as the kernel of the linear operator \(T_{2}^{12} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!68\)\( T_{2}^{10} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!48\)\( T_{2}^{8} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!96\)\( T_{2}^{6} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\( T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!00\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\( \) acting on \(S_{43}^{\mathrm{new}}(3, [\chi])\).