# 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 $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$33.5183121516$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 33864221333 x^{10} + 431067677694832840128 x^{8} + 2571749156180178266268142478336 x^{6} + 7347696986799808047021630608244043939840 x^{4} + 8856020548466312086094523118699705898743155916800 x^{2} + 2517175751433570113519145367463712046356978756810178560000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: multiple of $$2^{73}\cdot 3^{104}\cdot 5^{6}\cdot 7^{4}$$ Twist minimal: yes 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.

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

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.43.b.b 12
3.b odd 2 1 inner 3.43.b.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.43.b.b 12 1.a even 1 1 trivial
3.43.b.b 12 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} +$$43888030847568

'>$$43\!\cdots\!68$$$$T_{2}^{10} +$$724028168539084355604430848
'>$$72\!\cdots\!48$$$$T_{2}^{8} +$$5598138135795917283343597186373067472896'>$$55\!\cdots\!96$$$$T_{2}^{6} +$$20728660766445536533131756421653571789785220031447040'>$$20\!\cdots\!40$$$$T_{2}^{4} +$$32379014273646252739308711808142940235130807576123506779016396800'>$$32\!\cdots\!00$$$$T_{2}^{2} +$$11927338605868521624476192285007600722320157488318930112854941769672949760000'>$$11\!\cdots\!00$$ acting on $$S_{43}^{\mathrm{new}}(3, [\chi])$$.