Properties

Label 3.41.b.a
Level 3
Weight 41
Character orbit 3.b
Analytic conductor 30.403
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 41 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(30.4026855589\)
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 \( 2^{63}\cdot 3^{98}\cdot 5^{6}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -31006881 - 471 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -398447804756 - 10 \beta_{1} + 34 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 7197249 \beta_{1} + 1832 \beta_{2} + \beta_{4} ) q^{5} \) \( + ( 705988540443996 + 239159389 \beta_{1} - 15902 \beta_{2} - 1013 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{6} \) \( + ( -7861812427063882 - 380314 \beta_{1} + 1279276 \beta_{2} + 17920 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{7} \) \( + ( -629251453319 \beta_{1} - 90796669 \beta_{2} + 18779 \beta_{3} + 520 \beta_{4} - 46 \beta_{5} + 50 \beta_{7} + \beta_{8} ) q^{8} \) \( + ( -1967125188503494935 - 569001114127 \beta_{1} + 70714943 \beta_{2} + 1319668 \beta_{3} - 1489 \beta_{4} - 268 \beta_{5} - 6 \beta_{6} + 270 \beta_{7} + \beta_{8} + \beta_{10} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(-31006881 - 471 \beta_{1} - \beta_{2}) q^{3}\) \(+(-398447804756 - 10 \beta_{1} + 34 \beta_{2} + \beta_{3}) q^{4}\) \(+(7197249 \beta_{1} + 1832 \beta_{2} + \beta_{4}) q^{5}\) \(+(705988540443996 + 239159389 \beta_{1} - 15902 \beta_{2} - 1013 \beta_{3} + 2 \beta_{4} + \beta_{5}) q^{6}\) \(+(-7861812427063882 - 380314 \beta_{1} + 1279276 \beta_{2} + 17920 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6}) q^{7}\) \(+(-629251453319 \beta_{1} - 90796669 \beta_{2} + 18779 \beta_{3} + 520 \beta_{4} - 46 \beta_{5} + 50 \beta_{7} + \beta_{8}) q^{8}\) \(+(-1967125188503494935 - 569001114127 \beta_{1} + 70714943 \beta_{2} + 1319668 \beta_{3} - 1489 \beta_{4} - 268 \beta_{5} - 6 \beta_{6} + 270 \beta_{7} + \beta_{8} + \beta_{10}) q^{9}\) \(+(-10782011362892190840 - 446577289 \beta_{1} + 1491606019 \beta_{2} + 5639517 \beta_{3} - 4168 \beta_{4} - 1259 \beta_{5} + 53 \beta_{6} + 965 \beta_{7} + \beta_{8} + 4 \beta_{10} + \beta_{11}) q^{10}\) \(+(-47082734038067 \beta_{1} + 3920245985 \beta_{2} - 748009 \beta_{3} + 146137 \beta_{4} - 13371 \beta_{5} + 55 \beta_{6} - 2198 \beta_{7} + 78 \beta_{8} + \beta_{9} - 30 \beta_{10} + 15 \beta_{11}) q^{11}\) \(+(-\)\(39\!\cdots\!56\)\( + 1531342679041676 \beta_{1} + 319207732488 \beta_{2} + 329177315 \beta_{3} - 1596149 \beta_{4} + 31857 \beta_{5} - 13653 \beta_{6} - 114697 \beta_{7} - 2391 \beta_{8} + 27 \beta_{9} + 12 \beta_{10} + 81 \beta_{11}) q^{12}\) \(+(58697403822012908498 - 20092254195 \beta_{1} + 65278394415 \beta_{2} - 2415481549 \beta_{3} - 255333 \beta_{4} + 910403 \beta_{5} + 112489 \beta_{6} + 172523 \beta_{7} + 520 \beta_{8} + 351 \beta_{9} + 676 \beta_{10} + 169 \beta_{11}) q^{13}\) \(+(-31882740474169644 \beta_{1} - 4716073320440 \beta_{2} + 982417764 \beta_{3} + 46039475 \beta_{4} - 7106728 \beta_{5} + 28622 \beta_{6} + 2571688 \beta_{7} + 49135 \beta_{8} + 2999 \beta_{9} + 912 \beta_{10} - 456 \beta_{11}) q^{14}\) \(+(\)\(26\!\cdots\!20\)\( + 11560935310456259 \beta_{1} - 760464750043 \beta_{2} + 539544053 \beta_{3} - 486537389 \beta_{4} + 8357699 \beta_{5} - 2727486 \beta_{6} - 2240403 \beta_{7} - 22158 \beta_{8} + 18873 \beta_{9} - 2394 \beta_{10} - 4617 \beta_{11}) q^{15}\) \(+(\)\(50\!\cdots\!88\)\( + 14031442072574 \beta_{1} - 47207727389582 \beta_{2} - 674822526982 \beta_{3} + 137141290 \beta_{4} + 277126466 \beta_{5} + 10044778 \beta_{6} - 9649274 \beta_{7} + 79496 \beta_{8} + 91962 \beta_{9} - 49864 \beta_{10} - 12466 \beta_{11}) q^{16}\) \(+(810831997353408687 \beta_{1} + 145865418493791 \beta_{2} - 26369552263 \beta_{3} + 9337037421 \beta_{4} - 536696807 \beta_{5} + 3533171 \beta_{6} - 76061079 \beta_{7} - 1372234 \beta_{8} + 354461 \beta_{9} + 7626 \beta_{10} - 3813 \beta_{11}) q^{17}\) \(+(\)\(85\!\cdots\!60\)\( - 3730184184641694684 \beta_{1} - 505095414912033 \beta_{2} - 1381030307499 \beta_{3} - 60439295508 \beta_{4} + 199118469 \beta_{5} + 27938025 \beta_{6} + 266078817 \beta_{7} + 5509689 \beta_{8} + 1070172 \beta_{9} + 76452 \beta_{10} + 120285 \beta_{11}) q^{18}\) \(+(\)\(31\!\cdots\!10\)\( - 85133609092608 \beta_{1} + 277780987679280 \beta_{2} - 8498941849062 \beta_{3} - 217124780 \beta_{4} + 5334600208 \beta_{5} - 187337986 \beta_{6} + 396732483 \beta_{7} + 2767008 \beta_{8} + 2386476 \beta_{9} + 1522128 \beta_{10} + 380532 \beta_{11}) q^{19}\) \(+(-10723163564114824210 \beta_{1} - 166071529070742 \beta_{2} + 308203413514 \beta_{3} + 731681390904 \beta_{4} - 6383446828 \beta_{5} + 30978808 \beta_{6} + 762812532 \beta_{7} + 24116470 \beta_{8} + 2957128 \beta_{9} - 938352 \beta_{10} + 469176 \beta_{11}) q^{20}\) \(+(-\)\(14\!\cdots\!94\)\( + 52249519028509242920 \beta_{1} + 4992157905674345 \beta_{2} + 49223705950991 \beta_{3} - 2093976588806 \beta_{4} + 7309512735 \beta_{5} + 1234414521 \beta_{6} - 6500923093 \beta_{7} - 107252358 \beta_{8} - 3687741 \beta_{9} - 1274010 \beta_{10} - 1851579 \beta_{11}) q^{21}\) \(+(\)\(70\!\cdots\!60\)\( + 5626757644622458 \beta_{1} - 18843779514139912 \beta_{2} - 144027736903878 \beta_{3} + 20551823233 \beta_{4} - 80845583110 \beta_{5} - 324419330 \beta_{6} - 11427380204 \beta_{7} - 41338315 \beta_{8} - 34370703 \beta_{9} - 27870448 \beta_{10} - 6967612 \beta_{11}) q^{22}\) \(+(\)\(17\!\cdots\!28\)\( \beta_{1} + 39872780474060580 \beta_{2} - 4594112549060 \beta_{3} + 9499800306072 \beta_{4} + 202026040224 \beta_{5} - 1169627296 \beta_{6} - 10412810578 \beta_{7} - 66259968 \beta_{8} - 113158336 \beta_{9} + 25362624 \beta_{10} - 12681312 \beta_{11}) q^{23}\) \(+(-\)\(15\!\cdots\!12\)\( - \)\(65\!\cdots\!85\)\( \beta_{1} + 42150870263116673 \beta_{2} + 3191913838011041 \beta_{3} - 7989665732870 \beta_{4} - 223339805068 \beta_{5} - 26479340910 \beta_{6} + 2948967528 \beta_{7} + 956008917 \beta_{8} - 235928862 \beta_{9} + 12646872 \beta_{10} + 17737542 \beta_{11}) q^{24}\) \(+(\)\(73\!\cdots\!25\)\( + 92021541305455935 \beta_{1} - 309725877502026435 \beta_{2} - 4614069186443455 \beta_{3} + 244514225945 \beta_{4} - 733479887615 \beta_{5} + 36729981455 \beta_{6} - 104305729075 \beta_{7} - 178715240 \beta_{8} - 264373875 \beta_{9} + 342634540 \beta_{10} + 85658635 \beta_{11}) q^{25}\) \(+(\)\(32\!\cdots\!78\)\( \beta_{1} + 1462948459901026384 \beta_{2} - 324546916349760 \beta_{3} - 66896543649052 \beta_{4} - 271103085552 \beta_{5} + 2807844312 \beta_{6} - 878683858416 \beta_{7} - 4602805116 \beta_{8} + 219363924 \beta_{9} - 409470048 \beta_{10} + 204735024 \beta_{11}) q^{26}\) \(+(-\)\(58\!\cdots\!21\)\( + \)\(57\!\cdots\!80\)\( \beta_{1} + 1692302936099462262 \beta_{2} + 18289137564440901 \beta_{3} + 214042923322443 \beta_{4} + 1536760160055 \beta_{5} + 167222786931 \beta_{6} - 1922434375968 \beta_{7} - 4543048782 \beta_{8} + 1944306423 \beta_{9} - 65953602 \beta_{10} - 91584999 \beta_{11}) q^{27}\) \(+(\)\(39\!\cdots\!08\)\( + 2726936481033860762 \beta_{1} - 9131504311320557690 \beta_{2} - 68408788620323284 \beta_{3} + 23775216526202 \beta_{4} + 14030636124114 \beta_{5} - 415416645318 \beta_{6} - 5017486042058 \beta_{7} + 4586624840 \beta_{8} + 5313764106 \beta_{9} - 2908557064 \beta_{10} - 727139266 \beta_{11}) q^{28}\) \(+(-\)\(42\!\cdots\!83\)\( \beta_{1} + 10153559307556709222 \beta_{2} - 2382204007330622 \beta_{3} - 809006064224723 \beta_{4} - 17548139770878 \beta_{5} + 81619566038 \beta_{6} - 6630619212862 \beta_{7} + 81689694924 \beta_{8} + 8857110410 \beta_{9} + 4634358708 \beta_{10} - 2317179354 \beta_{11}) q^{29}\) \(+(-\)\(17\!\cdots\!00\)\( + \)\(25\!\cdots\!82\)\( \beta_{1} + 9570047299352439872 \beta_{2} + 41832933003214258 \beta_{3} + 899037227785511 \beta_{4} + 4029924381114 \beta_{5} + 402349472226 \beta_{6} - 28019920440116 \beta_{7} - 3057493125 \beta_{8} + 6410014191 \beta_{9} - 69753504 \beta_{10} - 86597748 \beta_{11}) q^{30}\) \(+(-\)\(76\!\cdots\!18\)\( + 24289717904073901862 \beta_{1} - 80901310345248340208 \beta_{2} + 26134161445767832 \beta_{3} + 230392875733054 \beta_{4} - 35814842902010 \beta_{5} + 1729099832525 \beta_{6} - 59728985324514 \beta_{7} - 8944296096 \beta_{8} - 12951448236 \beta_{9} + 16028608560 \beta_{10} + 4007152140 \beta_{11}) q^{31}\) \(+(\)\(71\!\cdots\!48\)\( \beta_{1} + \)\(33\!\cdots\!84\)\( \beta_{2} - 66045776612376424 \beta_{3} + 4665108240964864 \beta_{4} + 124785079361104 \beta_{5} - 552908823232 \beta_{6} - 177107979285936 \beta_{7} - 684331616632 \beta_{8} - 61149934912 \beta_{9} - 39060350592 \beta_{10} + 19530175296 \beta_{11}) q^{32}\) \(+(\)\(13\!\cdots\!00\)\( - \)\(14\!\cdots\!56\)\( \beta_{1} + 8879387024881324690 \beta_{2} - 456741562272251909 \beta_{3} - 9131393585961088 \beta_{4} - 104089129481669 \beta_{5} - 11769565726845 \beta_{6} - 257792578577031 \beta_{7} + 317156668497 \beta_{8} - 129203547525 \beta_{9} + 4394202525 \beta_{10} + 6019867917 \beta_{11}) q^{33}\) \(+(-\)\(12\!\cdots\!92\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} - \)\(66\!\cdots\!80\)\( \beta_{2} + 2350878135459852544 \beta_{3} + 2065908864026228 \beta_{4} - 423621099648704 \beta_{5} + 5246623789328 \beta_{6} - 547000856436472 \beta_{7} - 179778339236 \beta_{8} - 170946359820 \beta_{9} - 35327917664 \beta_{10} - 8831979416 \beta_{11}) q^{34}\) \(+(-\)\(29\!\cdots\!61\)\( \beta_{1} + \)\(14\!\cdots\!43\)\( \beta_{2} - 275990364246204327 \beta_{3} + 33918685933254669 \beta_{4} + 18383294005229 \beta_{5} - 988974378769 \beta_{6} - 746575627435001 \beta_{7} + 3331761072190 \beta_{8} - 61230676279 \beta_{9} + 251111743986 \beta_{10} - 125555871993 \beta_{11}) q^{35}\) \(+(\)\(34\!\cdots\!12\)\( + \)\(21\!\cdots\!76\)\( \beta_{1} + \)\(26\!\cdots\!88\)\( \beta_{2} - 8278558496957902829 \beta_{3} - 17017625855172628 \beta_{4} + 474523194770384 \beta_{5} + 63573285199404 \beta_{6} - 1812094572133224 \beta_{7} - 3193801546430 \beta_{8} + 323224856892 \beta_{9} - 39030264992 \beta_{10} - 55062720108 \beta_{11}) q^{36}\) \(+(\)\(47\!\cdots\!58\)\( + \)\(10\!\cdots\!55\)\( \beta_{1} - \)\(34\!\cdots\!91\)\( \beta_{2} + 2342781026935802997 \beta_{3} + 11152858447719181 \beta_{4} + 2781486837289861 \beta_{5} - 98485300096057 \beta_{6} - 2610132662995707 \beta_{7} + 1025005706040 \beta_{8} + 1101280560057 \beta_{9} - 305099416068 \beta_{10} - 76274854017 \beta_{11}) q^{37}\) \(+(\)\(14\!\cdots\!76\)\( \beta_{1} + \)\(76\!\cdots\!08\)\( \beta_{2} - 1624766277285326108 \beta_{3} - 132100191791684963 \beta_{4} - 3183869394820704 \beta_{5} + 21117459348074 \beta_{6} - 4420337647235728 \beta_{7} - 9132691000695 \beta_{8} + 1926996777209 \beta_{9} - 1231661050656 \beta_{10} + 615830525328 \beta_{11}) q^{38}\) \(+(-\)\(76\!\cdots\!34\)\( - \)\(13\!\cdots\!79\)\( \beta_{1} + \)\(72\!\cdots\!09\)\( \beta_{2} + 1077580172680526801 \beta_{3} + 315027944584196545 \beta_{4} + 101910503024889 \beta_{5} - 33661998554067 \beta_{6} - 3877784549232373 \beta_{7} + 14504215394370 \beta_{8} + 2272931233209 \beta_{9} + 174141630390 \beta_{10} + 260738775063 \beta_{11}) q^{39}\) \(+(\)\(42\!\cdots\!80\)\( + \)\(18\!\cdots\!28\)\( \beta_{1} - \)\(62\!\cdots\!88\)\( \beta_{2} - 33329372179070446244 \beta_{3} + 14920173203503356 \beta_{4} + 384833554294668 \beta_{5} + 452711528683644 \beta_{6} - 3542712521085660 \beta_{7} + 1349169565488 \beta_{8} + 332333824860 \beta_{9} + 4067342962512 \beta_{10} + 1016835740628 \beta_{11}) q^{40}\) \(+(\)\(35\!\cdots\!42\)\( \beta_{1} + 67494794109100542200 \beta_{2} - 204897152025114056 \beta_{3} - 510099871767817082 \beta_{4} + 8110948548637176 \beta_{5} - 49685090063320 \beta_{6} - 474170941079512 \beta_{7} + 8520969376272 \beta_{8} - 4295689609096 \beta_{9} + 4485462648240 \beta_{10} - 2242731324120 \beta_{11}) q^{41}\) \(+(-\)\(78\!\cdots\!60\)\( - \)\(82\!\cdots\!69\)\( \beta_{1} + \)\(50\!\cdots\!17\)\( \beta_{2} + \)\(17\!\cdots\!47\)\( \beta_{3} + 266567900272189588 \beta_{4} - 8566461310328533 \beta_{5} - 1246275466114437 \beta_{6} + 6105325195137651 \beta_{7} - 902053117341 \beta_{8} - 13662935489844 \beta_{9} - 147657740004 \beta_{10} - 349184465001 \beta_{11}) q^{42}\) \(+(\)\(84\!\cdots\!98\)\( - \)\(54\!\cdots\!72\)\( \beta_{1} + \)\(18\!\cdots\!84\)\( \beta_{2} - \)\(12\!\cdots\!58\)\( \beta_{3} - 77659796236499044 \beta_{4} - 46190491427937720 \beta_{5} - 315433068454710 \beta_{6} + 15774564802675999 \beta_{7} - 27387719740000 \beta_{8} - 20866380562548 \beta_{9} - 26085356709808 \beta_{10} - 6521339177452 \beta_{11}) q^{43}\) \(+(\)\(21\!\cdots\!02\)\( \beta_{1} - \)\(88\!\cdots\!62\)\( \beta_{2} + 19153566178171980778 \beta_{3} + 2610028277760810008 \beta_{4} + 39389458023550516 \beta_{5} - 236670193131240 \beta_{6} + 52136849945451604 \beta_{7} + 13736428598966 \beta_{8} - 25342002974424 \beta_{9} - 11166557742000 \beta_{10} + 5583278871000 \beta_{11}) q^{44}\) \(+(\)\(16\!\cdots\!00\)\( - \)\(36\!\cdots\!93\)\( \beta_{1} - \)\(57\!\cdots\!40\)\( \beta_{2} + 59125373589661792122 \beta_{3} - 5575811425354427925 \beta_{4} + 33780628955897466 \beta_{5} + 6436682247165714 \beta_{6} + 82416807842135526 \beta_{7} - 377091537456750 \beta_{8} - 1284639232086 \beta_{9} - 3636933975366 \beta_{10} - 4617152160282 \beta_{11}) q^{45}\) \(+(-\)\(25\!\cdots\!28\)\( - \)\(85\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!32\)\( \beta_{2} + \)\(27\!\cdots\!24\)\( \beta_{3} - 792883014490364546 \beta_{4} + 45212823259767812 \beta_{5} - 5099467616926724 \beta_{6} + 199920498566638288 \beta_{7} + 62874762952046 \beta_{8} + 35531021364318 \beta_{9} + 109374966350912 \beta_{10} + 27343741587728 \beta_{11}) q^{46}\) \(+(\)\(93\!\cdots\!34\)\( \beta_{1} - \)\(48\!\cdots\!30\)\( \beta_{2} + \)\(10\!\cdots\!54\)\( \beta_{3} + 4728007542255422170 \beta_{4} - 227864411294166054 \beta_{5} + 1294600523957982 \beta_{6} + 269863941167403030 \beta_{7} + 179370137998332 \beta_{8} + 131586241806162 \beta_{9} + 14174596069092 \beta_{10} - 7087298034546 \beta_{11}) q^{47}\) \(+(\)\(54\!\cdots\!24\)\( - \)\(40\!\cdots\!94\)\( \beta_{1} - \)\(27\!\cdots\!30\)\( \beta_{2} - \)\(12\!\cdots\!42\)\( \beta_{3} - 2834962278236686878 \beta_{4} - 82506475716525270 \beta_{5} - 11161528169741982 \beta_{6} + 469132447148737566 \beta_{7} + 2251330343840592 \beta_{8} + 170789228659314 \beta_{9} + 28617274459224 \beta_{10} + 41515081125750 \beta_{11}) q^{48}\) \(+(\)\(70\!\cdots\!91\)\( - \)\(14\!\cdots\!19\)\( \beta_{1} + \)\(48\!\cdots\!07\)\( \beta_{2} + \)\(89\!\cdots\!07\)\( \beta_{3} - 1142759825013972865 \beta_{4} + 781891184983058199 \beta_{5} + 13871402227684617 \beta_{6} + 350146671345756859 \beta_{7} + 177927147028904 \beta_{8} + 250822373892363 \beta_{9} - 291580907453836 \beta_{10} - 72895226863459 \beta_{11}) q^{49}\) \(+(\)\(69\!\cdots\!85\)\( \beta_{1} - \)\(52\!\cdots\!40\)\( \beta_{2} + 94652895894781046240 \beta_{3} - 34869483910788170420 \beta_{4} + 113763871814760720 \beta_{5} + 17591940541480 \beta_{6} + 257708691980498320 \beta_{7} - 2535453546059700 \beta_{8} + 2976628513180 \beta_{9} + 8116229726880 \beta_{10} - 4058114863440 \beta_{11}) q^{50}\) \(+(\)\(22\!\cdots\!36\)\( - \)\(52\!\cdots\!15\)\( \beta_{1} + \)\(30\!\cdots\!45\)\( \beta_{2} - \)\(42\!\cdots\!01\)\( \beta_{3} + 56470630969983358935 \beta_{4} + 363219131931647919 \beta_{5} - 9227065156617087 \beta_{6} - 175670628352175841 \beta_{7} - 6278759029421982 \beta_{8} - 147169585316133 \beta_{9} - 114813890382978 \beta_{10} - 185652180195435 \beta_{11}) q^{51}\) \(+(-\)\(47\!\cdots\!92\)\( + \)\(61\!\cdots\!36\)\( \beta_{1} - \)\(20\!\cdots\!88\)\( \beta_{2} + \)\(67\!\cdots\!14\)\( \beta_{3} + 550506061546254424 \beta_{4} - 2693942790434222600 \beta_{5} + 65193550916039000 \beta_{6} - 320591519628308760 \beta_{7} - 973360175392800 \beta_{8} - 1044771454510056 \beta_{9} + 285645116469024 \beta_{10} + 71411279117256 \beta_{11}) q^{52}\) \(+(\)\(13\!\cdots\!33\)\( \beta_{1} + \)\(97\!\cdots\!40\)\( \beta_{2} - \)\(22\!\cdots\!68\)\( \beta_{3} - 53689325616661576555 \beta_{4} + 1084488014099023728 \beta_{5} - 10376108592225584 \beta_{6} - 588745895039438000 \beta_{7} + 14720066726850336 \beta_{8} - 1040166488537744 \beta_{9} - 17037528767904 \beta_{10} + 8518764383952 \beta_{11}) q^{53}\) \(+(-\)\(85\!\cdots\!16\)\( - \)\(30\!\cdots\!93\)\( \beta_{1} + \)\(95\!\cdots\!38\)\( \beta_{2} + \)\(12\!\cdots\!13\)\( \beta_{3} + 30700171600402265829 \beta_{4} - 1726339265577814761 \beta_{5} - 6341403035861622 \beta_{6} - 2498445130442085396 \beta_{7} + 5968161071900703 \beta_{8} - 2224232071017021 \beta_{9} + 238071311710992 \beta_{10} + 464611348285020 \beta_{11}) q^{54}\) \(+(-\)\(80\!\cdots\!00\)\( + \)\(19\!\cdots\!40\)\( \beta_{1} - \)\(65\!\cdots\!40\)\( \beta_{2} - \)\(99\!\cdots\!80\)\( \beta_{3} + 18040711147838824000 \beta_{4} - 1336674005692332760 \beta_{5} - 515286617815989530 \beta_{6} - 4790942304851189930 \beta_{7} + 415588554733280 \beta_{8} + 65705058402660 \beta_{9} + 1399533985322480 \beta_{10} + 349883496330620 \beta_{11}) q^{55}\) \(+(\)\(96\!\cdots\!38\)\( \beta_{1} + \)\(19\!\cdots\!02\)\( \beta_{2} - \)\(38\!\cdots\!10\)\( \beta_{3} + \)\(33\!\cdots\!36\)\( \beta_{4} + 3046015278944068660 \beta_{5} - 4662230983726272 \beta_{6} - 10394373288110410508 \beta_{7} - 49117391073265750 \beta_{8} - 517218349685568 \beta_{9} - 339968342086272 \beta_{10} + 169984171043136 \beta_{11}) q^{56}\) \(+(-\)\(33\!\cdots\!22\)\( - \)\(67\!\cdots\!33\)\( \beta_{1} + \)\(81\!\cdots\!27\)\( \beta_{2} + \)\(18\!\cdots\!34\)\( \beta_{3} - \)\(45\!\cdots\!55\)\( \beta_{4} + 4113744962413562298 \beta_{5} + 642124052858904552 \beta_{6} - 10276039024145218200 \beta_{7} + 12979259070724287 \beta_{8} + 7056880774890390 \beta_{9} + 82942310538135 \beta_{10} - 177255447906726 \beta_{11}) q^{57}\) \(+(\)\(64\!\cdots\!60\)\( + \)\(75\!\cdots\!21\)\( \beta_{1} - \)\(25\!\cdots\!43\)\( \beta_{2} - \)\(11\!\cdots\!37\)\( \beta_{3} + 65845957254815049072 \beta_{4} + 16223641211722137087 \beta_{5} + 1324449151124693631 \beta_{6} - 14764314617124570273 \beta_{7} + 2266566299570715 \beta_{8} + 4193359976952648 \beta_{9} - 7707174709527732 \beta_{10} - 1926793677381933 \beta_{11}) q^{58}\) \(+(\)\(52\!\cdots\!46\)\( \beta_{1} + \)\(17\!\cdots\!18\)\( \beta_{2} - \)\(34\!\cdots\!70\)\( \beta_{3} + \)\(44\!\cdots\!16\)\( \beta_{4} - 31858872877577330244 \beta_{5} + 167432704897652244 \beta_{6} - 9621749716825391307 \beta_{7} + 74995233456795048 \beta_{8} + 16955917311284748 \beta_{9} + 1417645476796824 \beta_{10} - 708822738398412 \beta_{11}) q^{59}\) \(+(-\)\(95\!\cdots\!60\)\( - \)\(62\!\cdots\!14\)\( \beta_{1} + \)\(37\!\cdots\!90\)\( \beta_{2} + \)\(54\!\cdots\!22\)\( \beta_{3} - \)\(40\!\cdots\!00\)\( \beta_{4} - 2448963785821358704 \beta_{5} - 3060366604726966380 \beta_{6} - 11448640384504593528 \beta_{7} - 24104952267464466 \beta_{8} - 1344331417498812 \beta_{9} - 2182031293909536 \beta_{10} - 3666027692613780 \beta_{11}) q^{60}\) \(+(-\)\(41\!\cdots\!58\)\( - \)\(23\!\cdots\!85\)\( \beta_{1} + \)\(79\!\cdots\!21\)\( \beta_{2} + \)\(41\!\cdots\!25\)\( \beta_{3} - 11173500384194667351 \beta_{4} + 29846453832643805697 \beta_{5} - 901853566072054869 \beta_{6} + 4761697233976027953 \beta_{7} + 18698291236892376 \beta_{8} + 14383227296388933 \beta_{9} + 17260255762013772 \beta_{10} + 4315063940503443 \beta_{11}) q^{61}\) \(+(-\)\(88\!\cdots\!32\)\( \beta_{1} - \)\(46\!\cdots\!52\)\( \beta_{2} + \)\(90\!\cdots\!16\)\( \beta_{3} - \)\(11\!\cdots\!93\)\( \beta_{4} + 63036188700997702472 \beta_{5} - 400117388636509834 \beta_{6} + 25174116320845633432 \beta_{7} + 106805848208162347 \beta_{8} - 39917227430267869 \beta_{9} + 630076222554096 \beta_{10} - 315038111277048 \beta_{11}) q^{62}\) \(+(\)\(11\!\cdots\!98\)\( + \)\(16\!\cdots\!78\)\( \beta_{1} + \)\(52\!\cdots\!00\)\( \beta_{2} - \)\(35\!\cdots\!36\)\( \beta_{3} + \)\(43\!\cdots\!34\)\( \beta_{4} + 928992749432914762 \beta_{5} + 6048779334061354665 \beta_{6} + 45452273393172186072 \beta_{7} - 97197136379778952 \beta_{8} - 5696602683727608 \beta_{9} + 5972387201487512 \beta_{10} + 15637654542314808 \beta_{11}) q^{63}\) \(+(-\)\(51\!\cdots\!08\)\( - \)\(45\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2} + \)\(97\!\cdots\!72\)\( \beta_{3} - \)\(42\!\cdots\!32\)\( \beta_{4} - \)\(27\!\cdots\!76\)\( \beta_{5} - 1244508573421417648 \beta_{6} + 84407930778774207536 \beta_{7} - 107740805762583488 \beta_{8} - 105705941083547184 \beta_{9} - 8139458716145216 \beta_{10} - 2034864679036304 \beta_{11}) q^{64}\) \(+(-\)\(32\!\cdots\!16\)\( \beta_{1} - \)\(18\!\cdots\!82\)\( \beta_{2} + \)\(35\!\cdots\!18\)\( \beta_{3} - \)\(80\!\cdots\!56\)\( \beta_{4} + 9676347954632648614 \beta_{5} + 150083385159698146 \beta_{6} + 96379626064197214934 \beta_{7} - 771944495587592860 \beta_{8} + 11471784033403486 \beta_{9} - 23577029883775524 \beta_{10} + 11788514941887762 \beta_{11}) q^{65}\) \(+(\)\(22\!\cdots\!20\)\( + \)\(60\!\cdots\!77\)\( \beta_{1} - \)\(10\!\cdots\!87\)\( \beta_{2} - \)\(28\!\cdots\!85\)\( \beta_{3} - \)\(52\!\cdots\!48\)\( \beta_{4} - 55401869220476054097 \beta_{5} - 2031870500054296101 \beta_{6} + \)\(20\!\cdots\!35\)\( \beta_{7} + 248047969748198511 \beta_{8} - 130371777337669200 \beta_{9} - 1730394730562100 \beta_{10} - 27815124255698241 \beta_{11}) q^{66}\) \(+(\)\(45\!\cdots\!98\)\( - \)\(89\!\cdots\!96\)\( \beta_{1} + \)\(29\!\cdots\!92\)\( \beta_{2} - \)\(36\!\cdots\!86\)\( \beta_{3} - \)\(78\!\cdots\!16\)\( \beta_{4} + \)\(46\!\cdots\!64\)\( \beta_{5} - 6661484045922237568 \beta_{6} + \)\(22\!\cdots\!41\)\( \beta_{7} + 160776157736681920 \beta_{8} + 176508858972529224 \beta_{9} - 62930804943389216 \beta_{10} - 15732701235847304 \beta_{11}) q^{67}\) \(+(-\)\(32\!\cdots\!32\)\( \beta_{1} - \)\(58\!\cdots\!08\)\( \beta_{2} + \)\(12\!\cdots\!44\)\( \beta_{3} + \)\(17\!\cdots\!64\)\( \beta_{4} + 60294439746379168880 \beta_{5} - 705313238251805664 \beta_{6} + \)\(34\!\cdots\!84\)\( \beta_{7} + 1320982182049853704 \beta_{8} - 57381672485493024 \beta_{9} + 87664342264583616 \beta_{10} - 43832171132291808 \beta_{11}) q^{68}\) \(+(\)\(59\!\cdots\!44\)\( + \)\(22\!\cdots\!42\)\( \beta_{1} - \)\(13\!\cdots\!74\)\( \beta_{2} + \)\(37\!\cdots\!56\)\( \beta_{3} - \)\(12\!\cdots\!66\)\( \beta_{4} + \)\(19\!\cdots\!20\)\( \beta_{5} - 9475376466174792732 \beta_{6} + \)\(25\!\cdots\!00\)\( \beta_{7} + 978017114780820522 \beta_{8} + 508034926294396272 \beta_{9} - 27749388495617622 \beta_{10} - 13276849590771984 \beta_{11}) q^{69}\) \(+(\)\(43\!\cdots\!00\)\( - \)\(34\!\cdots\!80\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!80\)\( \beta_{3} - \)\(58\!\cdots\!70\)\( \beta_{4} - 98576975691483653880 \beta_{5} + 34535241958597666560 \beta_{6} + \)\(15\!\cdots\!40\)\( \beta_{7} + 1942483956878850 \beta_{8} - 41508419876110530 \beta_{9} + 173803615331957520 \beta_{10} + 43450903832989380 \beta_{11}) q^{70}\) \(+(-\)\(54\!\cdots\!46\)\( \beta_{1} + \)\(23\!\cdots\!66\)\( \beta_{2} - \)\(45\!\cdots\!22\)\( \beta_{3} + \)\(53\!\cdots\!42\)\( \beta_{4} - \)\(13\!\cdots\!34\)\( \beta_{5} + 7612624669238192302 \beta_{6} - \)\(13\!\cdots\!28\)\( \beta_{7} + 323189340836905436 \beta_{8} + 745885075341991714 \beta_{9} - 102515943878850108 \beta_{10} + 51257971939425054 \beta_{11}) q^{71}\) \(+(-\)\(22\!\cdots\!60\)\( + \)\(10\!\cdots\!97\)\( \beta_{1} + \)\(95\!\cdots\!39\)\( \beta_{2} + \)\(55\!\cdots\!03\)\( \beta_{3} - \)\(58\!\cdots\!84\)\( \beta_{4} - \)\(12\!\cdots\!46\)\( \beta_{5} - 7510917896691913836 \beta_{6} - \)\(77\!\cdots\!62\)\( \beta_{7} - 6376026725430051903 \beta_{8} - 512400342674175948 \beta_{9} + 34844168202398832 \beta_{10} + 199800127983637980 \beta_{11}) q^{72}\) \(+(\)\(25\!\cdots\!78\)\( + \)\(50\!\cdots\!48\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!00\)\( \beta_{3} + \)\(44\!\cdots\!28\)\( \beta_{4} + \)\(13\!\cdots\!24\)\( \beta_{5} + 3238380496158483612 \beta_{6} - \)\(10\!\cdots\!76\)\( \beta_{7} + 468616901902371360 \beta_{8} + 495485144823478140 \beta_{9} - 107472971684427120 \beta_{10} - 26868242921106780 \beta_{11}) q^{73}\) \(+(-\)\(16\!\cdots\!98\)\( \beta_{1} + \)\(34\!\cdots\!72\)\( \beta_{2} - \)\(69\!\cdots\!96\)\( \beta_{3} + \)\(51\!\cdots\!40\)\( \beta_{4} + \)\(29\!\cdots\!64\)\( \beta_{5} - 16686077257794828376 \beta_{6} - \)\(18\!\cdots\!40\)\( \beta_{7} - 3560148122829474084 \beta_{8} - 1718590727096418484 \beta_{9} - 333220008779570976 \beta_{10} + 166610004389785488 \beta_{11}) q^{74}\) \(+(\)\(36\!\cdots\!75\)\( + \)\(89\!\cdots\!70\)\( \beta_{1} - \)\(82\!\cdots\!80\)\( \beta_{2} + \)\(12\!\cdots\!55\)\( \beta_{3} + \)\(19\!\cdots\!35\)\( \beta_{4} - \)\(27\!\cdots\!85\)\( \beta_{5} + 49686501510657973035 \beta_{6} - \)\(16\!\cdots\!35\)\( \beta_{7} + 13901274163299729450 \beta_{8} - 91710341080219665 \beta_{9} + 213969175923903510 \beta_{10} - 463917481717034655 \beta_{11}) q^{75}\) \(+(-\)\(17\!\cdots\!96\)\( + \)\(98\!\cdots\!94\)\( \beta_{1} - \)\(32\!\cdots\!66\)\( \beta_{2} + \)\(21\!\cdots\!48\)\( \beta_{3} + \)\(98\!\cdots\!34\)\( \beta_{4} - \)\(80\!\cdots\!82\)\( \beta_{5} - \)\(28\!\cdots\!26\)\( \beta_{6} - \)\(31\!\cdots\!86\)\( \beta_{7} - 2947561855366261128 \beta_{8} - 2887733784058364250 \beta_{9} - 239312285231587512 \beta_{10} - 59828071307896878 \beta_{11}) q^{76}\) \(+(\)\(77\!\cdots\!26\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!92\)\( \beta_{3} - \)\(34\!\cdots\!30\)\( \beta_{4} - 76998986508389370968 \beta_{5} + 166133517778459704 \beta_{6} - \)\(30\!\cdots\!60\)\( \beta_{7} + 1590236627694134384 \beta_{8} + 263519125605446664 \beta_{9} + 1646038492184004624 \beta_{10} - 823019246092002312 \beta_{11}) q^{77}\) \(+(\)\(19\!\cdots\!80\)\( - \)\(22\!\cdots\!38\)\( \beta_{1} + \)\(87\!\cdots\!60\)\( \beta_{2} - \)\(25\!\cdots\!82\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4} - \)\(48\!\cdots\!70\)\( \beta_{5} + \)\(16\!\cdots\!28\)\( \beta_{6} - \)\(37\!\cdots\!08\)\( \beta_{7} - 3186632748694190796 \beta_{8} - 2112976991828622816 \beta_{9} - 783461184888270096 \beta_{10} + 230561556338468628 \beta_{11}) q^{78}\) \(+(-\)\(10\!\cdots\!70\)\( + \)\(12\!\cdots\!14\)\( \beta_{1} - \)\(40\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(13\!\cdots\!18\)\( \beta_{4} + \)\(91\!\cdots\!90\)\( \beta_{5} + \)\(56\!\cdots\!05\)\( \beta_{6} - \)\(25\!\cdots\!98\)\( \beta_{7} + 2993717318749233568 \beta_{8} + 3041738562133090188 \beta_{9} - 192084973535426480 \beta_{10} - 48021243383856620 \beta_{11}) q^{79}\) \(+(\)\(38\!\cdots\!08\)\( \beta_{1} + \)\(36\!\cdots\!16\)\( \beta_{2} - \)\(59\!\cdots\!84\)\( \beta_{3} + \)\(42\!\cdots\!28\)\( \beta_{4} - \)\(35\!\cdots\!32\)\( \beta_{5} + 21609338547072737152 \beta_{6} - \)\(16\!\cdots\!92\)\( \beta_{7} - 4069118354749725520 \beta_{8} + 1786271599095087232 \beta_{9} - 2497748370747909888 \beta_{10} + 1248874185373954944 \beta_{11}) q^{80}\) \(+(\)\(39\!\cdots\!61\)\( - \)\(13\!\cdots\!41\)\( \beta_{1} + \)\(67\!\cdots\!61\)\( \beta_{2} - \)\(18\!\cdots\!73\)\( \beta_{3} + \)\(20\!\cdots\!53\)\( \beta_{4} + \)\(30\!\cdots\!23\)\( \beta_{5} - \)\(95\!\cdots\!03\)\( \beta_{6} - \)\(51\!\cdots\!69\)\( \beta_{7} - 58277725397001134298 \beta_{8} + 6990369049252888119 \beta_{9} - 106546179040877790 \beta_{10} + 1267447363353383697 \beta_{11}) q^{81}\) \(+(-\)\(53\!\cdots\!60\)\( - \)\(33\!\cdots\!78\)\( \beta_{1} + \)\(11\!\cdots\!82\)\( \beta_{2} + \)\(31\!\cdots\!78\)\( \beta_{3} - \)\(26\!\cdots\!28\)\( \beta_{4} + \)\(15\!\cdots\!90\)\( \beta_{5} - 2481455641634305530 \beta_{6} + \)\(76\!\cdots\!74\)\( \beta_{7} + 6966511052594945310 \beta_{8} + 6243971425539174288 \beta_{9} + 2890158508223084088 \beta_{10} + 722539627055771022 \beta_{11}) q^{82}\) \(+(\)\(46\!\cdots\!25\)\( \beta_{1} - \)\(29\!\cdots\!87\)\( \beta_{2} + \)\(58\!\cdots\!19\)\( \beta_{3} - \)\(12\!\cdots\!91\)\( \beta_{4} - \)\(12\!\cdots\!27\)\( \beta_{5} + 60145875895967397139 \beta_{6} + \)\(15\!\cdots\!04\)\( \beta_{7} + 58016291555682070278 \beta_{8} + 5700109194125975749 \beta_{9} - 2096522636471759766 \beta_{10} + 1048261318235879883 \beta_{11}) q^{83}\) \(+(\)\(10\!\cdots\!96\)\( - \)\(25\!\cdots\!10\)\( \beta_{1} - \)\(25\!\cdots\!22\)\( \beta_{2} - \)\(31\!\cdots\!48\)\( \beta_{3} - \)\(21\!\cdots\!44\)\( \beta_{4} - \)\(87\!\cdots\!60\)\( \beta_{5} + \)\(11\!\cdots\!72\)\( \beta_{6} + \)\(27\!\cdots\!36\)\( \beta_{7} + \)\(13\!\cdots\!82\)\( \beta_{8} + 4923449200591814232 \beta_{9} + 6939056804865648048 \beta_{10} - 3134810285984571480 \beta_{11}) q^{84}\) \(+(-\)\(89\!\cdots\!40\)\( - \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(35\!\cdots\!04\)\( \beta_{2} - \)\(74\!\cdots\!08\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(24\!\cdots\!44\)\( \beta_{5} - \)\(72\!\cdots\!52\)\( \beta_{6} + \)\(27\!\cdots\!00\)\( \beta_{7} - 9682675644321885664 \beta_{8} - 9049176281292161220 \beta_{9} - 2533997452118897776 \beta_{10} - 633499363029724444 \beta_{11}) q^{85}\) \(+(\)\(24\!\cdots\!28\)\( \beta_{1} - \)\(49\!\cdots\!56\)\( \beta_{2} + \)\(11\!\cdots\!88\)\( \beta_{3} + \)\(21\!\cdots\!17\)\( \beta_{4} + \)\(17\!\cdots\!84\)\( \beta_{5} - 69771796023555245030 \beta_{6} + \)\(30\!\cdots\!00\)\( \beta_{7} - \)\(11\!\cdots\!15\)\( \beta_{8} - 4668154177939492703 \beta_{9} + 15393502829440212000 \beta_{10} - 7696751414720106000 \beta_{11}) q^{86}\) \(+(\)\(89\!\cdots\!80\)\( - \)\(19\!\cdots\!17\)\( \beta_{1} + \)\(11\!\cdots\!17\)\( \beta_{2} + \)\(34\!\cdots\!45\)\( \beta_{3} + \)\(23\!\cdots\!91\)\( \beta_{4} - \)\(74\!\cdots\!29\)\( \beta_{5} + \)\(13\!\cdots\!12\)\( \beta_{6} + \)\(31\!\cdots\!53\)\( \beta_{7} - 77025593871772468326 \beta_{8} - 36292738765277835567 \beta_{9} - 17446200902645580882 \beta_{10} + 1913647422982861791 \beta_{11}) q^{87}\) \(+(-\)\(24\!\cdots\!40\)\( - \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(38\!\cdots\!12\)\( \beta_{2} - \)\(56\!\cdots\!12\)\( \beta_{3} - \)\(12\!\cdots\!08\)\( \beta_{4} - \)\(20\!\cdots\!28\)\( \beta_{5} - \)\(14\!\cdots\!64\)\( \beta_{6} + \)\(28\!\cdots\!92\)\( \beta_{7} - 13861700818300163280 \beta_{8} - 9295149180944290212 \beta_{9} - 18266206549423492272 \beta_{10} - 4566551637355873068 \beta_{11}) q^{88}\) \(+(\)\(24\!\cdots\!97\)\( \beta_{1} - \)\(42\!\cdots\!97\)\( \beta_{2} + \)\(65\!\cdots\!41\)\( \beta_{3} - \)\(54\!\cdots\!73\)\( \beta_{4} + \)\(96\!\cdots\!77\)\( \beta_{5} - \)\(53\!\cdots\!17\)\( \beta_{6} + \)\(18\!\cdots\!65\)\( \beta_{7} - \)\(13\!\cdots\!18\)\( \beta_{8} - 56547635160107218371 \beta_{9} - 22791985631156942262 \beta_{10} + 11395992815578471131 \beta_{11}) q^{89}\) \(+(\)\(54\!\cdots\!40\)\( - \)\(62\!\cdots\!21\)\( \beta_{1} + \)\(20\!\cdots\!31\)\( \beta_{2} - \)\(18\!\cdots\!87\)\( \beta_{3} + \)\(25\!\cdots\!28\)\( \beta_{4} + \)\(55\!\cdots\!89\)\( \beta_{5} - \)\(33\!\cdots\!63\)\( \beta_{6} - \)\(60\!\cdots\!55\)\( \beta_{7} - \)\(16\!\cdots\!51\)\( \beta_{8} + 20919318174437266080 \beta_{9} - 775264865820395724 \beta_{10} + 2570612660674248069 \beta_{11}) q^{90}\) \(+(-\)\(82\!\cdots\!88\)\( + \)\(38\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!80\)\( \beta_{2} - \)\(18\!\cdots\!96\)\( \beta_{3} - \)\(13\!\cdots\!92\)\( \beta_{4} - \)\(12\!\cdots\!84\)\( \beta_{5} + \)\(26\!\cdots\!38\)\( \beta_{6} - \)\(52\!\cdots\!12\)\( \beta_{7} - 28298465645023167776 \beta_{8} - 42735483608052378780 \beta_{9} + 57748071852116844016 \beta_{10} + 14437017963029211004 \beta_{11}) q^{91}\) \(+(-\)\(50\!\cdots\!00\)\( \beta_{1} + \)\(69\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!76\)\( \beta_{3} - \)\(27\!\cdots\!28\)\( \beta_{4} - \)\(21\!\cdots\!20\)\( \beta_{5} + \)\(10\!\cdots\!16\)\( \beta_{6} - \)\(40\!\cdots\!88\)\( \beta_{7} + \)\(75\!\cdots\!24\)\( \beta_{8} + \)\(10\!\cdots\!56\)\( \beta_{9} - 7638029627095727904 \beta_{10} + 3819014813547863952 \beta_{11}) q^{92}\) \(+(\)\(95\!\cdots\!18\)\( + \)\(40\!\cdots\!74\)\( \beta_{1} + \)\(51\!\cdots\!81\)\( \beta_{2} + \)\(25\!\cdots\!13\)\( \beta_{3} + \)\(86\!\cdots\!44\)\( \beta_{4} - \)\(28\!\cdots\!67\)\( \beta_{5} - \)\(18\!\cdots\!69\)\( \beta_{6} - \)\(81\!\cdots\!35\)\( \beta_{7} + \)\(11\!\cdots\!24\)\( \beta_{8} + 1424477467706228937 \beta_{9} + \)\(10\!\cdots\!32\)\( \beta_{10} - 765588808050446097 \beta_{11}) q^{93}\) \(+(-\)\(14\!\cdots\!12\)\( + \)\(83\!\cdots\!92\)\( \beta_{1} - \)\(27\!\cdots\!68\)\( \beta_{2} + \)\(27\!\cdots\!88\)\( \beta_{3} + \)\(98\!\cdots\!08\)\( \beta_{4} + \)\(57\!\cdots\!32\)\( \beta_{5} + \)\(11\!\cdots\!16\)\( \beta_{6} - \)\(19\!\cdots\!16\)\( \beta_{7} + \)\(19\!\cdots\!20\)\( \beta_{8} + \)\(20\!\cdots\!72\)\( \beta_{9} - 30596433138358829408 \beta_{10} - 7649108284589707352 \beta_{11}) q^{94}\) \(+(-\)\(51\!\cdots\!36\)\( \beta_{1} + \)\(41\!\cdots\!20\)\( \beta_{2} - \)\(83\!\cdots\!96\)\( \beta_{3} + \)\(19\!\cdots\!20\)\( \beta_{4} + \)\(19\!\cdots\!92\)\( \beta_{5} - 10744026988229825912 \beta_{6} - \)\(22\!\cdots\!98\)\( \beta_{7} - \)\(36\!\cdots\!80\)\( \beta_{8} + 8935741579072883608 \beta_{9} + 66734295185972441328 \beta_{10} - 33367147592986220664 \beta_{11}) q^{95}\) \(+(\)\(44\!\cdots\!88\)\( + \)\(16\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} - \)\(34\!\cdots\!88\)\( \beta_{3} - \)\(15\!\cdots\!36\)\( \beta_{4} + \)\(16\!\cdots\!16\)\( \beta_{5} - \)\(14\!\cdots\!24\)\( \beta_{6} - \)\(36\!\cdots\!88\)\( \beta_{7} + \)\(17\!\cdots\!92\)\( \beta_{8} + \)\(22\!\cdots\!96\)\( \beta_{9} - \)\(21\!\cdots\!56\)\( \beta_{10} - 6125621254729497360 \beta_{11}) q^{96}\) \(+(-\)\(34\!\cdots\!22\)\( + \)\(11\!\cdots\!69\)\( \beta_{1} - \)\(39\!\cdots\!61\)\( \beta_{2} + \)\(59\!\cdots\!67\)\( \beta_{3} + \)\(11\!\cdots\!31\)\( \beta_{4} - \)\(31\!\cdots\!37\)\( \beta_{5} - \)\(30\!\cdots\!31\)\( \beta_{6} - \)\(32\!\cdots\!29\)\( \beta_{7} - \)\(16\!\cdots\!40\)\( \beta_{8} - \)\(11\!\cdots\!53\)\( \beta_{9} - \)\(17\!\cdots\!48\)\( \beta_{10} - 43232983876306648087 \beta_{11}) q^{97}\) \(+(-\)\(60\!\cdots\!53\)\( \beta_{1} + \)\(94\!\cdots\!92\)\( \beta_{2} - \)\(18\!\cdots\!32\)\( \beta_{3} + \)\(24\!\cdots\!76\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} + \)\(63\!\cdots\!92\)\( \beta_{6} - \)\(49\!\cdots\!64\)\( \beta_{7} - \)\(23\!\cdots\!52\)\( \beta_{8} + 54384199363688986372 \beta_{9} - 59583395143782054048 \beta_{10} + 29791697571891027024 \beta_{11}) q^{98}\) \(+(\)\(45\!\cdots\!40\)\( + \)\(23\!\cdots\!96\)\( \beta_{1} - \)\(16\!\cdots\!96\)\( \beta_{2} + \)\(16\!\cdots\!40\)\( \beta_{3} - \)\(18\!\cdots\!62\)\( \beta_{4} - \)\(21\!\cdots\!54\)\( \beta_{5} + \)\(41\!\cdots\!40\)\( \beta_{6} - \)\(78\!\cdots\!67\)\( \beta_{7} + \)\(16\!\cdots\!72\)\( \beta_{8} - \)\(37\!\cdots\!02\)\( \beta_{9} + 8626813286837176932 \beta_{10} - 29471156958180255138 \beta_{11}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 372082572q^{3} \) \(\mathstrut -\mathstrut 4781373657072q^{4} \) \(\mathstrut +\mathstrut 8471862485327952q^{6} \) \(\mathstrut -\mathstrut 94341749124766584q^{7} \) \(\mathstrut -\mathstrut 23605502262041939220q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 372082572q^{3} \) \(\mathstrut -\mathstrut 4781373657072q^{4} \) \(\mathstrut +\mathstrut 8471862485327952q^{6} \) \(\mathstrut -\mathstrut 94341749124766584q^{7} \) \(\mathstrut -\mathstrut 23605502262041939220q^{9} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!72\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!76\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!40\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!56\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!20\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!20\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!28\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!20\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!44\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!52\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!96\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!16\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!04\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!44\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!96\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!08\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!60\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!20\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!36\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!88\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!92\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!32\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!04\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!92\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!64\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!20\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!20\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!96\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!76\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!96\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!76\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!28\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!36\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!52\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!60\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!32\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!20\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!60\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!80\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!80\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!56\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!16\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!44\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!56\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!64\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut +\mathstrut \) \(249659905422\) \(x^{10}\mathstrut +\mathstrut \) \(21357900831219724160448\) \(x^{8}\mathstrut +\mathstrut \) \(757005714875983264074654770397184\) \(x^{6}\mathstrut +\mathstrut \) \(10196659004511185116272142565764285511761920\) \(x^{4}\mathstrut +\mathstrut \) \(42071301184141474816096210166597335428414342325862400\) \(x^{2}\mathstrut +\mathstrut \) \(12606130133749306378798090199763505554693706712963960799232000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(55\!\cdots\!75\) \(\nu^{11}\mathstrut -\mathstrut \) \(17\!\cdots\!72\) \(\nu^{10}\mathstrut -\mathstrut \) \(13\!\cdots\!50\) \(\nu^{9}\mathstrut -\mathstrut \) \(37\!\cdots\!24\) \(\nu^{8}\mathstrut -\mathstrut \) \(11\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(23\!\cdots\!36\) \(\nu^{6}\mathstrut -\mathstrut \) \(38\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(46\!\cdots\!68\) \(\nu^{4}\mathstrut -\mathstrut \) \(47\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!00\)\()/\)\(80\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(94\!\cdots\!75\) \(\nu^{11}\mathstrut +\mathstrut \) \(30\!\cdots\!24\) \(\nu^{10}\mathstrut +\mathstrut \) \(23\!\cdots\!50\) \(\nu^{9}\mathstrut +\mathstrut \) \(64\!\cdots\!08\) \(\nu^{8}\mathstrut +\mathstrut \) \(19\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(40\!\cdots\!12\) \(\nu^{6}\mathstrut +\mathstrut \) \(65\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(79\!\cdots\!56\) \(\nu^{4}\mathstrut +\mathstrut \) \(80\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(57\!\cdots\!00\)\()/\)\(40\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(16\!\cdots\!75\) \(\nu^{11}\mathstrut +\mathstrut \) \(14\!\cdots\!32\) \(\nu^{10}\mathstrut +\mathstrut \) \(42\!\cdots\!50\) \(\nu^{9}\mathstrut +\mathstrut \) \(31\!\cdots\!44\) \(\nu^{8}\mathstrut +\mathstrut \) \(41\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(39\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(37\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!00\)\()/\)\(36\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(69\!\cdots\!27\) \(\nu^{11}\mathstrut -\mathstrut \) \(78\!\cdots\!08\) \(\nu^{10}\mathstrut +\mathstrut \) \(14\!\cdots\!34\) \(\nu^{9}\mathstrut -\mathstrut \) \(17\!\cdots\!36\) \(\nu^{8}\mathstrut +\mathstrut \) \(91\!\cdots\!76\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!04\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!88\) \(\nu^{5}\mathstrut -\mathstrut \) \(34\!\cdots\!52\) \(\nu^{4}\mathstrut -\mathstrut \) \(42\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!00\)\()/\)\(50\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(74\!\cdots\!83\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!64\) \(\nu^{10}\mathstrut -\mathstrut \) \(17\!\cdots\!86\) \(\nu^{9}\mathstrut +\mathstrut \) \(30\!\cdots\!88\) \(\nu^{8}\mathstrut -\mathstrut \) \(13\!\cdots\!04\) \(\nu^{7}\mathstrut +\mathstrut \) \(24\!\cdots\!32\) \(\nu^{6}\mathstrut -\mathstrut \) \(40\!\cdots\!52\) \(\nu^{5}\mathstrut +\mathstrut \) \(77\!\cdots\!16\) \(\nu^{4}\mathstrut -\mathstrut \) \(39\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(83\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(84\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!00\)\()/\)\(10\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(31\!\cdots\!75\) \(\nu^{11}\mathstrut -\mathstrut \) \(13\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(78\!\cdots\!50\) \(\nu^{9}\mathstrut -\mathstrut \) \(28\!\cdots\!16\) \(\nu^{8}\mathstrut +\mathstrut \) \(65\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(18\!\cdots\!24\) \(\nu^{6}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(35\!\cdots\!12\) \(\nu^{4}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(90\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(33\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(28\!\cdots\!51\) \(\nu^{11}\mathstrut -\mathstrut \) \(20\!\cdots\!76\) \(\nu^{10}\mathstrut -\mathstrut \) \(70\!\cdots\!42\) \(\nu^{9}\mathstrut -\mathstrut \) \(43\!\cdots\!92\) \(\nu^{8}\mathstrut -\mathstrut \) \(60\!\cdots\!88\) \(\nu^{7}\mathstrut -\mathstrut \) \(30\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(21\!\cdots\!44\) \(\nu^{5}\mathstrut -\mathstrut \) \(85\!\cdots\!44\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(80\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(46\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(38\!\cdots\!00\)\()/\)\(26\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(77\!\cdots\!39\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(15\!\cdots\!38\) \(\nu^{9}\mathstrut -\mathstrut \) \(24\!\cdots\!84\) \(\nu^{8}\mathstrut -\mathstrut \) \(86\!\cdots\!32\) \(\nu^{7}\mathstrut -\mathstrut \) \(17\!\cdots\!76\) \(\nu^{6}\mathstrut -\mathstrut \) \(94\!\cdots\!16\) \(\nu^{5}\mathstrut -\mathstrut \) \(51\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(51\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(30\!\cdots\!00\)\()/\)\(26\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(90\!\cdots\!53\) \(\nu^{11}\mathstrut +\mathstrut \) \(30\!\cdots\!12\) \(\nu^{10}\mathstrut -\mathstrut \) \(22\!\cdots\!26\) \(\nu^{9}\mathstrut +\mathstrut \) \(79\!\cdots\!04\) \(\nu^{8}\mathstrut -\mathstrut \) \(19\!\cdots\!64\) \(\nu^{7}\mathstrut +\mathstrut \) \(69\!\cdots\!56\) \(\nu^{6}\mathstrut -\mathstrut \) \(66\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(22\!\cdots\!28\) \(\nu^{4}\mathstrut -\mathstrut \) \(83\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(29\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(91\!\cdots\!00\)\()/\)\(40\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(62\!\cdots\!61\) \(\nu^{11}\mathstrut +\mathstrut \) \(93\!\cdots\!04\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!62\) \(\nu^{9}\mathstrut +\mathstrut \) \(24\!\cdots\!68\) \(\nu^{8}\mathstrut +\mathstrut \) \(13\!\cdots\!68\) \(\nu^{7}\mathstrut +\mathstrut \) \(21\!\cdots\!52\) \(\nu^{6}\mathstrut +\mathstrut \) \(45\!\cdots\!84\) \(\nu^{5}\mathstrut +\mathstrut \) \(71\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(56\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(55\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(32\!\cdots\!00\)\()/\)\(66\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut -\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(1497959432532\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8}\mathstrut +\mathstrut \) \(50\) \(\beta_{7}\mathstrut -\mathstrut \) \(46\) \(\beta_{5}\mathstrut +\mathstrut \) \(520\) \(\beta_{4}\mathstrut +\mathstrut \) \(18779\) \(\beta_{3}\mathstrut -\mathstrut \) \(90796669\) \(\beta_{2}\mathstrut -\mathstrut \) \(2828274708871\) \(\beta_{1}\)\()/216\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(6233\) \(\beta_{11}\mathstrut -\mathstrut \) \(24932\) \(\beta_{10}\mathstrut +\mathstrut \) \(45981\) \(\beta_{9}\mathstrut +\mathstrut \) \(39748\) \(\beta_{8}\mathstrut -\mathstrut \) \(4824637\) \(\beta_{7}\mathstrut +\mathstrut \) \(5022389\) \(\beta_{6}\mathstrut +\mathstrut \) \(138563233\) \(\beta_{5}\mathstrut +\mathstrut \) \(68570645\) \(\beta_{4}\mathstrut -\mathstrut \) \(1986678705155\) \(\beta_{3}\mathstrut -\mathstrut \) \(79678956711367\) \(\beta_{2}\mathstrut +\mathstrut \) \(23508395452927\) \(\beta_{1}\mathstrut +\mathstrut \) \(2118340804991313961136304\)\()/648\)
\(\nu^{5}\)\(=\)\((\)\(2441271912\) \(\beta_{11}\mathstrut -\mathstrut \) \(4882543824\) \(\beta_{10}\mathstrut -\mathstrut \) \(7643741864\) \(\beta_{9}\mathstrut -\mathstrut \) \(635297265967\) \(\beta_{8}\mathstrut -\mathstrut \) \(49626288105142\) \(\beta_{7}\mathstrut -\mathstrut \) \(69113602904\) \(\beta_{6}\mathstrut +\mathstrut \) \(40886902358986\) \(\beta_{5}\mathstrut +\mathstrut \) \(297265506898848\) \(\beta_{4}\mathstrut -\mathstrut \) \(18579586505549805\) \(\beta_{3}\mathstrut +\mathstrut \) \(91318589410386160395\) \(\beta_{2}\mathstrut +\mathstrut \) \(1191127653018580490440513\) \(\beta_{1}\)\()/972\)
\(\nu^{6}\)\(=\)\((\)\(1385368647505037\) \(\beta_{11}\mathstrut +\mathstrut \) \(5541474590020148\) \(\beta_{10}\mathstrut -\mathstrut \) \(12734841305233953\) \(\beta_{9}\mathstrut -\mathstrut \) \(11349472657728916\) \(\beta_{8}\mathstrut +\mathstrut \) \(2863653658161612097\) \(\beta_{7}\mathstrut -\mathstrut \) \(1176380408761753881\) \(\beta_{6}\mathstrut -\mathstrut \) \(37481724828028500597\) \(\beta_{5}\mathstrut -\mathstrut \) \(24530586590508502009\) \(\beta_{4}\mathstrut +\mathstrut \) \(324363262253487363672767\) \(\beta_{3}\mathstrut +\mathstrut \) \(16316200239948364561726867\) \(\beta_{2}\mathstrut -\mathstrut \) \(4830885755094780983020075\) \(\beta_{1}\mathstrut -\mathstrut \) \(297383661719637063898142061478844784\)\()/972\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(689593161036855895176\) \(\beta_{11}\mathstrut +\mathstrut \) \(1379186322073711790352\) \(\beta_{10}\mathstrut +\mathstrut \) \(2028845819222206115272\) \(\beta_{9}\mathstrut +\mathstrut \) \(116260987782215378878411\) \(\beta_{8}\mathstrut +\mathstrut \) \(10783260511698560410442190\) \(\beta_{7}\mathstrut +\mathstrut \) \(18219678709111493467192\) \(\beta_{6}\mathstrut -\mathstrut \) \(8421294925222348084638898\) \(\beta_{5}\mathstrut -\mathstrut \) \(85038433564476345196096160\) \(\beta_{4}\mathstrut +\mathstrut \) \(4032918957872030054838105025\) \(\beta_{3}\mathstrut -\mathstrut \) \(19859276235983523452399564757271\) \(\beta_{2}\mathstrut -\mathstrut \) \(182578801518053293102585599465000677\) \(\beta_{1}\)\()/1458\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(252489906245020050740605281\) \(\beta_{11}\mathstrut -\mathstrut \) \(1009959624980080202962421124\) \(\beta_{10}\mathstrut +\mathstrut \) \(2622537515885674037519197125\) \(\beta_{9}\mathstrut +\mathstrut \) \(2370047609640653986778591844\) \(\beta_{8}\mathstrut -\mathstrut \) \(667410104744131895194565517157\) \(\beta_{7}\mathstrut +\mathstrut \) \(225569382859098537056660823453\) \(\beta_{6}\mathstrut +\mathstrut \) \(7640610404204258681299352351721\) \(\beta_{5}\mathstrut +\mathstrut \) \(5315472406041045163800994410493\) \(\beta_{4}\mathstrut -\mathstrut \) \(53901158356196696351863652891055259\) \(\beta_{3}\mathstrut -\mathstrut \) \(3015432870502543946247719701061752127\) \(\beta_{2}\mathstrut +\mathstrut \) \(894050806508782730010532915644022583\) \(\beta_{1}\mathstrut +\mathstrut \) \(45584094815384035507481253779161468630420598064\)\()/1458\)
\(\nu^{9}\)\(=\)\((\)\(47919312685515930419370072450360\) \(\beta_{11}\mathstrut -\mathstrut \) \(95838625371031860838740144900720\) \(\beta_{10}\mathstrut -\mathstrut \) \(133223110957051430455987160252920\) \(\beta_{9}\mathstrut -\mathstrut \) \(6810838122238804819690662660755485\) \(\beta_{8}\mathstrut -\mathstrut \) \(682603133493941999405071195175134786\) \(\beta_{7}\mathstrut -\mathstrut \) \(1188473171513966513301761385178120\) \(\beta_{6}\mathstrut +\mathstrut \) \(513682886490565264123996211453466430\) \(\beta_{5}\mathstrut +\mathstrut \) \(5494254210179092822216298240502202464\) \(\beta_{4}\mathstrut -\mathstrut \) \(255157828286487487814620592438589442295\) \(\beta_{3}\mathstrut +\mathstrut \) \(1256677742897561786456521405410844812969329\) \(\beta_{2}\mathstrut +\mathstrut \) \(9800732260678350040451397003697969909123528627\) \(\beta_{1}\)\()/729\)
\(\nu^{10}\)\(=\)\((\)\(14\!\cdots\!55\) \(\beta_{11}\mathstrut +\mathstrut \) \(58\!\cdots\!20\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\!\cdots\!99\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\!\cdots\!44\) \(\beta_{8}\mathstrut +\mathstrut \) \(42\!\cdots\!39\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\!\cdots\!35\) \(\beta_{6}\mathstrut -\mathstrut \) \(47\!\cdots\!35\) \(\beta_{5}\mathstrut -\mathstrut \) \(33\!\cdots\!99\) \(\beta_{4}\mathstrut +\mathstrut \) \(30\!\cdots\!77\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\!\cdots\!49\) \(\beta_{2}\mathstrut -\mathstrut \) \(52\!\cdots\!17\) \(\beta_{1}\mathstrut -\mathstrut \) \(24\!\cdots\!48\)\()/729\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(59\!\cdots\!72\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\!\cdots\!44\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\!\cdots\!84\) \(\beta_{9}\mathstrut +\mathstrut \) \(78\!\cdots\!22\) \(\beta_{8}\mathstrut +\mathstrut \) \(81\!\cdots\!88\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\!\cdots\!24\) \(\beta_{6}\mathstrut -\mathstrut \) \(59\!\cdots\!36\) \(\beta_{5}\mathstrut -\mathstrut \) \(64\!\cdots\!32\) \(\beta_{4}\mathstrut +\mathstrut \) \(30\!\cdots\!30\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\!\cdots\!54\) \(\beta_{2}\mathstrut -\mathstrut \) \(10\!\cdots\!50\) \(\beta_{1}\)\()/729\)

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
337096.i
245022.i
229859.i
127140.i
81647.3i
18015.5i
18015.5i
81647.3i
127140.i
229859.i
245022.i
337096.i
2.02257e6i 8.90215e8 + 3.37123e9i −2.99130e12 2.53853e13i 6.81856e15 1.80053e15i −1.01945e17 3.82628e18i −1.05727e19 + 6.00224e18i −5.13437e19
2.2 1.47013e6i 1.53875e9 3.12889e9i −1.06178e12 3.18557e13i −4.59988e15 2.26216e15i 7.85404e14 5.54756e16i −7.42219e18 9.62912e18i −4.68322e19
2.3 1.37915e6i −3.48296e9 1.63234e8i −8.02549e11 4.92196e13i −2.25124e14 + 4.80353e15i 5.39353e16 4.09556e17i 1.21044e19 + 1.13707e18i 6.78814e19
2.4 762842.i 3.30429e9 + 1.11325e9i 5.17583e11 7.70076e13i 8.49232e14 2.52065e15i −3.01966e15 1.23359e18i 9.67903e18 + 7.35699e18i 5.87447e19
2.5 489884.i −3.83248e8 + 3.46566e9i 8.59526e11 1.65087e14i 1.69777e15 + 1.87747e14i 1.22230e17 9.59700e17i −1.18639e19 2.65641e18i −8.08734e19
2.6 108093.i −2.05308e9 2.81825e9i 1.08783e12 1.13503e14i −3.04633e14 + 2.21924e14i −1.19157e17 2.36436e17i −3.72735e18 + 1.15722e19i −1.22689e19
2.7 108093.i −2.05308e9 + 2.81825e9i 1.08783e12 1.13503e14i −3.04633e14 2.21924e14i −1.19157e17 2.36436e17i −3.72735e18 1.15722e19i −1.22689e19
2.8 489884.i −3.83248e8 3.46566e9i 8.59526e11 1.65087e14i 1.69777e15 1.87747e14i 1.22230e17 9.59700e17i −1.18639e19 + 2.65641e18i −8.08734e19
2.9 762842.i 3.30429e9 1.11325e9i 5.17583e11 7.70076e13i 8.49232e14 + 2.52065e15i −3.01966e15 1.23359e18i 9.67903e18 7.35699e18i 5.87447e19
2.10 1.37915e6i −3.48296e9 + 1.63234e8i −8.02549e11 4.92196e13i −2.25124e14 4.80353e15i 5.39353e16 4.09556e17i 1.21044e19 1.13707e18i 6.78814e19
2.11 1.47013e6i 1.53875e9 + 3.12889e9i −1.06178e12 3.18557e13i −4.59988e15 + 2.26216e15i 7.85404e14 5.54756e16i −7.42219e18 + 9.62912e18i −4.68322e19
2.12 2.02257e6i 8.90215e8 3.37123e9i −2.99130e12 2.53853e13i 6.81856e15 + 1.80053e15i −1.01945e17 3.82628e18i −1.05727e19 6.00224e18i −5.13437e19
\(n\): 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

There are no other newforms in \(S_{41}^{\mathrm{new}}(3, [\chi])\).