LMFDB - Newform orbit 1.96.a.a

Newspace parameters

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 96 $
Character orbit:	$[\chi]$	$=$	1.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$57.1535908815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$x^{8} - x^{7} - $$45\!\cdots\!12$$ x^{6} + $$33\!\cdots\!60$$ x^{5} + $$59\!\cdots\!16$$ x^{4} - $$12\!\cdots\!20$$ x^{3} - $$20\!\cdots\!88$$ x^{2} + $$46\!\cdots\!68$$ x + $$12\!\cdots\!76$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{104}\cdot 3^{38}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

$f(q)$	$=$	$ q +(-729457392285 + \beta_{1}) q^{2} +(-$$11\!\cdots\!60$$ + 20242501 \beta_{1} - \beta_{2}) q^{3} +($$26\!\cdots\!48$$ - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +($$24\!\cdots\!70$$ - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +($$13\!\cdots\!72$$ + $$13\!\cdots\!06$$ \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +($$39\!\cdots\!00$$ + $$16\!\cdots\!42$$ \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-$$18\!\cdots\!20$$ + $$33\!\cdots\!36$$ \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +($$11\!\cdots\!17$$ - $$33\!\cdots\!72$$ \beta_{1} - $$34\!\cdots\!86$$ \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +O(q^{10})$	$ q +(-729457392285 + \beta_{1}) q^{2} +(-$$11\!\cdots\!60$$ + 20242501 \beta_{1} - \beta_{2}) q^{3} +($$26\!\cdots\!48$$ - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +($$24\!\cdots\!70$$ - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +($$13\!\cdots\!72$$ + $$13\!\cdots\!06$$ \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +($$39\!\cdots\!00$$ + $$16\!\cdots\!42$$ \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-$$18\!\cdots\!20$$ + $$33\!\cdots\!36$$ \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +($$11\!\cdots\!17$$ - $$33\!\cdots\!72$$ \beta_{1} - $$34\!\cdots\!86$$ \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +(-$$44\!\cdots\!30$$ + $$40\!\cdots\!26$$ \beta_{1} + $$26\!\cdots\!56$$ \beta_{2} + 1823205551639504480 \beta_{3} + 31201730362844 \beta_{4} + 32408570756 \beta_{5} - 8596448 \beta_{6} + 18144 \beta_{7}) q^{10} +($$66\!\cdots\!52$$ - $$38\!\cdots\!81$$ \beta_{1} + $$40\!\cdots\!65$$ \beta_{2} + 1971098222936635454 \beta_{3} - 2281046769584042 \beta_{4} + 1373746304736 \beta_{5} - 186517774 \beta_{6} - 1124608 \beta_{7}) q^{11} +($$13\!\cdots\!80$$ + $$13\!\cdots\!36$$ \beta_{1} - $$18\!\cdots\!36$$ \beta_{2} + $$34\!\cdots\!28$$ \beta_{3} - 1437017081453628624 \beta_{4} - 130671188070048 \beta_{5} + 46052813520 \beta_{6} + 51404976 \beta_{7}) q^{12} +($$14\!\cdots\!30$$ - $$13\!\cdots\!44$$ \beta_{1} - $$76\!\cdots\!69$$ \beta_{2} + $$43\!\cdots\!38$$ \beta_{3} - 22836391586997725369 \beta_{4} + 1277203065211264 \beta_{5} - 2768620169896 \beta_{6} - 1847131776 \beta_{7}) q^{13} +($$11\!\cdots\!76$$ + $$29\!\cdots\!16$$ \beta_{1} + $$68\!\cdots\!26$$ \beta_{2} + $$36\!\cdots\!08$$ \beta_{3} + $$67\!\cdots\!10$$ \beta_{4} + 113268416831067546 \beta_{5} + 101942184647552 \beta_{6} + 54313261184 \beta_{7}) q^{14} +($$23\!\cdots\!40$$ - $$11\!\cdots\!78$$ \beta_{1} - $$15\!\cdots\!68$$ \beta_{2} + $$13\!\cdots\!85$$ \beta_{3} + $$21\!\cdots\!93$$ \beta_{4} - 4760217497661248368 \beta_{5} - 2699459604617481 \beta_{6} - 1343169464832 \beta_{7}) q^{15} +($$11\!\cdots\!16$$ + $$63\!\cdots\!84$$ \beta_{1} - $$37\!\cdots\!80$$ \beta_{2} + $$35\!\cdots\!16$$ \beta_{3} - $$51\!\cdots\!44$$ \beta_{4} + 86123790987775929904 \beta_{5} + 54813613926115496 \beta_{6} + 28494357719832 \beta_{7}) q^{16} +(-$$17\!\cdots\!30$$ - $$31\!\cdots\!64$$ \beta_{1} - $$10\!\cdots\!74$$ \beta_{2} - $$76\!\cdots\!44$$ \beta_{3} - $$11\!\cdots\!38$$ \beta_{4} - $$63\!\cdots\!64$$ \beta_{5} - 878472313378704156 \beta_{6} - 526281973855680 \beta_{7}) q^{17} +(-$$22\!\cdots\!85$$ + $$79\!\cdots\!33$$ \beta_{1} - $$21\!\cdots\!60$$ \beta_{2} - $$75\!\cdots\!36$$ \beta_{3} + $$10\!\cdots\!28$$ \beta_{4} - $$62\!\cdots\!56$$ \beta_{5} + 11177988501349285056 \beta_{6} + 8559395719398720 \beta_{7}) q^{18} +(-$$65\!\cdots\!80$$ - $$31\!\cdots\!75$$ \beta_{1} - $$57\!\cdots\!09$$ \beta_{2} - $$39\!\cdots\!34$$ \beta_{3} - $$25\!\cdots\!86$$ \beta_{4} + $$23\!\cdots\!48$$ \beta_{5} - $$11\!\cdots\!22$$ \beta_{6} - 123672945181895424 \beta_{7}) q^{19} +($$17\!\cdots\!60$$ + $$47\!\cdots\!28$$ \beta_{1} - $$15\!\cdots\!54$$ \beta_{2} + $$88\!\cdots\!70$$ \beta_{3} - $$20\!\cdots\!88$$ \beta_{4} - $$32\!\cdots\!00$$ \beta_{5} + $$76\!\cdots\!00$$ \beta_{6} + 1598525547088209600 \beta_{7}) q^{20} +($$16\!\cdots\!92$$ - $$29\!\cdots\!24$$ \beta_{1} - $$93\!\cdots\!80$$ \beta_{2} + $$48\!\cdots\!08$$ \beta_{3} + $$30\!\cdots\!80$$ \beta_{4} + $$24\!\cdots\!52$$ \beta_{5} - $$20\!\cdots\!96$$ \beta_{6} - 18582876201795196032 \beta_{7}) q^{21} +(-$$25\!\cdots\!20$$ + $$76\!\cdots\!86$$ \beta_{1} - $$63\!\cdots\!89$$ \beta_{2} - $$60\!\cdots\!52$$ \beta_{3} - $$13\!\cdots\!99$$ \beta_{4} - $$42\!\cdots\!61$$ \beta_{5} - $$37\!\cdots\!76$$ \beta_{6} + $$19\!\cdots\!84$$ \beta_{7}) q^{22} +(-$$89\!\cdots\!80$$ - $$36\!\cdots\!22$$ \beta_{1} + $$49\!\cdots\!56$$ \beta_{2} - $$50\!\cdots\!35$$ \beta_{3} - $$86\!\cdots\!95$$ \beta_{4} - $$13\!\cdots\!60$$ \beta_{5} + $$75\!\cdots\!35$$ \beta_{6} - $$18\!\cdots\!00$$ \beta_{7}) q^{23} +($$36\!\cdots\!60$$ + $$25\!\cdots\!36$$ \beta_{1} + $$77\!\cdots\!24$$ \beta_{2} + $$55\!\cdots\!04$$ \beta_{3} + $$14\!\cdots\!24$$ \beta_{4} + $$18\!\cdots\!32$$ \beta_{5} - $$83\!\cdots\!04$$ \beta_{6} + $$16\!\cdots\!32$$ \beta_{7}) q^{24} +($$10\!\cdots\!75$$ - $$26\!\cdots\!40$$ \beta_{1} + $$10\!\cdots\!40$$ \beta_{2} + $$63\!\cdots\!00$$ \beta_{3} - $$54\!\cdots\!60$$ \beta_{4} - $$13\!\cdots\!20$$ \beta_{5} + $$68\!\cdots\!60$$ \beta_{6} - $$12\!\cdots\!80$$ \beta_{7}) q^{25} +(-$$86\!\cdots\!38$$ + $$38\!\cdots\!38$$ \beta_{1} - $$37\!\cdots\!80$$ \beta_{2} - $$13\!\cdots\!24$$ \beta_{3} - $$24\!\cdots\!92$$ \beta_{4} + $$54\!\cdots\!04$$ \beta_{5} - $$43\!\cdots\!48$$ \beta_{6} + $$89\!\cdots\!84$$ \beta_{7}) q^{26} +($$80\!\cdots\!20$$ + $$84\!\cdots\!74$$ \beta_{1} - $$12\!\cdots\!10$$ \beta_{2} - $$77\!\cdots\!58$$ \beta_{3} + $$35\!\cdots\!14$$ \beta_{4} + $$11\!\cdots\!48$$ \beta_{5} + $$21\!\cdots\!70$$ \beta_{6} - $$57\!\cdots\!56$$ \beta_{7}) q^{27} +($$37\!\cdots\!80$$ + $$20\!\cdots\!88$$ \beta_{1} - $$58\!\cdots\!44$$ \beta_{2} + $$85\!\cdots\!72$$ \beta_{3} - $$13\!\cdots\!76$$ \beta_{4} - $$16\!\cdots\!92$$ \beta_{5} - $$64\!\cdots\!00$$ \beta_{6} + $$33\!\cdots\!64$$ \beta_{7}) q^{28} +($$96\!\cdots\!30$$ + $$45\!\cdots\!36$$ \beta_{1} + $$22\!\cdots\!31$$ \beta_{2} - $$10\!\cdots\!70$$ \beta_{3} - $$36\!\cdots\!37$$ \beta_{4} + $$11\!\cdots\!04$$ \beta_{5} - $$82\!\cdots\!08$$ \beta_{6} - $$17\!\cdots\!36$$ \beta_{7}) q^{29} +(-$$78\!\cdots\!60$$ + $$92\!\cdots\!92$$ \beta_{1} + $$20\!\cdots\!94$$ \beta_{2} - $$16\!\cdots\!20$$ \beta_{3} + $$18\!\cdots\!18$$ \beta_{4} - $$26\!\cdots\!50$$ \beta_{5} + $$15\!\cdots\!00$$ \beta_{6} + $$77\!\cdots\!00$$ \beta_{7}) q^{30} +($$60\!\cdots\!52$$ + $$20\!\cdots\!72$$ \beta_{1} + $$75\!\cdots\!92$$ \beta_{2} - $$10\!\cdots\!88$$ \beta_{3} + $$17\!\cdots\!64$$ \beta_{4} - $$12\!\cdots\!64$$ \beta_{5} - $$10\!\cdots\!56$$ \beta_{6} - $$29\!\cdots\!52$$ \beta_{7}) q^{31} +($$11\!\cdots\!40$$ + $$14\!\cdots\!24$$ \beta_{1} - $$31\!\cdots\!56$$ \beta_{2} + $$47\!\cdots\!76$$ \beta_{3} - $$65\!\cdots\!48$$ \beta_{4} + $$14\!\cdots\!56$$ \beta_{5} + $$30\!\cdots\!24$$ \beta_{6} + $$91\!\cdots\!20$$ \beta_{7}) q^{32} +(-$$19\!\cdots\!20$$ + $$33\!\cdots\!20$$ \beta_{1} - $$14\!\cdots\!54$$ \beta_{2} - $$68\!\cdots\!44$$ \beta_{3} + $$20\!\cdots\!42$$ \beta_{4} - $$63\!\cdots\!48$$ \beta_{5} + $$67\!\cdots\!96$$ \beta_{6} - $$17\!\cdots\!96$$ \beta_{7}) q^{33} +(-$$20\!\cdots\!54$$ - $$87\!\cdots\!54$$ \beta_{1} + $$43\!\cdots\!88$$ \beta_{2} - $$61\!\cdots\!76$$ \beta_{3} + $$79\!\cdots\!24$$ \beta_{4} + $$10\!\cdots\!48$$ \beta_{5} - $$14\!\cdots\!92$$ \beta_{6} - $$23\!\cdots\!64$$ \beta_{7}) q^{34} +($$73\!\cdots\!20$$ - $$29\!\cdots\!84$$ \beta_{1} + $$18\!\cdots\!96$$ \beta_{2} + $$12\!\cdots\!80$$ \beta_{3} - $$77\!\cdots\!96$$ \beta_{4} + $$35\!\cdots\!96$$ \beta_{5} + $$10\!\cdots\!32$$ \beta_{6} + $$40\!\cdots\!04$$ \beta_{7}) q^{35} +($$64\!\cdots\!16$$ - $$45\!\cdots\!56$$ \beta_{1} + $$35\!\cdots\!13$$ \beta_{2} + $$76\!\cdots\!01$$ \beta_{3} + $$20\!\cdots\!56$$ \beta_{4} - $$23\!\cdots\!24$$ \beta_{5} - $$46\!\cdots\!00$$ \beta_{6} - $$21\!\cdots\!00$$ \beta_{7}) q^{36} +(-$$22\!\cdots\!90$$ - $$39\!\cdots\!24$$ \beta_{1} - $$29\!\cdots\!93$$ \beta_{2} - $$15\!\cdots\!06$$ \beta_{3} + $$74\!\cdots\!23$$ \beta_{4} + $$90\!\cdots\!56$$ \beta_{5} + $$14\!\cdots\!80$$ \beta_{6} + $$76\!\cdots\!88$$ \beta_{7}) q^{37} +(-$$20\!\cdots\!80$$ - $$22\!\cdots\!54$$ \beta_{1} - $$67\!\cdots\!87$$ \beta_{2} - $$95\!\cdots\!64$$ \beta_{3} - $$85\!\cdots\!33$$ \beta_{4} + $$50\!\cdots\!45$$ \beta_{5} - $$25\!\cdots\!48$$ \beta_{6} - $$16\!\cdots\!84$$ \beta_{7}) q^{38} +($$58\!\cdots\!04$$ + $$11\!\cdots\!30$$ \beta_{1} - $$90\!\cdots\!28$$ \beta_{2} + $$15\!\cdots\!57$$ \beta_{3} - $$58\!\cdots\!07$$ \beta_{4} - $$30\!\cdots\!44$$ \beta_{5} - $$37\!\cdots\!29$$ \beta_{6} - $$15\!\cdots\!68$$ \beta_{7}) q^{39} +($$48\!\cdots\!00$$ + $$45\!\cdots\!20$$ \beta_{1} + $$20\!\cdots\!20$$ \beta_{2} + $$13\!\cdots\!50$$ \beta_{3} + $$41\!\cdots\!30$$ \beta_{4} + $$98\!\cdots\!20$$ \beta_{5} + $$46\!\cdots\!90$$ \beta_{6} + $$20\!\cdots\!30$$ \beta_{7}) q^{40} +(-$$10\!\cdots\!98$$ + $$61\!\cdots\!32$$ \beta_{1} + $$62\!\cdots\!84$$ \beta_{2} - $$30\!\cdots\!68$$ \beta_{3} - $$63\!\cdots\!56$$ \beta_{4} - $$17\!\cdots\!76$$ \beta_{5} - $$18\!\cdots\!00$$ \beta_{6} - $$10\!\cdots\!00$$ \beta_{7}) q^{41} +(-$$19\!\cdots\!80$$ + $$43\!\cdots\!80$$ \beta_{1} - $$11\!\cdots\!28$$ \beta_{2} - $$99\!\cdots\!36$$ \beta_{3} - $$20\!\cdots\!72$$ \beta_{4} + $$14\!\cdots\!04$$ \beta_{5} + $$41\!\cdots\!76$$ \beta_{6} + $$29\!\cdots\!60$$ \beta_{7}) q^{42} +($$45\!\cdots\!00$$ - $$29\!\cdots\!45$$ \beta_{1} - $$34\!\cdots\!31$$ \beta_{2} + $$20\!\cdots\!00$$ \beta_{3} + $$55\!\cdots\!20$$ \beta_{4} - $$44\!\cdots\!56$$ \beta_{5} - $$18\!\cdots\!32$$ \beta_{6} - $$34\!\cdots\!44$$ \beta_{7}) q^{43} +($$24\!\cdots\!96$$ - $$38\!\cdots\!44$$ \beta_{1} + $$73\!\cdots\!20$$ \beta_{2} + $$10\!\cdots\!04$$ \beta_{3} + $$33\!\cdots\!44$$ \beta_{4} + $$41\!\cdots\!36$$ \beta_{5} - $$24\!\cdots\!16$$ \beta_{6} - $$13\!\cdots\!72$$ \beta_{7}) q^{44} +(-$$20\!\cdots\!10$$ - $$66\!\cdots\!68$$ \beta_{1} + $$10\!\cdots\!99$$ \beta_{2} - $$16\!\cdots\!70$$ \beta_{3} - $$20\!\cdots\!97$$ \beta_{4} - $$14\!\cdots\!00$$ \beta_{5} + $$10\!\cdots\!00$$ \beta_{6} + $$92\!\cdots\!00$$ \beta_{7}) q^{45} +(-$$24\!\cdots\!48$$ - $$23\!\cdots\!68$$ \beta_{1} + $$83\!\cdots\!74$$ \beta_{2} - $$10\!\cdots\!88$$ \beta_{3} + $$38\!\cdots\!14$$ \beta_{4} + $$91\!\cdots\!54$$ \beta_{5} - $$19\!\cdots\!80$$ \beta_{6} - $$28\!\cdots\!60$$ \beta_{7}) q^{46} +($$48\!\cdots\!80$$ + $$29\!\cdots\!12$$ \beta_{1} - $$11\!\cdots\!32$$ \beta_{2} + $$17\!\cdots\!50$$ \beta_{3} + $$75\!\cdots\!90$$ \beta_{4} + $$12\!\cdots\!48$$ \beta_{5} + $$49\!\cdots\!06$$ \beta_{6} + $$36\!\cdots\!52$$ \beta_{7}) q^{47} +($$11\!\cdots\!20$$ + $$20\!\cdots\!96$$ \beta_{1} - $$10\!\cdots\!20$$ \beta_{2} + $$60\!\cdots\!76$$ \beta_{3} - $$69\!\cdots\!48$$ \beta_{4} - $$54\!\cdots\!44$$ \beta_{5} + $$76\!\cdots\!24$$ \beta_{6} + $$92\!\cdots\!20$$ \beta_{7}) q^{48} +($$33\!\cdots\!93$$ + $$27\!\cdots\!12$$ \beta_{1} + $$25\!\cdots\!04$$ \beta_{2} - $$85\!\cdots\!48$$ \beta_{3} - $$47\!\cdots\!36$$ \beta_{4} + $$90\!\cdots\!84$$ \beta_{5} - $$17\!\cdots\!20$$ \beta_{6} - $$67\!\cdots\!40$$ \beta_{7}) q^{49} +(-$$17\!\cdots\!75$$ + $$13\!\cdots\!35$$ \beta_{1} + $$72\!\cdots\!40$$ \beta_{2} - $$39\!\cdots\!00$$ \beta_{3} + $$22\!\cdots\!40$$ \beta_{4} + $$26\!\cdots\!80$$ \beta_{5} + $$14\!\cdots\!60$$ \beta_{6} + $$18\!\cdots\!20$$ \beta_{7}) q^{50} +($$30\!\cdots\!32$$ - $$21\!\cdots\!10$$ \beta_{1} + $$29\!\cdots\!74$$ \beta_{2} + $$49\!\cdots\!34$$ \beta_{3} - $$30\!\cdots\!94$$ \beta_{4} - $$33\!\cdots\!88$$ \beta_{5} + $$32\!\cdots\!02$$ \beta_{6} - $$17\!\cdots\!16$$ \beta_{7}) q^{51} +($$19\!\cdots\!00$$ - $$10\!\cdots\!20$$ \beta_{1} - $$20\!\cdots\!74$$ \beta_{2} + $$36\!\cdots\!50$$ \beta_{3} + $$26\!\cdots\!40$$ \beta_{4} + $$10\!\cdots\!68$$ \beta_{5} + $$20\!\cdots\!96$$ \beta_{6} - $$74\!\cdots\!68$$ \beta_{7}) q^{52} +(-$$36\!\cdots\!10$$ - $$85\!\cdots\!60$$ \beta_{1} - $$62\!\cdots\!33$$ \beta_{2} - $$59\!\cdots\!82$$ \beta_{3} - $$10\!\cdots\!69$$ \beta_{4} + $$14\!\cdots\!12$$ \beta_{5} - $$13\!\cdots\!80$$ \beta_{6} + $$38\!\cdots\!56$$ \beta_{7}) q^{53} +($$55\!\cdots\!20$$ - $$26\!\cdots\!20$$ \beta_{1} + $$42\!\cdots\!38$$ \beta_{2} - $$17\!\cdots\!24$$ \beta_{3} + $$65\!\cdots\!14$$ \beta_{4} + $$12\!\cdots\!66$$ \beta_{5} + $$25\!\cdots\!04$$ \beta_{6} - $$81\!\cdots\!32$$ \beta_{7}) q^{54} +($$97\!\cdots\!40$$ + $$68\!\cdots\!62$$ \beta_{1} + $$19\!\cdots\!84$$ \beta_{2} - $$33\!\cdots\!45$$ \beta_{3} - $$71\!\cdots\!77$$ \beta_{4} - $$25\!\cdots\!00$$ \beta_{5} + $$37\!\cdots\!25$$ \beta_{6} + $$11\!\cdots\!00$$ \beta_{7}) q^{55} +($$90\!\cdots\!80$$ + $$27\!\cdots\!76$$ \beta_{1} + $$11\!\cdots\!08$$ \beta_{2} + $$22\!\cdots\!00$$ \beta_{3} - $$26\!\cdots\!92$$ \beta_{4} + $$67\!\cdots\!32$$ \beta_{5} - $$33\!\cdots\!52$$ \beta_{6} + $$48\!\cdots\!16$$ \beta_{7}) q^{56} +($$18\!\cdots\!40$$ + $$29\!\cdots\!56$$ \beta_{1} - $$45\!\cdots\!58$$ \beta_{2} - $$22\!\cdots\!00$$ \beta_{3} + $$88\!\cdots\!70$$ \beta_{4} - $$23\!\cdots\!36$$ \beta_{5} + $$76\!\cdots\!08$$ \beta_{6} - $$15\!\cdots\!64$$ \beta_{7}) q^{57} +($$29\!\cdots\!30$$ + $$76\!\cdots\!30$$ \beta_{1} - $$72\!\cdots\!76$$ \beta_{2} + $$48\!\cdots\!56$$ \beta_{3} - $$11\!\cdots\!28$$ \beta_{4} - $$28\!\cdots\!52$$ \beta_{5} - $$14\!\cdots\!92$$ \beta_{6} + $$19\!\cdots\!08$$ \beta_{7}) q^{58} +($$30\!\cdots\!60$$ - $$99\!\cdots\!21$$ \beta_{1} + $$18\!\cdots\!25$$ \beta_{2} - $$16\!\cdots\!68$$ \beta_{3} + $$23\!\cdots\!20$$ \beta_{4} + $$68\!\cdots\!88$$ \beta_{5} - $$38\!\cdots\!24$$ \beta_{6} + $$33\!\cdots\!92$$ \beta_{7}) q^{59} +($$51\!\cdots\!20$$ - $$13\!\cdots\!64$$ \beta_{1} - $$42\!\cdots\!84$$ \beta_{2} + $$41\!\cdots\!80$$ \beta_{3} - $$85\!\cdots\!16$$ \beta_{4} - $$45\!\cdots\!84$$ \beta_{5} + $$10\!\cdots\!72$$ \beta_{6} - $$20\!\cdots\!16$$ \beta_{7}) q^{60} +($$32\!\cdots\!02$$ - $$41\!\cdots\!60$$ \beta_{1} + $$75\!\cdots\!19$$ \beta_{2} - $$71\!\cdots\!90$$ \beta_{3} + $$97\!\cdots\!75$$ \beta_{4} - $$52\!\cdots\!04$$ \beta_{5} - $$77\!\cdots\!28$$ \beta_{6} + $$39\!\cdots\!24$$ \beta_{7}) q^{61} +($$13\!\cdots\!80$$ - $$46\!\cdots\!36$$ \beta_{1} + $$38\!\cdots\!96$$ \beta_{2} + $$90\!\cdots\!44$$ \beta_{3} + $$21\!\cdots\!88$$ \beta_{4} - $$22\!\cdots\!96$$ \beta_{5} - $$22\!\cdots\!64$$ \beta_{6} - $$22\!\cdots\!60$$ \beta_{7}) q^{62} +($$21\!\cdots\!60$$ + $$29\!\cdots\!22$$ \beta_{1} - $$25\!\cdots\!72$$ \beta_{2} - $$66\!\cdots\!29$$ \beta_{3} - $$51\!\cdots\!93$$ \beta_{4} + $$43\!\cdots\!04$$ \beta_{5} + $$67\!\cdots\!45$$ \beta_{6} - $$18\!\cdots\!08$$ \beta_{7}) q^{63} +($$50\!\cdots\!28$$ + $$27\!\cdots\!68$$ \beta_{1} - $$30\!\cdots\!64$$ \beta_{2} + $$98\!\cdots\!76$$ \beta_{3} - $$14\!\cdots\!92$$ \beta_{4} + $$19\!\cdots\!88$$ \beta_{5} - $$53\!\cdots\!60$$ \beta_{6} + $$49\!\cdots\!80$$ \beta_{7}) q^{64} +($$88\!\cdots\!40$$ + $$49\!\cdots\!32$$ \beta_{1} - $$69\!\cdots\!08$$ \beta_{2} - $$23\!\cdots\!40$$ \beta_{3} - $$13\!\cdots\!92$$ \beta_{4} - $$88\!\cdots\!08$$ \beta_{5} - $$25\!\cdots\!36$$ \beta_{6} - $$37\!\cdots\!92$$ \beta_{7}) q^{65} +($$21\!\cdots\!44$$ - $$55\!\cdots\!20$$ \beta_{1} + $$35\!\cdots\!12$$ \beta_{2} + $$16\!\cdots\!72$$ \beta_{3} + $$12\!\cdots\!28$$ \beta_{4} + $$95\!\cdots\!76$$ \beta_{5} - $$92\!\cdots\!84$$ \beta_{6} - $$12\!\cdots\!28$$ \beta_{7}) q^{66} +(-$$80\!\cdots\!80$$ + $$21\!\cdots\!01$$ \beta_{1} + $$33\!\cdots\!87$$ \beta_{2} - $$34\!\cdots\!34$$ \beta_{3} - $$21\!\cdots\!98$$ \beta_{4} + $$16\!\cdots\!40$$ \beta_{5} + $$44\!\cdots\!02$$ \beta_{6} + $$43\!\cdots\!76$$ \beta_{7}) q^{67} +($$11\!\cdots\!40$$ - $$37\!\cdots\!28$$ \beta_{1} - $$54\!\cdots\!50$$ \beta_{2} - $$97\!\cdots\!78$$ \beta_{3} - $$11\!\cdots\!76$$ \beta_{4} - $$48\!\cdots\!72$$ \beta_{5} + $$79\!\cdots\!40$$ \beta_{6} - $$50\!\cdots\!56$$ \beta_{7}) q^{68} +(-$$20\!\cdots\!16$$ - $$45\!\cdots\!24$$ \beta_{1} - $$17\!\cdots\!56$$ \beta_{2} + $$15\!\cdots\!12$$ \beta_{3} + $$71\!\cdots\!96$$ \beta_{4} + $$10\!\cdots\!44$$ \beta_{5} - $$64\!\cdots\!04$$ \beta_{6} - $$50\!\cdots\!68$$ \beta_{7}) q^{69} +(-$$19\!\cdots\!80$$ + $$13\!\cdots\!76$$ \beta_{1} - $$13\!\cdots\!68$$ \beta_{2} + $$26\!\cdots\!40$$ \beta_{3} - $$56\!\cdots\!96$$ \beta_{4} + $$43\!\cdots\!00$$ \beta_{5} + $$10\!\cdots\!00$$ \beta_{6} + $$27\!\cdots\!00$$ \beta_{7}) q^{70} +(-$$18\!\cdots\!48$$ + $$85\!\cdots\!70$$ \beta_{1} + $$15\!\cdots\!52$$ \beta_{2} + $$32\!\cdots\!55$$ \beta_{3} + $$10\!\cdots\!75$$ \beta_{4} + $$22\!\cdots\!68$$ \beta_{5} + $$12\!\cdots\!01$$ \beta_{6} - $$38\!\cdots\!08$$ \beta_{7}) q^{71} +(-$$21\!\cdots\!40$$ + $$21\!\cdots\!28$$ \beta_{1} + $$16\!\cdots\!16$$ \beta_{2} - $$29\!\cdots\!37$$ \beta_{3} - $$55\!\cdots\!99$$ \beta_{4} - $$66\!\cdots\!82$$ \beta_{5} - $$69\!\cdots\!33$$ \beta_{6} - $$60\!\cdots\!55$$ \beta_{7}) q^{72} +(-$$20\!\cdots\!30$$ + $$66\!\cdots\!76$$ \beta_{1} + $$65\!\cdots\!26$$ \beta_{2} + $$21\!\cdots\!16$$ \beta_{3} + $$55\!\cdots\!02$$ \beta_{4} - $$27\!\cdots\!00$$ \beta_{5} + $$65\!\cdots\!72$$ \beta_{6} + $$10\!\cdots\!16$$ \beta_{7}) q^{73} +(-$$26\!\cdots\!14$$ - $$74\!\cdots\!62$$ \beta_{1} + $$17\!\cdots\!12$$ \beta_{2} - $$68\!\cdots\!72$$ \beta_{3} + $$12\!\cdots\!16$$ \beta_{4} + $$31\!\cdots\!20$$ \beta_{5} + $$15\!\cdots\!88$$ \beta_{6} - $$15\!\cdots\!04$$ \beta_{7}) q^{74} +(-$$39\!\cdots\!00$$ - $$32\!\cdots\!05$$ \beta_{1} - $$18\!\cdots\!95$$ \beta_{2} + $$15\!\cdots\!00$$ \beta_{3} - $$16\!\cdots\!20$$ \beta_{4} - $$29\!\cdots\!40$$ \beta_{5} - $$40\!\cdots\!80$$ \beta_{6} + $$72\!\cdots\!40$$ \beta_{7}) q^{75} +(-$$11\!\cdots\!40$$ - $$44\!\cdots\!48$$ \beta_{1} + $$62\!\cdots\!16$$ \beta_{2} - $$17\!\cdots\!40$$ \beta_{3} - $$20\!\cdots\!44$$ \beta_{4} - $$32\!\cdots\!96$$ \beta_{5} - $$16\!\cdots\!04$$ \beta_{6} - $$66\!\cdots\!68$$ \beta_{7}) q^{76} +(-$$44\!\cdots\!00$$ + $$18\!\cdots\!48$$ \beta_{1} - $$16\!\cdots\!48$$ \beta_{2} - $$32\!\cdots\!64$$ \beta_{3} - $$67\!\cdots\!28$$ \beta_{4} - $$15\!\cdots\!24$$ \beta_{5} + $$17\!\cdots\!84$$ \beta_{6} + $$12\!\cdots\!60$$ \beta_{7}) q^{77} +($$73\!\cdots\!00$$ + $$79\!\cdots\!56$$ \beta_{1} + $$16\!\cdots\!86$$ \beta_{2} + $$14\!\cdots\!52$$ \beta_{3} + $$34\!\cdots\!94$$ \beta_{4} + $$28\!\cdots\!90$$ \beta_{5} - $$11\!\cdots\!36$$ \beta_{6} + $$51\!\cdots\!12$$ \beta_{7}) q^{78} +($$58\!\cdots\!80$$ + $$80\!\cdots\!64$$ \beta_{1} - $$92\!\cdots\!44$$ \beta_{2} + $$43\!\cdots\!18$$ \beta_{3} - $$10\!\cdots\!06$$ \beta_{4} - $$45\!\cdots\!24$$ \beta_{5} - $$58\!\cdots\!86$$ \beta_{6} - $$37\!\cdots\!12$$ \beta_{7}) q^{79} +($$23\!\cdots\!20$$ + $$73\!\cdots\!16$$ \beta_{1} - $$12\!\cdots\!88$$ \beta_{2} + $$29\!\cdots\!40$$ \beta_{3} - $$10\!\cdots\!36$$ \beta_{4} - $$16\!\cdots\!00$$ \beta_{5} + $$13\!\cdots\!00$$ \beta_{6} + $$69\!\cdots\!00$$ \beta_{7}) q^{80} +($$15\!\cdots\!21$$ + $$63\!\cdots\!88$$ \beta_{1} - $$57\!\cdots\!22$$ \beta_{2} - $$51\!\cdots\!56$$ \beta_{3} + $$55\!\cdots\!30$$ \beta_{4} + $$28\!\cdots\!68$$ \beta_{5} + $$17\!\cdots\!16$$ \beta_{6} + $$18\!\cdots\!72$$ \beta_{7}) q^{81} +($$40\!\cdots\!30$$ - $$26\!\cdots\!06$$ \beta_{1} + $$91\!\cdots\!44$$ \beta_{2} - $$15\!\cdots\!16$$ \beta_{3} + $$22\!\cdots\!28$$ \beta_{4} + $$14\!\cdots\!56$$ \beta_{5} - $$43\!\cdots\!40$$ \beta_{6} - $$30\!\cdots\!72$$ \beta_{7}) q^{82} +($$42\!\cdots\!60$$ - $$26\!\cdots\!51$$ \beta_{1} - $$88\!\cdots\!73$$ \beta_{2} - $$99\!\cdots\!00$$ \beta_{3} - $$21\!\cdots\!00$$ \beta_{4} - $$54\!\cdots\!20$$ \beta_{5} + $$57\!\cdots\!60$$ \beta_{6} + $$47\!\cdots\!20$$ \beta_{7}) q^{83} +($$21\!\cdots\!16$$ - $$62\!\cdots\!64$$ \beta_{1} + $$39\!\cdots\!32$$ \beta_{2} + $$36\!\cdots\!00$$ \beta_{3} - $$78\!\cdots\!72$$ \beta_{4} + $$75\!\cdots\!08$$ \beta_{5} + $$19\!\cdots\!40$$ \beta_{6} + $$10\!\cdots\!80$$ \beta_{7}) q^{84} +($$73\!\cdots\!20$$ - $$69\!\cdots\!04$$ \beta_{1} - $$36\!\cdots\!74$$ \beta_{2} + $$71\!\cdots\!80$$ \beta_{3} + $$24\!\cdots\!74$$ \beta_{4} - $$13\!\cdots\!24$$ \beta_{5} - $$10\!\cdots\!08$$ \beta_{6} - $$12\!\cdots\!76$$ \beta_{7}) q^{85} +(-$$19\!\cdots\!68$$ + $$15\!\cdots\!58$$ \beta_{1} + $$36\!\cdots\!55$$ \beta_{2} - $$39\!\cdots\!88$$ \beta_{3} + $$73\!\cdots\!77$$ \beta_{4} + $$21\!\cdots\!63$$ \beta_{5} + $$73\!\cdots\!72$$ \beta_{6} + $$10\!\cdots\!24$$ \beta_{7}) q^{86} +(-$$68\!\cdots\!40$$ + $$36\!\cdots\!86$$ \beta_{1} - $$14\!\cdots\!04$$ \beta_{2} - $$50\!\cdots\!71$$ \beta_{3} + $$28\!\cdots\!53$$ \beta_{4} + $$31\!\cdots\!08$$ \beta_{5} - $$90\!\cdots\!81$$ \beta_{6} + $$88\!\cdots\!96$$ \beta_{7}) q^{87} +(-$$15\!\cdots\!40$$ + $$51\!\cdots\!12$$ \beta_{1} - $$51\!\cdots\!32$$ \beta_{2} - $$19\!\cdots\!84$$ \beta_{3} - $$48\!\cdots\!68$$ \beta_{4} - $$14\!\cdots\!44$$ \beta_{5} + $$52\!\cdots\!04$$ \beta_{6} + $$21\!\cdots\!60$$ \beta_{7}) q^{88} +(-$$42\!\cdots\!10$$ - $$43\!\cdots\!92$$ \beta_{1} - $$85\!\cdots\!98$$ \beta_{2} + $$60\!\cdots\!20$$ \beta_{3} - $$94\!\cdots\!26$$ \beta_{4} + $$19\!\cdots\!48$$ \beta_{5} - $$68\!\cdots\!92$$ \beta_{6} - $$72\!\cdots\!64$$ \beta_{7}) q^{89} +(-$$43\!\cdots\!10$$ - $$83\!\cdots\!78$$ \beta_{1} + $$11\!\cdots\!32$$ \beta_{2} + $$95\!\cdots\!60$$ \beta_{3} + $$25\!\cdots\!68$$ \beta_{4} + $$44\!\cdots\!32$$ \beta_{5} - $$12\!\cdots\!56$$ \beta_{6} + $$31\!\cdots\!68$$ \beta_{7}) q^{90} +(-$$43\!\cdots\!68$$ - $$71\!\cdots\!04$$ \beta_{1} - $$16\!\cdots\!52$$ \beta_{2} + $$34\!\cdots\!20$$ \beta_{3} - $$19\!\cdots\!52$$ \beta_{4} - $$24\!\cdots\!28$$ \beta_{5} + $$48\!\cdots\!48$$ \beta_{6} + $$36\!\cdots\!16$$ \beta_{7}) q^{91} +(-$$15\!\cdots\!80$$ - $$52\!\cdots\!80$$ \beta_{1} - $$18\!\cdots\!84$$ \beta_{2} - $$30\!\cdots\!56$$ \beta_{3} + $$21\!\cdots\!48$$ \beta_{4} - $$14\!\cdots\!04$$ \beta_{5} - $$51\!\cdots\!40$$ \beta_{6} - $$69\!\cdots\!52$$ \beta_{7}) q^{92} +($$24\!\cdots\!80$$ - $$22\!\cdots\!24$$ \beta_{1} + $$89\!\cdots\!88$$ \beta_{2} + $$18\!\cdots\!68$$ \beta_{3} + $$61\!\cdots\!96$$ \beta_{4} + $$23\!\cdots\!60$$ \beta_{5} - $$78\!\cdots\!24$$ \beta_{6} - $$22\!\cdots\!92$$ \beta_{7}) q^{93} +($$19\!\cdots\!56$$ + $$12\!\cdots\!00$$ \beta_{1} - $$57\!\cdots\!32$$ \beta_{2} + $$59\!\cdots\!36$$ \beta_{3} - $$18\!\cdots\!36$$ \beta_{4} + $$11\!\cdots\!76$$ \beta_{5} + $$98\!\cdots\!24$$ \beta_{6} + $$21\!\cdots\!08$$ \beta_{7}) q^{94} +($$19\!\cdots\!00$$ + $$63\!\cdots\!30$$ \beta_{1} + $$82\!\cdots\!60$$ \beta_{2} + $$17\!\cdots\!25$$ \beta_{3} + $$23\!\cdots\!45$$ \beta_{4} - $$57\!\cdots\!00$$ \beta_{5} - $$23\!\cdots\!25$$ \beta_{6} - $$16\!\cdots\!00$$ \beta_{7}) q^{95} +($$12\!\cdots\!72$$ + $$26\!\cdots\!92$$ \beta_{1} + $$33\!\cdots\!64$$ \beta_{2} + $$64\!\cdots\!08$$ \beta_{3} + $$20\!\cdots\!68$$ \beta_{4} + $$65\!\cdots\!48$$ \beta_{5} + $$44\!\cdots\!40$$ \beta_{6} - $$28\!\cdots\!20$$ \beta_{7}) q^{96} +($$12\!\cdots\!30$$ + $$19\!\cdots\!28$$ \beta_{1} + $$26\!\cdots\!18$$ \beta_{2} - $$35\!\cdots\!68$$ \beta_{3} - $$26\!\cdots\!06$$ \beta_{4} - $$80\!\cdots\!12$$ \beta_{5} - $$28\!\cdots\!20$$ \beta_{6} + $$29\!\cdots\!44$$ \beta_{7}) q^{97} +($$18\!\cdots\!95$$ - $$14\!\cdots\!95$$ \beta_{1} - $$38\!\cdots\!96$$ \beta_{2} + $$15\!\cdots\!84$$ \beta_{3} - $$68\!\cdots\!72$$ \beta_{4} + $$44\!\cdots\!16$$ \beta_{5} - $$85\!\cdots\!20$$ \beta_{6} + $$85\!\cdots\!68$$ \beta_{7}) q^{98} +($$38\!\cdots\!84$$ - $$10\!\cdots\!21$$ \beta_{1} + $$97\!\cdots\!57$$ \beta_{2} - $$83\!\cdots\!20$$ \beta_{3} + $$72\!\cdots\!52$$ \beta_{4} + $$31\!\cdots\!48$$ \beta_{5} + $$20\!\cdots\!92$$ \beta_{6} - $$79\!\cdots\!36$$ \beta_{7}) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8q - 5835659138280q^{2} - $$95\!\cdots\!80$$q^{3} + $$20\!\cdots\!84$$q^{4} + $$19\!\cdots\!60$$q^{5} + $$10\!\cdots\!76$$q^{6} + $$31\!\cdots\!00$$q^{7} - $$14\!\cdots\!60$$q^{8} + $$92\!\cdots\!36$$q^{9} + O(q^{10}) $	$ 8q - 5835659138280q^{2} - $$95\!\cdots\!80$$q^{3} + $$20\!\cdots\!84$$q^{4} + $$19\!\cdots\!60$$q^{5} + $$10\!\cdots\!76$$q^{6} + $$31\!\cdots\!00$$q^{7} - $$14\!\cdots\!60$$q^{8} + $$92\!\cdots\!36$$q^{9} - $$35\!\cdots\!40$$q^{10} + $$53\!\cdots\!16$$q^{11} + $$10\!\cdots\!40$$q^{12} + $$11\!\cdots\!40$$q^{13} + $$88\!\cdots\!08$$q^{14} + $$18\!\cdots\!20$$q^{15} + $$95\!\cdots\!28$$q^{16} - $$13\!\cdots\!40$$q^{17} - $$17\!\cdots\!80$$q^{18} - $$52\!\cdots\!40$$q^{19} + $$13\!\cdots\!80$$q^{20} + $$13\!\cdots\!36$$q^{21} - $$20\!\cdots\!60$$q^{22} - $$71\!\cdots\!40$$q^{23} + $$28\!\cdots\!80$$q^{24} + $$81\!\cdots\!00$$q^{25} - $$69\!\cdots\!04$$q^{26} + $$64\!\cdots\!60$$q^{27} + $$30\!\cdots\!40$$q^{28} + $$77\!\cdots\!40$$q^{29} - $$62\!\cdots\!80$$q^{30} + $$48\!\cdots\!16$$q^{31} + $$91\!\cdots\!20$$q^{32} - $$15\!\cdots\!60$$q^{33} - $$16\!\cdots\!32$$q^{34} + $$58\!\cdots\!60$$q^{35} + $$51\!\cdots\!28$$q^{36} - $$18\!\cdots\!20$$q^{37} - $$16\!\cdots\!40$$q^{38} + $$46\!\cdots\!32$$q^{39} + $$39\!\cdots\!00$$q^{40} - $$87\!\cdots\!84$$q^{41} - $$15\!\cdots\!40$$q^{42} + $$36\!\cdots\!00$$q^{43} + $$19\!\cdots\!68$$q^{44} - $$16\!\cdots\!80$$q^{45} - $$19\!\cdots\!84$$q^{46} + $$38\!\cdots\!40$$q^{47} + $$93\!\cdots\!60$$q^{48} + $$26\!\cdots\!44$$q^{49} - $$14\!\cdots\!00$$q^{50} + $$24\!\cdots\!56$$q^{51} + $$15\!\cdots\!00$$q^{52} - $$29\!\cdots\!80$$q^{53} + $$44\!\cdots\!60$$q^{54} + $$77\!\cdots\!20$$q^{55} + $$72\!\cdots\!40$$q^{56} + $$14\!\cdots\!20$$q^{57} + $$23\!\cdots\!40$$q^{58} + $$24\!\cdots\!80$$q^{59} + $$41\!\cdots\!60$$q^{60} + $$26\!\cdots\!16$$q^{61} + $$10\!\cdots\!40$$q^{62} + $$17\!\cdots\!80$$q^{63} + $$40\!\cdots\!24$$q^{64} + $$71\!\cdots\!20$$q^{65} + $$17\!\cdots\!52$$q^{66} - $$64\!\cdots\!40$$q^{67} + $$91\!\cdots\!20$$q^{68} - $$16\!\cdots\!28$$q^{69} - $$15\!\cdots\!40$$q^{70} - $$14\!\cdots\!84$$q^{71} - $$16\!\cdots\!20$$q^{72} - $$16\!\cdots\!40$$q^{73} - $$20\!\cdots\!12$$q^{74} - $$31\!\cdots\!00$$q^{75} - $$95\!\cdots\!20$$q^{76} - $$35\!\cdots\!00$$q^{77} + $$59\!\cdots\!00$$q^{78} + $$46\!\cdots\!40$$q^{79} + $$18\!\cdots\!60$$q^{80} + $$12\!\cdots\!68$$q^{81} + $$32\!\cdots\!40$$q^{82} + $$34\!\cdots\!80$$q^{83} + $$17\!\cdots\!28$$q^{84} + $$58\!\cdots\!60$$q^{85} - $$15\!\cdots\!44$$q^{86} - $$54\!\cdots\!20$$q^{87} - $$12\!\cdots\!20$$q^{88} - $$34\!\cdots\!80$$q^{89} - $$35\!\cdots\!80$$q^{90} - $$34\!\cdots\!44$$q^{91} - $$12\!\cdots\!40$$q^{92} + $$19\!\cdots\!40$$q^{93} + $$15\!\cdots\!48$$q^{94} + $$15\!\cdots\!00$$q^{95} + $$97\!\cdots\!76$$q^{96} + $$10\!\cdots\!40$$q^{97} + $$14\!\cdots\!60$$q^{98} + $$30\!\cdots\!72$$q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - x^{7} - $$45\!\cdots\!12$$ x^{6} + $$33\!\cdots\!60$$ x^{5} + $$59\!\cdots\!16$$ x^{4} - $$12\!\cdots\!20$$ x^{3} - $$20\!\cdots\!88$$ x^{2} + $$46\!\cdots\!68$$ x + $$12\!\cdots\!76$:

$\beta_{0}$	$=$	$ 1 $
$\beta_{1}$	$=$	$ 24 \nu - 3 $
$\beta_{2}$	$=$	$($$76\!\cdots\!51$$ \nu^{7} + $$14\!\cdots\!81$$ \nu^{6} - $$52\!\cdots\!16$$ \nu^{5} - $$40\!\cdots\!00$$ \nu^{4} + $$90\!\cdots\!96$$ \nu^{3} + $$14\!\cdots\!32$$ \nu^{2} - $$29\!\cdots\!00$$ \nu - $$69\!\cdots\!36$$)/ $$58\!\cdots\!72$
$\beta_{3}$	$=$	$($$52\!\cdots\!49$$ \nu^{7} + $$10\!\cdots\!19$$ \nu^{6} - $$36\!\cdots\!84$$ \nu^{5} - $$27\!\cdots\!00$$ \nu^{4} + $$62\!\cdots\!04$$ \nu^{3} + $$47\!\cdots\!76$$ \nu^{2} - $$16\!\cdots\!16$$ \nu - $$43\!\cdots\!12$$)/ $$64\!\cdots\!08$
$\beta_{4}$	$=$	$($$-$$13\!\cdots\!97$$ \nu^{7} - $$63\!\cdots\!15$$ \nu^{6} + $$51\!\cdots\!24$$ \nu^{5} + $$41\!\cdots\!84$$ \nu^{4} - $$50\!\cdots\!88$$ \nu^{3} - $$67\!\cdots\!08$$ \nu^{2} + $$12\!\cdots\!68$$ \nu - $$31\!\cdots\!68$$)/ $$73\!\cdots\!00$
$\beta_{5}$	$=$	$($$-$$60\!\cdots\!17$$ \nu^{7} + $$10\!\cdots\!85$$ \nu^{6} + $$17\!\cdots\!64$$ \nu^{5} - $$26\!\cdots\!76$$ \nu^{4} - $$11\!\cdots\!68$$ \nu^{3} + $$76\!\cdots\!12$$ \nu^{2} + $$33\!\cdots\!48$$ \nu + $$10\!\cdots\!52$$)/ $$18\!\cdots\!00$
$\beta_{6}$	$=$	$($$-$$14\!\cdots\!57$$ \nu^{7} - $$12\!\cdots\!15$$ \nu^{6} + $$58\!\cdots\!44$$ \nu^{5} + $$43\!\cdots\!04$$ \nu^{4} - $$62\!\cdots\!28$$ \nu^{3} - $$32\!\cdots\!48$$ \nu^{2} + $$15\!\cdots\!08$$ \nu + $$49\!\cdots\!92$$)/ $$12\!\cdots\!00$
$\beta_{7}$	$=$	$($$37\!\cdots\!87$$ \nu^{7} + $$30\!\cdots\!65$$ \nu^{6} - $$21\!\cdots\!04$$ \nu^{5} - $$12\!\cdots\!64$$ \nu^{4} + $$51\!\cdots\!48$$ \nu^{3} + $$11\!\cdots\!68$$ \nu^{2} - $$54\!\cdots\!28$$ \nu - $$15\!\cdots\!72$$)/ $$18\!\cdots\!00$

Display $\nu^j$ in terms of $\beta_i$

$1$	$=$	$\beta_0$
$\nu$	$=$	$($$\beta_{1} + 3$$)/24$
$\nu^{2}$	$=$	$($$\beta_{3} - 62091 \beta_{2} - 26761022262152 \beta_{1} + 65733123937758428360847400200$$)/576$
$\nu^{3}$	$=$	$($$\beta_{7} + 295 \beta_{6} + 160002 \beta_{5} - 2004356999 \beta_{4} + 51851570017067 \beta_{3} + 3276903991940428836 \beta_{2} + 112986517961279136356684832712 \beta_{1} - 1759085591639276897686588409592076569213504$$)/13824$
$\nu^{4}$	$=$	$($$3926523411123 \beta_{7} + 6959296706126917 \beta_{6} + 10823831194312423526 \beta_{5} - 64844235599751336866149 \beta_{4} + 19321453573813038341957137905 \beta_{3} - 1394410694473382624668011668515956 \beta_{2} - 298710901026801058381966898320637168298536 \beta_{1} + 928369598546200014171379235704963029862290826589385463872$$)/41472$
$\nu^{5}$	$=$	$($$2621067272289636342635857733 \beta_{7} + 781805958213198104016871956611 \beta_{6} + 626720668829570339117854559945610 \beta_{5} - 6015663869305187979638343702232227939 \beta_{4} + 205850449531108259889513399496641754951863 \beta_{3} - 41887807299957782555913661895122244965673155244 \beta_{2} + 229515919805726057201751531896339304137050604018359815144 \beta_{1} - 2454400080858624658714224866616934605524010680291289642149797035267648$$)/124416$
$\nu^{6}$	$=$	$($$2456698967573154102027892526992622040587 \beta_{7} + 2287336922187280642817740919038569124333229 \beta_{6} + 4173765778599191392471236993532318130284922646 \beta_{5} - 22719176458145430265227683260813234396944017761485 \beta_{4} + 4820608307462255271214311913264579460921736185013012921 \beta_{3} - 421824107166754602266817224392645212904577134947783234195476 \beta_{2} + 14896020318557401322538306359430870429741697951273163415269325560472 \beta_{1} + 209538866711613197149097879153829515723664322115869508558284660196534474158205620800$$)/41472$
$\nu^{7}$	$=$	$($$22\!\cdots\!91$$ \beta_{7} + $$67\!\cdots\!33$$ \beta_{6} + $$63\!\cdots\!18$$ \beta_{5} - $$57\!\cdots\!61$$ \beta_{4} + $$23\!\cdots\!85$$ \beta_{3} - $$59\!\cdots\!32$$ \beta_{2} + $$18\!\cdots\!88$$ \beta_{1} + $$40\!\cdots\!88$$)/41472$

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

−1.48943e13
−1.46162e13
−6.56924e12
−1.67307e12
4.79736e12
6.86315e12
9.80521e12
1.62871e13

−3.58194e14

−7.12911e22

8.86885e28

−1.64420e33

2.55360e37

−1.35904e40

−1.75781e43

2.96152e45

5.88943e47

1.2

−3.51519e14

6.56090e22

8.39514e28

2.79946e33

−2.30628e37

5.53054e39

−1.55854e43

2.18364e45

−9.84062e47

1.3

−1.58391e14

9.80327e21

−1.45263e28

−7.31691e32

−1.55275e36

−1.37055e39

8.57536e42

−2.02479e45

1.15893e47

1.4

−4.08831e13

−7.46060e22

−3.79427e28

2.52173e33

3.05013e36

2.11876e40

3.17076e42

3.44515e45

−1.03096e47

1.5

1.14407e14

8.75958e22

−2.65251e28

−1.07741e33

1.00216e37

1.86215e40

−7.56680e42

5.55214e45

−1.23264e47

1.6

1.63986e14

6.46739e21

−1.27226e28

1.20020e33

1.06056e36

−2.14893e40

−8.58249e42

−2.07907e45

1.96817e47

1.7

2.34596e14

−5.48368e22

1.54210e28

−2.52565e33

−1.28645e37

4.80687e39

−5.67559e42

8.86182e44

−5.92506e47

1.8

3.90162e14

2.16924e22

1.12612e29

1.39914e33

8.46353e36

1.75361e40

2.84812e43

−1.65034e45

5.45891e47

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1.96.a.a	✓	8
3.b	odd	2	1		9.96.a.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.96.a.a	✓	8	1.a	even	1	1	trivial
9.96.a.c		8	3.b	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{96}^{\mathrm{new}}(\Gamma_0(1))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$14\!\cdots\!36$$ + $$20\!\cdots\!20$$ T - $$40\!\cdots\!84$$ T^{2} - $$92\!\cdots\!60$$ T^{3} + $$19\!\cdots\!36$$ T^{4} + $$35\!\cdots\!60$$ T^{5} - $$26\!\cdots\!64$$ T^{6} + 5835659138280 T^{7} + T^{8} $
$3$	$ -$$23\!\cdots\!24$$ + $$65\!\cdots\!80$$ T - $$49\!\cdots\!96$$ T^{2} + $$23\!\cdots\!40$$ T^{3} + $$49\!\cdots\!56$$ T^{4} - $$13\!\cdots\!60$$ T^{5} - $$13\!\cdots\!96$$ T^{6} + $$95\!\cdots\!80$$ T^{7} + T^{8} $
$5$	$38\!\cdots\!00$$ + $$38\!\cdots\!00$$ T - $$72\!\cdots\!00$$ T^{2} - $$51\!\cdots\!00$$ T^{3} + $$48\!\cdots\!00$$ T^{4} + $$19\!\cdots\!00$$ T^{5} - $$12\!\cdots\!00$$ T^{6} - $$19\!\cdots\!60$$ T^{7} + T^{8} $
$7$	$ -$$73\!\cdots\!84$$ - $$22\!\cdots\!00$$ T + $$19\!\cdots\!76$$ T^{2} - $$25\!\cdots\!00$$ T^{3} - $$63\!\cdots\!24$$ T^{4} + $$20\!\cdots\!00$$ T^{5} - $$41\!\cdots\!44$$ T^{6} - $$31\!\cdots\!00$$ T^{7} + T^{8} $
$11$	$ -$$36\!\cdots\!44$$ + $$66\!\cdots\!76$$ T + $$16\!\cdots\!92$$ T^{2} - $$18\!\cdots\!92$$ T^{3} - $$30\!\cdots\!80$$ T^{4} + $$70\!\cdots\!52$$ T^{5} - $$89\!\cdots\!88$$ T^{6} - $$53\!\cdots\!16$$ T^{7} + T^{8} $
$13$	$ -$$54\!\cdots\!64$$ + $$66\!\cdots\!60$$ T - $$69\!\cdots\!16$$ T^{2} - $$71\!\cdots\!20$$ T^{3} + $$58\!\cdots\!36$$ T^{4} + $$18\!\cdots\!80$$ T^{5} - $$15\!\cdots\!36$$ T^{6} - $$11\!\cdots\!40$$ T^{7} + T^{8} $
$17$	$18\!\cdots\!76$$ + $$17\!\cdots\!60$$ T + $$57\!\cdots\!96$$ T^{2} + $$77\!\cdots\!20$$ T^{3} + $$24\!\cdots\!56$$ T^{4} - $$22\!\cdots\!20$$ T^{5} - $$12\!\cdots\!04$$ T^{6} + $$13\!\cdots\!40$$ T^{7} + T^{8} $
$19$	$21\!\cdots\!00$$ + $$45\!\cdots\!00$$ T + $$10\!\cdots\!00$$ T^{2} + $$80\!\cdots\!00$$ T^{3} + $$15\!\cdots\!00$$ T^{4} - $$41\!\cdots\!00$$ T^{5} - $$10\!\cdots\!00$$ T^{6} + $$52\!\cdots\!40$$ T^{7} + T^{8} $
$23$	$ -$$15\!\cdots\!04$$ + $$28\!\cdots\!40$$ T - $$28\!\cdots\!36$$ T^{2} - $$74\!\cdots\!80$$ T^{3} + $$10\!\cdots\!16$$ T^{4} - $$63\!\cdots\!80$$ T^{5} - $$71\!\cdots\!76$$ T^{6} + $$71\!\cdots\!40$$ T^{7} + T^{8} $
$29$	$26\!\cdots\!00$$ - $$53\!\cdots\!00$$ T + $$36\!\cdots\!00$$ T^{2} - $$51\!\cdots\!00$$ T^{3} - $$43\!\cdots\!00$$ T^{4} + $$17\!\cdots\!00$$ T^{5} - $$36\!\cdots\!00$$ T^{6} - $$77\!\cdots\!40$$ T^{7} + T^{8} $
$31$	$25\!\cdots\!56$$ + $$15\!\cdots\!76$$ T - $$10\!\cdots\!08$$ T^{2} - $$14\!\cdots\!92$$ T^{3} + $$72\!\cdots\!20$$ T^{4} + $$50\!\cdots\!52$$ T^{5} - $$15\!\cdots\!88$$ T^{6} - $$48\!\cdots\!16$$ T^{7} + T^{8} $
$37$	$73\!\cdots\!96$$ - $$70\!\cdots\!20$$ T - $$10\!\cdots\!64$$ T^{2} + $$29\!\cdots\!60$$ T^{3} + $$33\!\cdots\!16$$ T^{4} - $$39\!\cdots\!60$$ T^{5} - $$33\!\cdots\!24$$ T^{6} + $$18\!\cdots\!20$$ T^{7} + T^{8} $
$41$	$35\!\cdots\!56$$ + $$12\!\cdots\!76$$ T + $$13\!\cdots\!92$$ T^{2} + $$36\!\cdots\!08$$ T^{3} - $$98\!\cdots\!80$$ T^{4} - $$54\!\cdots\!48$$ T^{5} - $$40\!\cdots\!88$$ T^{6} + $$87\!\cdots\!84$$ T^{7} + T^{8} $
$43$	$71\!\cdots\!16$$ - $$12\!\cdots\!00$$ T + $$37\!\cdots\!24$$ T^{2} - $$65\!\cdots\!00$$ T^{3} - $$95\!\cdots\!24$$ T^{4} + $$10\!\cdots\!00$$ T^{5} - $$22\!\cdots\!56$$ T^{6} - $$36\!\cdots\!00$$ T^{7} + T^{8} $
$47$	$37\!\cdots\!56$$ + $$18\!\cdots\!40$$ T - $$60\!\cdots\!44$$ T^{2} - $$62\!\cdots\!20$$ T^{3} + $$46\!\cdots\!96$$ T^{4} + $$32\!\cdots\!20$$ T^{5} - $$12\!\cdots\!84$$ T^{6} - $$38\!\cdots\!40$$ T^{7} + T^{8} $
$53$	$69\!\cdots\!76$$ + $$50\!\cdots\!80$$ T - $$13\!\cdots\!96$$ T^{2} - $$24\!\cdots\!60$$ T^{3} + $$28\!\cdots\!56$$ T^{4} + $$18\!\cdots\!40$$ T^{5} - $$15\!\cdots\!96$$ T^{6} + $$29\!\cdots\!80$$ T^{7} + T^{8} $
$59$	$ -$$10\!\cdots\!00$$ + $$27\!\cdots\!00$$ T + $$47\!\cdots\!00$$ T^{2} - $$39\!\cdots\!00$$ T^{3} + $$88\!\cdots\!00$$ T^{4} + $$17\!\cdots\!00$$ T^{5} - $$62\!\cdots\!00$$ T^{6} - $$24\!\cdots\!80$$ T^{7} + T^{8} $
$61$	$79\!\cdots\!56$$ - $$75\!\cdots\!24$$ T + $$19\!\cdots\!92$$ T^{2} + $$18\!\cdots\!08$$ T^{3} - $$12\!\cdots\!80$$ T^{4} + $$10\!\cdots\!52$$ T^{5} + $$15\!\cdots\!12$$ T^{6} - $$26\!\cdots\!16$$ T^{7} + T^{8} $
$67$	$13\!\cdots\!76$$ - $$40\!\cdots\!40$$ T - $$39\!\cdots\!04$$ T^{2} + $$45\!\cdots\!20$$ T^{3} + $$55\!\cdots\!56$$ T^{4} - $$11\!\cdots\!20$$ T^{5} - $$15\!\cdots\!04$$ T^{6} + $$64\!\cdots\!40$$ T^{7} + T^{8} $
$71$	$48\!\cdots\!56$$ - $$73\!\cdots\!24$$ T - $$12\!\cdots\!08$$ T^{2} + $$15\!\cdots\!08$$ T^{3} + $$10\!\cdots\!20$$ T^{4} - $$91\!\cdots\!48$$ T^{5} - $$46\!\cdots\!88$$ T^{6} + $$14\!\cdots\!84$$ T^{7} + T^{8} $
$73$	$28\!\cdots\!96$$ + $$23\!\cdots\!40$$ T - $$19\!\cdots\!36$$ T^{2} - $$26\!\cdots\!80$$ T^{3} - $$70\!\cdots\!84$$ T^{4} + $$12\!\cdots\!20$$ T^{5} + $$94\!\cdots\!24$$ T^{6} + $$16\!\cdots\!40$$ T^{7} + T^{8} $
$79$	$99\!\cdots\!00$$ - $$18\!\cdots\!00$$ T + $$79\!\cdots\!00$$ T^{2} + $$11\!\cdots\!00$$ T^{3} - $$59\!\cdots\!00$$ T^{4} + $$28\!\cdots\!00$$ T^{5} + $$50\!\cdots\!00$$ T^{6} - $$46\!\cdots\!40$$ T^{7} + T^{8} $
$83$	$ -$$23\!\cdots\!44$$ - $$40\!\cdots\!80$$ T + $$23\!\cdots\!44$$ T^{2} - $$18\!\cdots\!40$$ T^{3} - $$12\!\cdots\!04$$ T^{4} + $$17\!\cdots\!60$$ T^{5} - $$19\!\cdots\!16$$ T^{6} - $$34\!\cdots\!80$$ T^{7} + T^{8} $
$89$	$56\!\cdots\!00$$ - $$15\!\cdots\!00$$ T - $$84\!\cdots\!00$$ T^{2} + $$30\!\cdots\!00$$ T^{3} + $$12\!\cdots\!00$$ T^{4} - $$18\!\cdots\!00$$ T^{5} - $$61\!\cdots\!00$$ T^{6} + $$34\!\cdots\!80$$ T^{7} + T^{8} $
$97$	$21\!\cdots\!56$$ + $$62\!\cdots\!40$$ T - $$14\!\cdots\!44$$ T^{2} - $$74\!\cdots\!20$$ T^{3} + $$32\!\cdots\!96$$ T^{4} + $$16\!\cdots\!20$$ T^{5} - $$30\!\cdots\!84$$ T^{6} - $$10\!\cdots\!40$$ T^{7} + T^{8} $
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1 \)
Weight:	\( k \)	\(=\)	\( 96 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\( q +(-729457392285 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3} +(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +O(q^{10})\)	\( q +(-729457392285 + \beta_{1}) q^{2} +(-\)\(11\!\cdots\!60\)\( + 20242501 \beta_{1} - \beta_{2}) q^{3} +(\)\(26\!\cdots\!48\)\( - 28219937046728 \beta_{1} - 62091 \beta_{2} + \beta_{3}) q^{4} +(\)\(24\!\cdots\!70\)\( - 673115573808572444 \beta_{1} - 7584146833 \beta_{2} + 2590 \beta_{3} - \beta_{4}) q^{5} +(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!06\)\( \beta_{1} + 69343623947177 \beta_{2} + 23148272 \beta_{3} + 2031 \beta_{4} + \beta_{5}) q^{6} +(\)\(39\!\cdots\!00\)\( + \)\(16\!\cdots\!42\)\( \beta_{1} - 46292780348852220 \beta_{2} - 13817544041 \beta_{3} - 1881757 \beta_{4} - 16 \beta_{5} + \beta_{6}) q^{7} +(-\)\(18\!\cdots\!20\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} + 3412782208774091460 \beta_{2} + 49663197840203 \beta_{3} - 2004356999 \beta_{4} + 160002 \beta_{5} + 295 \beta_{6} + \beta_{7}) q^{8} +(\)\(11\!\cdots\!17\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(34\!\cdots\!86\)\( \beta_{2} - 10116482261368704 \beta_{3} + 196679311398 \beta_{4} + 45983040 \beta_{5} + 121524 \beta_{6} - 192 \beta_{7}) q^{9} +(-\)\(44\!\cdots\!30\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} + \)\(26\!\cdots\!56\)\( \beta_{2} + 1823205551639504480 \beta_{3} + 31201730362844 \beta_{4} + 32408570756 \beta_{5} - 8596448 \beta_{6} + 18144 \beta_{7}) q^{10} +(\)\(66\!\cdots\!52\)\( - \)\(38\!\cdots\!81\)\( \beta_{1} + \)\(40\!\cdots\!65\)\( \beta_{2} + 1971098222936635454 \beta_{3} - 2281046769584042 \beta_{4} + 1373746304736 \beta_{5} - 186517774 \beta_{6} - 1124608 \beta_{7}) q^{11} +(\)\(13\!\cdots\!80\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!36\)\( \beta_{2} + \)\(34\!\cdots\!28\)\( \beta_{3} - 1437017081453628624 \beta_{4} - 130671188070048 \beta_{5} + 46052813520 \beta_{6} + 51404976 \beta_{7}) q^{12} +(\)\(14\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(76\!\cdots\!69\)\( \beta_{2} + \)\(43\!\cdots\!38\)\( \beta_{3} - 22836391586997725369 \beta_{4} + 1277203065211264 \beta_{5} - 2768620169896 \beta_{6} - 1847131776 \beta_{7}) q^{13} +(\)\(11\!\cdots\!76\)\( + \)\(29\!\cdots\!16\)\( \beta_{1} + \)\(68\!\cdots\!26\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3} + \)\(67\!\cdots\!10\)\( \beta_{4} + 113268416831067546 \beta_{5} + 101942184647552 \beta_{6} + 54313261184 \beta_{7}) q^{14} +(\)\(23\!\cdots\!40\)\( - \)\(11\!\cdots\!78\)\( \beta_{1} - \)\(15\!\cdots\!68\)\( \beta_{2} + \)\(13\!\cdots\!85\)\( \beta_{3} + \)\(21\!\cdots\!93\)\( \beta_{4} - 4760217497661248368 \beta_{5} - 2699459604617481 \beta_{6} - 1343169464832 \beta_{7}) q^{15} +(\)\(11\!\cdots\!16\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} + \)\(35\!\cdots\!16\)\( \beta_{3} - \)\(51\!\cdots\!44\)\( \beta_{4} + 86123790987775929904 \beta_{5} + 54813613926115496 \beta_{6} + 28494357719832 \beta_{7}) q^{16} +(-\)\(17\!\cdots\!30\)\( - \)\(31\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!74\)\( \beta_{2} - \)\(76\!\cdots\!44\)\( \beta_{3} - \)\(11\!\cdots\!38\)\( \beta_{4} - \)\(63\!\cdots\!64\)\( \beta_{5} - 878472313378704156 \beta_{6} - 526281973855680 \beta_{7}) q^{17} +(-\)\(22\!\cdots\!85\)\( + \)\(79\!\cdots\!33\)\( \beta_{1} - \)\(21\!\cdots\!60\)\( \beta_{2} - \)\(75\!\cdots\!36\)\( \beta_{3} + \)\(10\!\cdots\!28\)\( \beta_{4} - \)\(62\!\cdots\!56\)\( \beta_{5} + 11177988501349285056 \beta_{6} + 8559395719398720 \beta_{7}) q^{18} +(-\)\(65\!\cdots\!80\)\( - \)\(31\!\cdots\!75\)\( \beta_{1} - \)\(57\!\cdots\!09\)\( \beta_{2} - \)\(39\!\cdots\!34\)\( \beta_{3} - \)\(25\!\cdots\!86\)\( \beta_{4} + \)\(23\!\cdots\!48\)\( \beta_{5} - \)\(11\!\cdots\!22\)\( \beta_{6} - 123672945181895424 \beta_{7}) q^{19} +(\)\(17\!\cdots\!60\)\( + \)\(47\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!54\)\( \beta_{2} + \)\(88\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!88\)\( \beta_{4} - \)\(32\!\cdots\!00\)\( \beta_{5} + \)\(76\!\cdots\!00\)\( \beta_{6} + 1598525547088209600 \beta_{7}) q^{20} +(\)\(16\!\cdots\!92\)\( - \)\(29\!\cdots\!24\)\( \beta_{1} - \)\(93\!\cdots\!80\)\( \beta_{2} + \)\(48\!\cdots\!08\)\( \beta_{3} + \)\(30\!\cdots\!80\)\( \beta_{4} + \)\(24\!\cdots\!52\)\( \beta_{5} - \)\(20\!\cdots\!96\)\( \beta_{6} - 18582876201795196032 \beta_{7}) q^{21} +(-\)\(25\!\cdots\!20\)\( + \)\(76\!\cdots\!86\)\( \beta_{1} - \)\(63\!\cdots\!89\)\( \beta_{2} - \)\(60\!\cdots\!52\)\( \beta_{3} - \)\(13\!\cdots\!99\)\( \beta_{4} - \)\(42\!\cdots\!61\)\( \beta_{5} - \)\(37\!\cdots\!76\)\( \beta_{6} + \)\(19\!\cdots\!84\)\( \beta_{7}) q^{22} +(-\)\(89\!\cdots\!80\)\( - \)\(36\!\cdots\!22\)\( \beta_{1} + \)\(49\!\cdots\!56\)\( \beta_{2} - \)\(50\!\cdots\!35\)\( \beta_{3} - \)\(86\!\cdots\!95\)\( \beta_{4} - \)\(13\!\cdots\!60\)\( \beta_{5} + \)\(75\!\cdots\!35\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7}) q^{23} +(\)\(36\!\cdots\!60\)\( + \)\(25\!\cdots\!36\)\( \beta_{1} + \)\(77\!\cdots\!24\)\( \beta_{2} + \)\(55\!\cdots\!04\)\( \beta_{3} + \)\(14\!\cdots\!24\)\( \beta_{4} + \)\(18\!\cdots\!32\)\( \beta_{5} - \)\(83\!\cdots\!04\)\( \beta_{6} + \)\(16\!\cdots\!32\)\( \beta_{7}) q^{24} +(\)\(10\!\cdots\!75\)\( - \)\(26\!\cdots\!40\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(63\!\cdots\!00\)\( \beta_{3} - \)\(54\!\cdots\!60\)\( \beta_{4} - \)\(13\!\cdots\!20\)\( \beta_{5} + \)\(68\!\cdots\!60\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7}) q^{25} +(-\)\(86\!\cdots\!38\)\( + \)\(38\!\cdots\!38\)\( \beta_{1} - \)\(37\!\cdots\!80\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(24\!\cdots\!92\)\( \beta_{4} + \)\(54\!\cdots\!04\)\( \beta_{5} - \)\(43\!\cdots\!48\)\( \beta_{6} + \)\(89\!\cdots\!84\)\( \beta_{7}) q^{26} +(\)\(80\!\cdots\!20\)\( + \)\(84\!\cdots\!74\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(77\!\cdots\!58\)\( \beta_{3} + \)\(35\!\cdots\!14\)\( \beta_{4} + \)\(11\!\cdots\!48\)\( \beta_{5} + \)\(21\!\cdots\!70\)\( \beta_{6} - \)\(57\!\cdots\!56\)\( \beta_{7}) q^{27} +(\)\(37\!\cdots\!80\)\( + \)\(20\!\cdots\!88\)\( \beta_{1} - \)\(58\!\cdots\!44\)\( \beta_{2} + \)\(85\!\cdots\!72\)\( \beta_{3} - \)\(13\!\cdots\!76\)\( \beta_{4} - \)\(16\!\cdots\!92\)\( \beta_{5} - \)\(64\!\cdots\!00\)\( \beta_{6} + \)\(33\!\cdots\!64\)\( \beta_{7}) q^{28} +(\)\(96\!\cdots\!30\)\( + \)\(45\!\cdots\!36\)\( \beta_{1} + \)\(22\!\cdots\!31\)\( \beta_{2} - \)\(10\!\cdots\!70\)\( \beta_{3} - \)\(36\!\cdots\!37\)\( \beta_{4} + \)\(11\!\cdots\!04\)\( \beta_{5} - \)\(82\!\cdots\!08\)\( \beta_{6} - \)\(17\!\cdots\!36\)\( \beta_{7}) q^{29} +(-\)\(78\!\cdots\!60\)\( + \)\(92\!\cdots\!92\)\( \beta_{1} + \)\(20\!\cdots\!94\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} + \)\(18\!\cdots\!18\)\( \beta_{4} - \)\(26\!\cdots\!50\)\( \beta_{5} + \)\(15\!\cdots\!00\)\( \beta_{6} + \)\(77\!\cdots\!00\)\( \beta_{7}) q^{30} +(\)\(60\!\cdots\!52\)\( + \)\(20\!\cdots\!72\)\( \beta_{1} + \)\(75\!\cdots\!92\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(17\!\cdots\!64\)\( \beta_{4} - \)\(12\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!56\)\( \beta_{6} - \)\(29\!\cdots\!52\)\( \beta_{7}) q^{31} +(\)\(11\!\cdots\!40\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - \)\(31\!\cdots\!56\)\( \beta_{2} + \)\(47\!\cdots\!76\)\( \beta_{3} - \)\(65\!\cdots\!48\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(91\!\cdots\!20\)\( \beta_{7}) q^{32} +(-\)\(19\!\cdots\!20\)\( + \)\(33\!\cdots\!20\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(68\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!42\)\( \beta_{4} - \)\(63\!\cdots\!48\)\( \beta_{5} + \)\(67\!\cdots\!96\)\( \beta_{6} - \)\(17\!\cdots\!96\)\( \beta_{7}) q^{33} +(-\)\(20\!\cdots\!54\)\( - \)\(87\!\cdots\!54\)\( \beta_{1} + \)\(43\!\cdots\!88\)\( \beta_{2} - \)\(61\!\cdots\!76\)\( \beta_{3} + \)\(79\!\cdots\!24\)\( \beta_{4} + \)\(10\!\cdots\!48\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} - \)\(23\!\cdots\!64\)\( \beta_{7}) q^{34} +(\)\(73\!\cdots\!20\)\( - \)\(29\!\cdots\!84\)\( \beta_{1} + \)\(18\!\cdots\!96\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} - \)\(77\!\cdots\!96\)\( \beta_{4} + \)\(35\!\cdots\!96\)\( \beta_{5} + \)\(10\!\cdots\!32\)\( \beta_{6} + \)\(40\!\cdots\!04\)\( \beta_{7}) q^{35} +(\)\(64\!\cdots\!16\)\( - \)\(45\!\cdots\!56\)\( \beta_{1} + \)\(35\!\cdots\!13\)\( \beta_{2} + \)\(76\!\cdots\!01\)\( \beta_{3} + \)\(20\!\cdots\!56\)\( \beta_{4} - \)\(23\!\cdots\!24\)\( \beta_{5} - \)\(46\!\cdots\!00\)\( \beta_{6} - \)\(21\!\cdots\!00\)\( \beta_{7}) q^{36} +(-\)\(22\!\cdots\!90\)\( - \)\(39\!\cdots\!24\)\( \beta_{1} - \)\(29\!\cdots\!93\)\( \beta_{2} - \)\(15\!\cdots\!06\)\( \beta_{3} + \)\(74\!\cdots\!23\)\( \beta_{4} + \)\(90\!\cdots\!56\)\( \beta_{5} + \)\(14\!\cdots\!80\)\( \beta_{6} + \)\(76\!\cdots\!88\)\( \beta_{7}) q^{37} +(-\)\(20\!\cdots\!80\)\( - \)\(22\!\cdots\!54\)\( \beta_{1} - \)\(67\!\cdots\!87\)\( \beta_{2} - \)\(95\!\cdots\!64\)\( \beta_{3} - \)\(85\!\cdots\!33\)\( \beta_{4} + \)\(50\!\cdots\!45\)\( \beta_{5} - \)\(25\!\cdots\!48\)\( \beta_{6} - \)\(16\!\cdots\!84\)\( \beta_{7}) q^{38} +(\)\(58\!\cdots\!04\)\( + \)\(11\!\cdots\!30\)\( \beta_{1} - \)\(90\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!57\)\( \beta_{3} - \)\(58\!\cdots\!07\)\( \beta_{4} - \)\(30\!\cdots\!44\)\( \beta_{5} - \)\(37\!\cdots\!29\)\( \beta_{6} - \)\(15\!\cdots\!68\)\( \beta_{7}) q^{39} +(\)\(48\!\cdots\!00\)\( + \)\(45\!\cdots\!20\)\( \beta_{1} + \)\(20\!\cdots\!20\)\( \beta_{2} + \)\(13\!\cdots\!50\)\( \beta_{3} + \)\(41\!\cdots\!30\)\( \beta_{4} + \)\(98\!\cdots\!20\)\( \beta_{5} + \)\(46\!\cdots\!90\)\( \beta_{6} + \)\(20\!\cdots\!30\)\( \beta_{7}) q^{40} +(-\)\(10\!\cdots\!98\)\( + \)\(61\!\cdots\!32\)\( \beta_{1} + \)\(62\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!68\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4} - \)\(17\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!00\)\( \beta_{6} - \)\(10\!\cdots\!00\)\( \beta_{7}) q^{41} +(-\)\(19\!\cdots\!80\)\( + \)\(43\!\cdots\!80\)\( \beta_{1} - \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(99\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!72\)\( \beta_{4} + \)\(14\!\cdots\!04\)\( \beta_{5} + \)\(41\!\cdots\!76\)\( \beta_{6} + \)\(29\!\cdots\!60\)\( \beta_{7}) q^{42} +(\)\(45\!\cdots\!00\)\( - \)\(29\!\cdots\!45\)\( \beta_{1} - \)\(34\!\cdots\!31\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(55\!\cdots\!20\)\( \beta_{4} - \)\(44\!\cdots\!56\)\( \beta_{5} - \)\(18\!\cdots\!32\)\( \beta_{6} - \)\(34\!\cdots\!44\)\( \beta_{7}) q^{43} +(\)\(24\!\cdots\!96\)\( - \)\(38\!\cdots\!44\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(33\!\cdots\!44\)\( \beta_{4} + \)\(41\!\cdots\!36\)\( \beta_{5} - \)\(24\!\cdots\!16\)\( \beta_{6} - \)\(13\!\cdots\!72\)\( \beta_{7}) q^{44} +(-\)\(20\!\cdots\!10\)\( - \)\(66\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!99\)\( \beta_{2} - \)\(16\!\cdots\!70\)\( \beta_{3} - \)\(20\!\cdots\!97\)\( \beta_{4} - \)\(14\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(92\!\cdots\!00\)\( \beta_{7}) q^{45} +(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!68\)\( \beta_{1} + \)\(83\!\cdots\!74\)\( \beta_{2} - \)\(10\!\cdots\!88\)\( \beta_{3} + \)\(38\!\cdots\!14\)\( \beta_{4} + \)\(91\!\cdots\!54\)\( \beta_{5} - \)\(19\!\cdots\!80\)\( \beta_{6} - \)\(28\!\cdots\!60\)\( \beta_{7}) q^{46} +(\)\(48\!\cdots\!80\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(17\!\cdots\!50\)\( \beta_{3} + \)\(75\!\cdots\!90\)\( \beta_{4} + \)\(12\!\cdots\!48\)\( \beta_{5} + \)\(49\!\cdots\!06\)\( \beta_{6} + \)\(36\!\cdots\!52\)\( \beta_{7}) q^{47} +(\)\(11\!\cdots\!20\)\( + \)\(20\!\cdots\!96\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} + \)\(60\!\cdots\!76\)\( \beta_{3} - \)\(69\!\cdots\!48\)\( \beta_{4} - \)\(54\!\cdots\!44\)\( \beta_{5} + \)\(76\!\cdots\!24\)\( \beta_{6} + \)\(92\!\cdots\!20\)\( \beta_{7}) q^{48} +(\)\(33\!\cdots\!93\)\( + \)\(27\!\cdots\!12\)\( \beta_{1} + \)\(25\!\cdots\!04\)\( \beta_{2} - \)\(85\!\cdots\!48\)\( \beta_{3} - \)\(47\!\cdots\!36\)\( \beta_{4} + \)\(90\!\cdots\!84\)\( \beta_{5} - \)\(17\!\cdots\!20\)\( \beta_{6} - \)\(67\!\cdots\!40\)\( \beta_{7}) q^{49} +(-\)\(17\!\cdots\!75\)\( + \)\(13\!\cdots\!35\)\( \beta_{1} + \)\(72\!\cdots\!40\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + \)\(22\!\cdots\!40\)\( \beta_{4} + \)\(26\!\cdots\!80\)\( \beta_{5} + \)\(14\!\cdots\!60\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7}) q^{50} +(\)\(30\!\cdots\!32\)\( - \)\(21\!\cdots\!10\)\( \beta_{1} + \)\(29\!\cdots\!74\)\( \beta_{2} + \)\(49\!\cdots\!34\)\( \beta_{3} - \)\(30\!\cdots\!94\)\( \beta_{4} - \)\(33\!\cdots\!88\)\( \beta_{5} + \)\(32\!\cdots\!02\)\( \beta_{6} - \)\(17\!\cdots\!16\)\( \beta_{7}) q^{51} +(\)\(19\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(20\!\cdots\!74\)\( \beta_{2} + \)\(36\!\cdots\!50\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4} + \)\(10\!\cdots\!68\)\( \beta_{5} + \)\(20\!\cdots\!96\)\( \beta_{6} - \)\(74\!\cdots\!68\)\( \beta_{7}) q^{52} +(-\)\(36\!\cdots\!10\)\( - \)\(85\!\cdots\!60\)\( \beta_{1} - \)\(62\!\cdots\!33\)\( \beta_{2} - \)\(59\!\cdots\!82\)\( \beta_{3} - \)\(10\!\cdots\!69\)\( \beta_{4} + \)\(14\!\cdots\!12\)\( \beta_{5} - \)\(13\!\cdots\!80\)\( \beta_{6} + \)\(38\!\cdots\!56\)\( \beta_{7}) q^{53} +(\)\(55\!\cdots\!20\)\( - \)\(26\!\cdots\!20\)\( \beta_{1} + \)\(42\!\cdots\!38\)\( \beta_{2} - \)\(17\!\cdots\!24\)\( \beta_{3} + \)\(65\!\cdots\!14\)\( \beta_{4} + \)\(12\!\cdots\!66\)\( \beta_{5} + \)\(25\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!32\)\( \beta_{7}) q^{54} +(\)\(97\!\cdots\!40\)\( + \)\(68\!\cdots\!62\)\( \beta_{1} + \)\(19\!\cdots\!84\)\( \beta_{2} - \)\(33\!\cdots\!45\)\( \beta_{3} - \)\(71\!\cdots\!77\)\( \beta_{4} - \)\(25\!\cdots\!00\)\( \beta_{5} + \)\(37\!\cdots\!25\)\( \beta_{6} + \)\(11\!\cdots\!00\)\( \beta_{7}) q^{55} +(\)\(90\!\cdots\!80\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!92\)\( \beta_{4} + \)\(67\!\cdots\!32\)\( \beta_{5} - \)\(33\!\cdots\!52\)\( \beta_{6} + \)\(48\!\cdots\!16\)\( \beta_{7}) q^{56} +(\)\(18\!\cdots\!40\)\( + \)\(29\!\cdots\!56\)\( \beta_{1} - \)\(45\!\cdots\!58\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} + \)\(88\!\cdots\!70\)\( \beta_{4} - \)\(23\!\cdots\!36\)\( \beta_{5} + \)\(76\!\cdots\!08\)\( \beta_{6} - \)\(15\!\cdots\!64\)\( \beta_{7}) q^{57} +(\)\(29\!\cdots\!30\)\( + \)\(76\!\cdots\!30\)\( \beta_{1} - \)\(72\!\cdots\!76\)\( \beta_{2} + \)\(48\!\cdots\!56\)\( \beta_{3} - \)\(11\!\cdots\!28\)\( \beta_{4} - \)\(28\!\cdots\!52\)\( \beta_{5} - \)\(14\!\cdots\!92\)\( \beta_{6} + \)\(19\!\cdots\!08\)\( \beta_{7}) q^{58} +(\)\(30\!\cdots\!60\)\( - \)\(99\!\cdots\!21\)\( \beta_{1} + \)\(18\!\cdots\!25\)\( \beta_{2} - \)\(16\!\cdots\!68\)\( \beta_{3} + \)\(23\!\cdots\!20\)\( \beta_{4} + \)\(68\!\cdots\!88\)\( \beta_{5} - \)\(38\!\cdots\!24\)\( \beta_{6} + \)\(33\!\cdots\!92\)\( \beta_{7}) q^{59} +(\)\(51\!\cdots\!20\)\( - \)\(13\!\cdots\!64\)\( \beta_{1} - \)\(42\!\cdots\!84\)\( \beta_{2} + \)\(41\!\cdots\!80\)\( \beta_{3} - \)\(85\!\cdots\!16\)\( \beta_{4} - \)\(45\!\cdots\!84\)\( \beta_{5} + \)\(10\!\cdots\!72\)\( \beta_{6} - \)\(20\!\cdots\!16\)\( \beta_{7}) q^{60} +(\)\(32\!\cdots\!02\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(75\!\cdots\!19\)\( \beta_{2} - \)\(71\!\cdots\!90\)\( \beta_{3} + \)\(97\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!04\)\( \beta_{5} - \)\(77\!\cdots\!28\)\( \beta_{6} + \)\(39\!\cdots\!24\)\( \beta_{7}) q^{61} +(\)\(13\!\cdots\!80\)\( - \)\(46\!\cdots\!36\)\( \beta_{1} + \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(90\!\cdots\!44\)\( \beta_{3} + \)\(21\!\cdots\!88\)\( \beta_{4} - \)\(22\!\cdots\!96\)\( \beta_{5} - \)\(22\!\cdots\!64\)\( \beta_{6} - \)\(22\!\cdots\!60\)\( \beta_{7}) q^{62} +(\)\(21\!\cdots\!60\)\( + \)\(29\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} - \)\(66\!\cdots\!29\)\( \beta_{3} - \)\(51\!\cdots\!93\)\( \beta_{4} + \)\(43\!\cdots\!04\)\( \beta_{5} + \)\(67\!\cdots\!45\)\( \beta_{6} - \)\(18\!\cdots\!08\)\( \beta_{7}) q^{63} +(\)\(50\!\cdots\!28\)\( + \)\(27\!\cdots\!68\)\( \beta_{1} - \)\(30\!\cdots\!64\)\( \beta_{2} + \)\(98\!\cdots\!76\)\( \beta_{3} - \)\(14\!\cdots\!92\)\( \beta_{4} + \)\(19\!\cdots\!88\)\( \beta_{5} - \)\(53\!\cdots\!60\)\( \beta_{6} + \)\(49\!\cdots\!80\)\( \beta_{7}) q^{64} +(\)\(88\!\cdots\!40\)\( + \)\(49\!\cdots\!32\)\( \beta_{1} - \)\(69\!\cdots\!08\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3} - \)\(13\!\cdots\!92\)\( \beta_{4} - \)\(88\!\cdots\!08\)\( \beta_{5} - \)\(25\!\cdots\!36\)\( \beta_{6} - \)\(37\!\cdots\!92\)\( \beta_{7}) q^{65} +(\)\(21\!\cdots\!44\)\( - \)\(55\!\cdots\!20\)\( \beta_{1} + \)\(35\!\cdots\!12\)\( \beta_{2} + \)\(16\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(95\!\cdots\!76\)\( \beta_{5} - \)\(92\!\cdots\!84\)\( \beta_{6} - \)\(12\!\cdots\!28\)\( \beta_{7}) q^{66} +(-\)\(80\!\cdots\!80\)\( + \)\(21\!\cdots\!01\)\( \beta_{1} + \)\(33\!\cdots\!87\)\( \beta_{2} - \)\(34\!\cdots\!34\)\( \beta_{3} - \)\(21\!\cdots\!98\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(44\!\cdots\!02\)\( \beta_{6} + \)\(43\!\cdots\!76\)\( \beta_{7}) q^{67} +(\)\(11\!\cdots\!40\)\( - \)\(37\!\cdots\!28\)\( \beta_{1} - \)\(54\!\cdots\!50\)\( \beta_{2} - \)\(97\!\cdots\!78\)\( \beta_{3} - \)\(11\!\cdots\!76\)\( \beta_{4} - \)\(48\!\cdots\!72\)\( \beta_{5} + \)\(79\!\cdots\!40\)\( \beta_{6} - \)\(50\!\cdots\!56\)\( \beta_{7}) q^{68} +(-\)\(20\!\cdots\!16\)\( - \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!56\)\( \beta_{2} + \)\(15\!\cdots\!12\)\( \beta_{3} + \)\(71\!\cdots\!96\)\( \beta_{4} + \)\(10\!\cdots\!44\)\( \beta_{5} - \)\(64\!\cdots\!04\)\( \beta_{6} - \)\(50\!\cdots\!68\)\( \beta_{7}) q^{69} +(-\)\(19\!\cdots\!80\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(56\!\cdots\!96\)\( \beta_{4} + \)\(43\!\cdots\!00\)\( \beta_{5} + \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(27\!\cdots\!00\)\( \beta_{7}) q^{70} +(-\)\(18\!\cdots\!48\)\( + \)\(85\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!55\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} + \)\(22\!\cdots\!68\)\( \beta_{5} + \)\(12\!\cdots\!01\)\( \beta_{6} - \)\(38\!\cdots\!08\)\( \beta_{7}) q^{71} +(-\)\(21\!\cdots\!40\)\( + \)\(21\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2} - \)\(29\!\cdots\!37\)\( \beta_{3} - \)\(55\!\cdots\!99\)\( \beta_{4} - \)\(66\!\cdots\!82\)\( \beta_{5} - \)\(69\!\cdots\!33\)\( \beta_{6} - \)\(60\!\cdots\!55\)\( \beta_{7}) q^{72} +(-\)\(20\!\cdots\!30\)\( + \)\(66\!\cdots\!76\)\( \beta_{1} + \)\(65\!\cdots\!26\)\( \beta_{2} + \)\(21\!\cdots\!16\)\( \beta_{3} + \)\(55\!\cdots\!02\)\( \beta_{4} - \)\(27\!\cdots\!00\)\( \beta_{5} + \)\(65\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!16\)\( \beta_{7}) q^{73} +(-\)\(26\!\cdots\!14\)\( - \)\(74\!\cdots\!62\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} - \)\(68\!\cdots\!72\)\( \beta_{3} + \)\(12\!\cdots\!16\)\( \beta_{4} + \)\(31\!\cdots\!20\)\( \beta_{5} + \)\(15\!\cdots\!88\)\( \beta_{6} - \)\(15\!\cdots\!04\)\( \beta_{7}) q^{74} +(-\)\(39\!\cdots\!00\)\( - \)\(32\!\cdots\!05\)\( \beta_{1} - \)\(18\!\cdots\!95\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(16\!\cdots\!20\)\( \beta_{4} - \)\(29\!\cdots\!40\)\( \beta_{5} - \)\(40\!\cdots\!80\)\( \beta_{6} + \)\(72\!\cdots\!40\)\( \beta_{7}) q^{75} +(-\)\(11\!\cdots\!40\)\( - \)\(44\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!16\)\( \beta_{2} - \)\(17\!\cdots\!40\)\( \beta_{3} - \)\(20\!\cdots\!44\)\( \beta_{4} - \)\(32\!\cdots\!96\)\( \beta_{5} - \)\(16\!\cdots\!04\)\( \beta_{6} - \)\(66\!\cdots\!68\)\( \beta_{7}) q^{76} +(-\)\(44\!\cdots\!00\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(32\!\cdots\!64\)\( \beta_{3} - \)\(67\!\cdots\!28\)\( \beta_{4} - \)\(15\!\cdots\!24\)\( \beta_{5} + \)\(17\!\cdots\!84\)\( \beta_{6} + \)\(12\!\cdots\!60\)\( \beta_{7}) q^{77} +(\)\(73\!\cdots\!00\)\( + \)\(79\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3} + \)\(34\!\cdots\!94\)\( \beta_{4} + \)\(28\!\cdots\!90\)\( \beta_{5} - \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(51\!\cdots\!12\)\( \beta_{7}) q^{78} +(\)\(58\!\cdots\!80\)\( + \)\(80\!\cdots\!64\)\( \beta_{1} - \)\(92\!\cdots\!44\)\( \beta_{2} + \)\(43\!\cdots\!18\)\( \beta_{3} - \)\(10\!\cdots\!06\)\( \beta_{4} - \)\(45\!\cdots\!24\)\( \beta_{5} - \)\(58\!\cdots\!86\)\( \beta_{6} - \)\(37\!\cdots\!12\)\( \beta_{7}) q^{79} +(\)\(23\!\cdots\!20\)\( + \)\(73\!\cdots\!16\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(29\!\cdots\!40\)\( \beta_{3} - \)\(10\!\cdots\!36\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} + \)\(13\!\cdots\!00\)\( \beta_{6} + \)\(69\!\cdots\!00\)\( \beta_{7}) q^{80} +(\)\(15\!\cdots\!21\)\( + \)\(63\!\cdots\!88\)\( \beta_{1} - \)\(57\!\cdots\!22\)\( \beta_{2} - \)\(51\!\cdots\!56\)\( \beta_{3} + \)\(55\!\cdots\!30\)\( \beta_{4} + \)\(28\!\cdots\!68\)\( \beta_{5} + \)\(17\!\cdots\!16\)\( \beta_{6} + \)\(18\!\cdots\!72\)\( \beta_{7}) q^{81} +(\)\(40\!\cdots\!30\)\( - \)\(26\!\cdots\!06\)\( \beta_{1} + \)\(91\!\cdots\!44\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + \)\(22\!\cdots\!28\)\( \beta_{4} + \)\(14\!\cdots\!56\)\( \beta_{5} - \)\(43\!\cdots\!40\)\( \beta_{6} - \)\(30\!\cdots\!72\)\( \beta_{7}) q^{82} +(\)\(42\!\cdots\!60\)\( - \)\(26\!\cdots\!51\)\( \beta_{1} - \)\(88\!\cdots\!73\)\( \beta_{2} - \)\(99\!\cdots\!00\)\( \beta_{3} - \)\(21\!\cdots\!00\)\( \beta_{4} - \)\(54\!\cdots\!20\)\( \beta_{5} + \)\(57\!\cdots\!60\)\( \beta_{6} + \)\(47\!\cdots\!20\)\( \beta_{7}) q^{83} +(\)\(21\!\cdots\!16\)\( - \)\(62\!\cdots\!64\)\( \beta_{1} + \)\(39\!\cdots\!32\)\( \beta_{2} + \)\(36\!\cdots\!00\)\( \beta_{3} - \)\(78\!\cdots\!72\)\( \beta_{4} + \)\(75\!\cdots\!08\)\( \beta_{5} + \)\(19\!\cdots\!40\)\( \beta_{6} + \)\(10\!\cdots\!80\)\( \beta_{7}) q^{84} +(\)\(73\!\cdots\!20\)\( - \)\(69\!\cdots\!04\)\( \beta_{1} - \)\(36\!\cdots\!74\)\( \beta_{2} + \)\(71\!\cdots\!80\)\( \beta_{3} + \)\(24\!\cdots\!74\)\( \beta_{4} - \)\(13\!\cdots\!24\)\( \beta_{5} - \)\(10\!\cdots\!08\)\( \beta_{6} - \)\(12\!\cdots\!76\)\( \beta_{7}) q^{85} +(-\)\(19\!\cdots\!68\)\( + \)\(15\!\cdots\!58\)\( \beta_{1} + \)\(36\!\cdots\!55\)\( \beta_{2} - \)\(39\!\cdots\!88\)\( \beta_{3} + \)\(73\!\cdots\!77\)\( \beta_{4} + \)\(21\!\cdots\!63\)\( \beta_{5} + \)\(73\!\cdots\!72\)\( \beta_{6} + \)\(10\!\cdots\!24\)\( \beta_{7}) q^{86} +(-\)\(68\!\cdots\!40\)\( + \)\(36\!\cdots\!86\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} - \)\(50\!\cdots\!71\)\( \beta_{3} + \)\(28\!\cdots\!53\)\( \beta_{4} + \)\(31\!\cdots\!08\)\( \beta_{5} - \)\(90\!\cdots\!81\)\( \beta_{6} + \)\(88\!\cdots\!96\)\( \beta_{7}) q^{87} +(-\)\(15\!\cdots\!40\)\( + \)\(51\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!32\)\( \beta_{2} - \)\(19\!\cdots\!84\)\( \beta_{3} - \)\(48\!\cdots\!68\)\( \beta_{4} - \)\(14\!\cdots\!44\)\( \beta_{5} + \)\(52\!\cdots\!04\)\( \beta_{6} + \)\(21\!\cdots\!60\)\( \beta_{7}) q^{88} +(-\)\(42\!\cdots\!10\)\( - \)\(43\!\cdots\!92\)\( \beta_{1} - \)\(85\!\cdots\!98\)\( \beta_{2} + \)\(60\!\cdots\!20\)\( \beta_{3} - \)\(94\!\cdots\!26\)\( \beta_{4} + \)\(19\!\cdots\!48\)\( \beta_{5} - \)\(68\!\cdots\!92\)\( \beta_{6} - \)\(72\!\cdots\!64\)\( \beta_{7}) q^{89} +(-\)\(43\!\cdots\!10\)\( - \)\(83\!\cdots\!78\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2} + \)\(95\!\cdots\!60\)\( \beta_{3} + \)\(25\!\cdots\!68\)\( \beta_{4} + \)\(44\!\cdots\!32\)\( \beta_{5} - \)\(12\!\cdots\!56\)\( \beta_{6} + \)\(31\!\cdots\!68\)\( \beta_{7}) q^{90} +(-\)\(43\!\cdots\!68\)\( - \)\(71\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!52\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3} - \)\(19\!\cdots\!52\)\( \beta_{4} - \)\(24\!\cdots\!28\)\( \beta_{5} + \)\(48\!\cdots\!48\)\( \beta_{6} + \)\(36\!\cdots\!16\)\( \beta_{7}) q^{91} +(-\)\(15\!\cdots\!80\)\( - \)\(52\!\cdots\!80\)\( \beta_{1} - \)\(18\!\cdots\!84\)\( \beta_{2} - \)\(30\!\cdots\!56\)\( \beta_{3} + \)\(21\!\cdots\!48\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(51\!\cdots\!40\)\( \beta_{6} - \)\(69\!\cdots\!52\)\( \beta_{7}) q^{92} +(\)\(24\!\cdots\!80\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} + \)\(89\!\cdots\!88\)\( \beta_{2} + \)\(18\!\cdots\!68\)\( \beta_{3} + \)\(61\!\cdots\!96\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - \)\(78\!\cdots\!24\)\( \beta_{6} - \)\(22\!\cdots\!92\)\( \beta_{7}) q^{93} +(\)\(19\!\cdots\!56\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} + \)\(59\!\cdots\!36\)\( \beta_{3} - \)\(18\!\cdots\!36\)\( \beta_{4} + \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(98\!\cdots\!24\)\( \beta_{6} + \)\(21\!\cdots\!08\)\( \beta_{7}) q^{94} +(\)\(19\!\cdots\!00\)\( + \)\(63\!\cdots\!30\)\( \beta_{1} + \)\(82\!\cdots\!60\)\( \beta_{2} + \)\(17\!\cdots\!25\)\( \beta_{3} + \)\(23\!\cdots\!45\)\( \beta_{4} - \)\(57\!\cdots\!00\)\( \beta_{5} - \)\(23\!\cdots\!25\)\( \beta_{6} - \)\(16\!\cdots\!00\)\( \beta_{7}) q^{95} +(\)\(12\!\cdots\!72\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} + \)\(33\!\cdots\!64\)\( \beta_{2} + \)\(64\!\cdots\!08\)\( \beta_{3} + \)\(20\!\cdots\!68\)\( \beta_{4} + \)\(65\!\cdots\!48\)\( \beta_{5} + \)\(44\!\cdots\!40\)\( \beta_{6} - \)\(28\!\cdots\!20\)\( \beta_{7}) q^{96} +(\)\(12\!\cdots\!30\)\( + \)\(19\!\cdots\!28\)\( \beta_{1} + \)\(26\!\cdots\!18\)\( \beta_{2} - \)\(35\!\cdots\!68\)\( \beta_{3} - \)\(26\!\cdots\!06\)\( \beta_{4} - \)\(80\!\cdots\!12\)\( \beta_{5} - \)\(28\!\cdots\!20\)\( \beta_{6} + \)\(29\!\cdots\!44\)\( \beta_{7}) q^{97} +(\)\(18\!\cdots\!95\)\( - \)\(14\!\cdots\!95\)\( \beta_{1} - \)\(38\!\cdots\!96\)\( \beta_{2} + \)\(15\!\cdots\!84\)\( \beta_{3} - \)\(68\!\cdots\!72\)\( \beta_{4} + \)\(44\!\cdots\!16\)\( \beta_{5} - \)\(85\!\cdots\!20\)\( \beta_{6} + \)\(85\!\cdots\!68\)\( \beta_{7}) q^{98} +(\)\(38\!\cdots\!84\)\( - \)\(10\!\cdots\!21\)\( \beta_{1} + \)\(97\!\cdots\!57\)\( \beta_{2} - \)\(83\!\cdots\!20\)\( \beta_{3} + \)\(72\!\cdots\!52\)\( \beta_{4} + \)\(31\!\cdots\!48\)\( \beta_{5} + \)\(20\!\cdots\!92\)\( \beta_{6} - \)\(79\!\cdots\!36\)\( \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - 5835659138280q^{2} - \)\(95\!\cdots\!80\)\(q^{3} + \)\(20\!\cdots\!84\)\(q^{4} + \)\(19\!\cdots\!60\)\(q^{5} + \)\(10\!\cdots\!76\)\(q^{6} + \)\(31\!\cdots\!00\)\(q^{7} - \)\(14\!\cdots\!60\)\(q^{8} + \)\(92\!\cdots\!36\)\(q^{9} + O(q^{10}) \)	\( 8q - 5835659138280q^{2} - \)\(95\!\cdots\!80\)\(q^{3} + \)\(20\!\cdots\!84\)\(q^{4} + \)\(19\!\cdots\!60\)\(q^{5} + \)\(10\!\cdots\!76\)\(q^{6} + \)\(31\!\cdots\!00\)\(q^{7} - \)\(14\!\cdots\!60\)\(q^{8} + \)\(92\!\cdots\!36\)\(q^{9} - \)\(35\!\cdots\!40\)\(q^{10} + \)\(53\!\cdots\!16\)\(q^{11} + \)\(10\!\cdots\!40\)\(q^{12} + \)\(11\!\cdots\!40\)\(q^{13} + \)\(88\!\cdots\!08\)\(q^{14} + \)\(18\!\cdots\!20\)\(q^{15} + \)\(95\!\cdots\!28\)\(q^{16} - \)\(13\!\cdots\!40\)\(q^{17} - \)\(17\!\cdots\!80\)\(q^{18} - \)\(52\!\cdots\!40\)\(q^{19} + \)\(13\!\cdots\!80\)\(q^{20} + \)\(13\!\cdots\!36\)\(q^{21} - \)\(20\!\cdots\!60\)\(q^{22} - \)\(71\!\cdots\!40\)\(q^{23} + \)\(28\!\cdots\!80\)\(q^{24} + \)\(81\!\cdots\!00\)\(q^{25} - \)\(69\!\cdots\!04\)\(q^{26} + \)\(64\!\cdots\!60\)\(q^{27} + \)\(30\!\cdots\!40\)\(q^{28} + \)\(77\!\cdots\!40\)\(q^{29} - \)\(62\!\cdots\!80\)\(q^{30} + \)\(48\!\cdots\!16\)\(q^{31} + \)\(91\!\cdots\!20\)\(q^{32} - \)\(15\!\cdots\!60\)\(q^{33} - \)\(16\!\cdots\!32\)\(q^{34} + \)\(58\!\cdots\!60\)\(q^{35} + \)\(51\!\cdots\!28\)\(q^{36} - \)\(18\!\cdots\!20\)\(q^{37} - \)\(16\!\cdots\!40\)\(q^{38} + \)\(46\!\cdots\!32\)\(q^{39} + \)\(39\!\cdots\!00\)\(q^{40} - \)\(87\!\cdots\!84\)\(q^{41} - \)\(15\!\cdots\!40\)\(q^{42} + \)\(36\!\cdots\!00\)\(q^{43} + \)\(19\!\cdots\!68\)\(q^{44} - \)\(16\!\cdots\!80\)\(q^{45} - \)\(19\!\cdots\!84\)\(q^{46} + \)\(38\!\cdots\!40\)\(q^{47} + \)\(93\!\cdots\!60\)\(q^{48} + \)\(26\!\cdots\!44\)\(q^{49} - \)\(14\!\cdots\!00\)\(q^{50} + \)\(24\!\cdots\!56\)\(q^{51} + \)\(15\!\cdots\!00\)\(q^{52} - \)\(29\!\cdots\!80\)\(q^{53} + \)\(44\!\cdots\!60\)\(q^{54} + \)\(77\!\cdots\!20\)\(q^{55} + \)\(72\!\cdots\!40\)\(q^{56} + \)\(14\!\cdots\!20\)\(q^{57} + \)\(23\!\cdots\!40\)\(q^{58} + \)\(24\!\cdots\!80\)\(q^{59} + \)\(41\!\cdots\!60\)\(q^{60} + \)\(26\!\cdots\!16\)\(q^{61} + \)\(10\!\cdots\!40\)\(q^{62} + \)\(17\!\cdots\!80\)\(q^{63} + \)\(40\!\cdots\!24\)\(q^{64} + \)\(71\!\cdots\!20\)\(q^{65} + \)\(17\!\cdots\!52\)\(q^{66} - \)\(64\!\cdots\!40\)\(q^{67} + \)\(91\!\cdots\!20\)\(q^{68} - \)\(16\!\cdots\!28\)\(q^{69} - \)\(15\!\cdots\!40\)\(q^{70} - \)\(14\!\cdots\!84\)\(q^{71} - \)\(16\!\cdots\!20\)\(q^{72} - \)\(16\!\cdots\!40\)\(q^{73} - \)\(20\!\cdots\!12\)\(q^{74} - \)\(31\!\cdots\!00\)\(q^{75} - \)\(95\!\cdots\!20\)\(q^{76} - \)\(35\!\cdots\!00\)\(q^{77} + \)\(59\!\cdots\!00\)\(q^{78} + \)\(46\!\cdots\!40\)\(q^{79} + \)\(18\!\cdots\!60\)\(q^{80} + \)\(12\!\cdots\!68\)\(q^{81} + \)\(32\!\cdots\!40\)\(q^{82} + \)\(34\!\cdots\!80\)\(q^{83} + \)\(17\!\cdots\!28\)\(q^{84} + \)\(58\!\cdots\!60\)\(q^{85} - \)\(15\!\cdots\!44\)\(q^{86} - \)\(54\!\cdots\!20\)\(q^{87} - \)\(12\!\cdots\!20\)\(q^{88} - \)\(34\!\cdots\!80\)\(q^{89} - \)\(35\!\cdots\!80\)\(q^{90} - \)\(34\!\cdots\!44\)\(q^{91} - \)\(12\!\cdots\!40\)\(q^{92} + \)\(19\!\cdots\!40\)\(q^{93} + \)\(15\!\cdots\!48\)\(q^{94} + \)\(15\!\cdots\!00\)\(q^{95} + \)\(97\!\cdots\!76\)\(q^{96} + \)\(10\!\cdots\!40\)\(q^{97} + \)\(14\!\cdots\!60\)\(q^{98} + \)\(30\!\cdots\!72\)\(q^{99} + O(q^{100}) \)

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