LMFDB - Newform orbit 1.108.a.a

Newspace parameters

Level:	$ N $	$=$	$ 1 $
Weight:	$ k $	$=$	$ 108 $
Character orbit:	$[\chi]$	$=$	1.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$72.5037502298$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
Defining polynomial:	$x^{9} - 4 x^{8} - $$18\!\cdots\!64$$ x^{7} - $$73\!\cdots\!04$$ x^{6} + $$10\!\cdots\!46$$ x^{5} + $$37\!\cdots\!92$$ x^{4} - $$21\!\cdots\!48$$ x^{3} - $$45\!\cdots\!88$$ x^{2} + $$13\!\cdots\!17$$ x + $$27\!\cdots\!52$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{143}\cdot 3^{48}\cdot 5^{18}\cdot 7^{8}\cdot 11^{2}\cdot 13^{2} $
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 $1,\beta_1,\ldots,\beta_{8}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q +(607308845937277 - \beta_{1}) q^{2} +($$17\!\cdots\!33$$ + 62113396 \beta_{1} + \beta_{2}) q^{3} +($$71\!\cdots\!70$$ + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +($$27\!\cdots\!09$$ + $$50\!\cdots\!29$$ \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-$$13\!\cdots\!31$$ + $$25\!\cdots\!77$$ \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-$$10\!\cdots\!24$$ + $$32\!\cdots\!71$$ \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-$$11\!\cdots\!77$$ - $$63\!\cdots\!41$$ \beta_{1} + $$51\!\cdots\!47$$ \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +($$41\!\cdots\!23$$ - $$10\!\cdots\!03$$ \beta_{1} + $$17\!\cdots\!19$$ \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})$	$ q +(607308845937277 - \beta_{1}) q^{2} +($$17\!\cdots\!33$$ + 62113396 \beta_{1} + \beta_{2}) q^{3} +($$71\!\cdots\!70$$ + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +($$27\!\cdots\!09$$ + $$50\!\cdots\!29$$ \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-$$13\!\cdots\!31$$ + $$25\!\cdots\!77$$ \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-$$10\!\cdots\!24$$ + $$32\!\cdots\!71$$ \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-$$11\!\cdots\!77$$ - $$63\!\cdots\!41$$ \beta_{1} + $$51\!\cdots\!47$$ \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +($$41\!\cdots\!23$$ - $$10\!\cdots\!03$$ \beta_{1} + $$17\!\cdots\!19$$ \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +(-$$11\!\cdots\!14$$ - $$85\!\cdots\!98$$ \beta_{1} + $$34\!\cdots\!28$$ \beta_{2} - $$42\!\cdots\!40$$ \beta_{3} + 3035644428618332 \beta_{4} + 137787538404 \beta_{5} + 147169648 \beta_{6} - 120312 \beta_{7} - 216 \beta_{8}) q^{10} +($$83\!\cdots\!59$$ + $$10\!\cdots\!50$$ \beta_{1} + $$26\!\cdots\!77$$ \beta_{2} - $$30\!\cdots\!24$$ \beta_{3} + 39655237059879084 \beta_{4} + 18943495258882 \beta_{5} + 20412816178 \beta_{6} + 12038260 \beta_{7} + 23004 \beta_{8}) q^{11} +(-$$88\!\cdots\!96$$ + $$84\!\cdots\!16$$ \beta_{1} + $$19\!\cdots\!48$$ \beta_{2} + $$29\!\cdots\!32$$ \beta_{3} + 17417687351454383040 \beta_{4} + 8277620636103552 \beta_{5} + 1126938004992 \beta_{6} - 91849536 \beta_{7} - 1609920 \beta_{8}) q^{12} +($$42\!\cdots\!13$$ - $$29\!\cdots\!41$$ \beta_{1} + $$84\!\cdots\!80$$ \beta_{2} + $$12\!\cdots\!40$$ \beta_{3} - 1292007267852162005 \beta_{4} + 688583074305208092 \beta_{5} - 165811929566128 \beta_{6} - 46020832614 \beta_{7} + 83255310 \beta_{8}) q^{13} +(-$$82\!\cdots\!78$$ + $$43\!\cdots\!78$$ \beta_{1} + $$78\!\cdots\!18$$ \beta_{2} + $$22\!\cdots\!84$$ \beta_{3} + $$34\!\cdots\!22$$ \beta_{4} + 17161982422390714858 \beta_{5} + 4494593484494656 \beta_{6} + 3819461803360 \beta_{7} - 3391893792 \beta_{8}) q^{14} +($$12\!\cdots\!72$$ - $$49\!\cdots\!31$$ \beta_{1} - $$43\!\cdots\!19$$ \beta_{2} + $$11\!\cdots\!20$$ \beta_{3} + $$28\!\cdots\!84$$ \beta_{4} - $$14\!\cdots\!57$$ \beta_{5} + 71321094534085191 \beta_{6} - 177279761520504 \beta_{7} + 113342680728 \beta_{8}) q^{15} +($$32\!\cdots\!36$$ + $$77\!\cdots\!76$$ \beta_{1} - $$67\!\cdots\!24$$ \beta_{2} + $$88\!\cdots\!92$$ \beta_{3} + $$73\!\cdots\!32$$ \beta_{4} + $$23\!\cdots\!60$$ \beta_{5} - 7893052997101223272 \beta_{6} + 5829377304694680 \beta_{7} - 3193371357696 \beta_{8}) q^{16} +($$61\!\cdots\!04$$ - $$51\!\cdots\!51$$ \beta_{1} + $$30\!\cdots\!87$$ \beta_{2} - $$44\!\cdots\!23$$ \beta_{3} + $$45\!\cdots\!85$$ \beta_{4} + $$86\!\cdots\!14$$ \beta_{5} + $$25\!\cdots\!84$$ \beta_{6} - 147577032261572145 \beta_{7} + 77391112194765 \beta_{8}) q^{17} +($$26\!\cdots\!33$$ - $$54\!\cdots\!45$$ \beta_{1} - $$12\!\cdots\!88$$ \beta_{2} + $$56\!\cdots\!08$$ \beta_{3} - $$44\!\cdots\!20$$ \beta_{4} - $$14\!\cdots\!04$$ \beta_{5} - $$45\!\cdots\!64$$ \beta_{6} + 2995387893492815280 \beta_{7} - 1637786543560080 \beta_{8}) q^{18} +($$95\!\cdots\!49$$ - $$36\!\cdots\!30$$ \beta_{1} - $$75\!\cdots\!37$$ \beta_{2} + $$95\!\cdots\!52$$ \beta_{3} - $$10\!\cdots\!68$$ \beta_{4} - $$67\!\cdots\!94$$ \beta_{5} + $$45\!\cdots\!14$$ \beta_{6} - 49760167465115702820 \beta_{7} + 30620433423116052 \beta_{8}) q^{19} +($$14\!\cdots\!52$$ + $$10\!\cdots\!32$$ \beta_{1} + $$59\!\cdots\!86$$ \beta_{2} + $$26\!\cdots\!70$$ \beta_{3} + $$90\!\cdots\!88$$ \beta_{4} + $$51\!\cdots\!80$$ \beta_{5} + $$96\!\cdots\!60$$ \beta_{6} + $$68\!\cdots\!60$$ \beta_{7} - 510445792877080320 \beta_{8}) q^{20} +($$10\!\cdots\!08$$ - $$50\!\cdots\!94$$ \beta_{1} - $$12\!\cdots\!98$$ \beta_{2} + $$67\!\cdots\!70$$ \beta_{3} + $$16\!\cdots\!94$$ \beta_{4} - $$69\!\cdots\!64$$ \beta_{5} - $$89\!\cdots\!68$$ \beta_{6} - $$76\!\cdots\!30$$ \beta_{7} + 7643011086349644726 \beta_{8}) q^{21} +(-$$23\!\cdots\!17$$ - $$35\!\cdots\!21$$ \beta_{1} + $$45\!\cdots\!75$$ \beta_{2} - $$25\!\cdots\!70$$ \beta_{3} - $$13\!\cdots\!85$$ \beta_{4} + $$23\!\cdots\!47$$ \beta_{5} + $$17\!\cdots\!32$$ \beta_{6} + $$66\!\cdots\!56$$ \beta_{7} - $$10\!\cdots\!60$$ \beta_{8}) q^{22} +(-$$11\!\cdots\!80$$ + $$57\!\cdots\!65$$ \beta_{1} + $$88\!\cdots\!29$$ \beta_{2} - $$25\!\cdots\!00$$ \beta_{3} + $$11\!\cdots\!40$$ \beta_{4} + $$44\!\cdots\!95$$ \beta_{5} - $$20\!\cdots\!45$$ \beta_{6} - $$39\!\cdots\!00$$ \beta_{7} + $$12\!\cdots\!60$$ \beta_{8}) q^{23} +(-$$32\!\cdots\!60$$ + $$54\!\cdots\!52$$ \beta_{1} + $$11\!\cdots\!08$$ \beta_{2} + $$46\!\cdots\!44$$ \beta_{3} + $$15\!\cdots\!60$$ \beta_{4} - $$63\!\cdots\!52$$ \beta_{5} + $$16\!\cdots\!08$$ \beta_{6} + $$18\!\cdots\!20$$ \beta_{7} - $$14\!\cdots\!56$$ \beta_{8}) q^{24} +($$23\!\cdots\!75$$ + $$76\!\cdots\!50$$ \beta_{1} + $$12\!\cdots\!50$$ \beta_{2} + $$20\!\cdots\!50$$ \beta_{3} - $$96\!\cdots\!50$$ \beta_{4} + $$21\!\cdots\!00$$ \beta_{5} - $$67\!\cdots\!00$$ \beta_{6} + $$32\!\cdots\!50$$ \beta_{7} + $$14\!\cdots\!50$$ \beta_{8}) q^{25} +($$33\!\cdots\!98$$ - $$58\!\cdots\!06$$ \beta_{1} + $$12\!\cdots\!00$$ \beta_{2} + $$65\!\cdots\!56$$ \beta_{3} - $$55\!\cdots\!60$$ \beta_{4} + $$29\!\cdots\!96$$ \beta_{5} - $$14\!\cdots\!04$$ \beta_{6} - $$53\!\cdots\!60$$ \beta_{7} - $$13\!\cdots\!72$$ \beta_{8}) q^{26} +($$13\!\cdots\!82$$ + $$74\!\cdots\!46$$ \beta_{1} + $$91\!\cdots\!08$$ \beta_{2} - $$84\!\cdots\!72$$ \beta_{3} + $$70\!\cdots\!40$$ \beta_{4} - $$42\!\cdots\!42$$ \beta_{5} + $$48\!\cdots\!38$$ \beta_{6} + $$58\!\cdots\!36$$ \beta_{7} + $$11\!\cdots\!40$$ \beta_{8}) q^{27} +(-$$85\!\cdots\!36$$ - $$32\!\cdots\!80$$ \beta_{1} + $$22\!\cdots\!92$$ \beta_{2} - $$63\!\cdots\!32$$ \beta_{3} + $$65\!\cdots\!00$$ \beta_{4} + $$21\!\cdots\!48$$ \beta_{5} - $$38\!\cdots\!32$$ \beta_{6} - $$48\!\cdots\!24$$ \beta_{7} - $$91\!\cdots\!20$$ \beta_{8}) q^{28} +($$39\!\cdots\!69$$ + $$80\!\cdots\!49$$ \beta_{1} - $$17\!\cdots\!98$$ \beta_{2} + $$19\!\cdots\!10$$ \beta_{3} - $$36\!\cdots\!15$$ \beta_{4} + $$12\!\cdots\!56$$ \beta_{5} + $$11\!\cdots\!36$$ \beta_{6} + $$33\!\cdots\!20$$ \beta_{7} + $$66\!\cdots\!48$$ \beta_{8}) q^{29} +($$19\!\cdots\!58$$ - $$14\!\cdots\!22$$ \beta_{1} - $$25\!\cdots\!06$$ \beta_{2} + $$33\!\cdots\!80$$ \beta_{3} + $$12\!\cdots\!02$$ \beta_{4} - $$79\!\cdots\!30$$ \beta_{5} + $$90\!\cdots\!40$$ \beta_{6} - $$18\!\cdots\!60$$ \beta_{7} - $$43\!\cdots\!80$$ \beta_{8}) q^{30} +($$88\!\cdots\!68$$ + $$88\!\cdots\!24$$ \beta_{1} - $$76\!\cdots\!52$$ \beta_{2} - $$10\!\cdots\!04$$ \beta_{3} + $$56\!\cdots\!48$$ \beta_{4} + $$36\!\cdots\!20$$ \beta_{5} - $$14\!\cdots\!28$$ \beta_{6} + $$75\!\cdots\!40$$ \beta_{7} + $$26\!\cdots\!96$$ \beta_{8}) q^{31} +(-$$16\!\cdots\!28$$ - $$69\!\cdots\!40$$ \beta_{1} + $$84\!\cdots\!44$$ \beta_{2} - $$13\!\cdots\!08$$ \beta_{3} - $$36\!\cdots\!20$$ \beta_{4} + $$96\!\cdots\!04$$ \beta_{5} + $$10\!\cdots\!04$$ \beta_{6} - $$19\!\cdots\!20$$ \beta_{7} - $$14\!\cdots\!80$$ \beta_{8}) q^{32} +($$43\!\cdots\!10$$ - $$26\!\cdots\!29$$ \beta_{1} + $$21\!\cdots\!69$$ \beta_{2} + $$86\!\cdots\!91$$ \beta_{3} - $$15\!\cdots\!25$$ \beta_{4} - $$19\!\cdots\!86$$ \beta_{5} - $$49\!\cdots\!96$$ \beta_{6} - $$20\!\cdots\!79$$ \beta_{7} + $$69\!\cdots\!55$$ \beta_{8}) q^{33} +($$12\!\cdots\!46$$ - $$37\!\cdots\!70$$ \beta_{1} + $$15\!\cdots\!12$$ \beta_{2} + $$36\!\cdots\!80$$ \beta_{3} + $$16\!\cdots\!92$$ \beta_{4} + $$10\!\cdots\!96$$ \beta_{5} + $$11\!\cdots\!44$$ \beta_{6} + $$52\!\cdots\!60$$ \beta_{7} - $$29\!\cdots\!08$$ \beta_{8}) q^{34} +($$54\!\cdots\!76$$ - $$14\!\cdots\!28$$ \beta_{1} + $$11\!\cdots\!48$$ \beta_{2} - $$26\!\cdots\!40$$ \beta_{3} - $$11\!\cdots\!68$$ \beta_{4} - $$11\!\cdots\!76$$ \beta_{5} + $$30\!\cdots\!88$$ \beta_{6} - $$27\!\cdots\!72$$ \beta_{7} + $$10\!\cdots\!04$$ \beta_{8}) q^{35} +(-$$52\!\cdots\!14$$ - $$10\!\cdots\!58$$ \beta_{1} - $$29\!\cdots\!21$$ \beta_{2} - $$51\!\cdots\!11$$ \beta_{3} - $$32\!\cdots\!04$$ \beta_{4} - $$13\!\cdots\!44$$ \beta_{5} - $$39\!\cdots\!00$$ \beta_{6} + $$41\!\cdots\!80$$ \beta_{7} - $$29\!\cdots\!00$$ \beta_{8}) q^{36} +(-$$23\!\cdots\!87$$ - $$34\!\cdots\!45$$ \beta_{1} - $$84\!\cdots\!56$$ \beta_{2} - $$51\!\cdots\!04$$ \beta_{3} + $$94\!\cdots\!35$$ \beta_{4} + $$75\!\cdots\!56$$ \beta_{5} + $$17\!\cdots\!36$$ \beta_{6} + $$57\!\cdots\!82$$ \beta_{7} + $$39\!\cdots\!50$$ \beta_{8}) q^{37} +($$91\!\cdots\!89$$ - $$23\!\cdots\!71$$ \beta_{1} - $$13\!\cdots\!27$$ \beta_{2} + $$81\!\cdots\!98$$ \beta_{3} + $$38\!\cdots\!25$$ \beta_{4} - $$78\!\cdots\!31$$ \beta_{5} - $$26\!\cdots\!36$$ \beta_{6} - $$63\!\cdots\!76$$ \beta_{7} + $$18\!\cdots\!20$$ \beta_{8}) q^{38} +($$13\!\cdots\!32$$ - $$79\!\cdots\!55$$ \beta_{1} + $$11\!\cdots\!41$$ \beta_{2} + $$29\!\cdots\!84$$ \beta_{3} - $$24\!\cdots\!36$$ \beta_{4} - $$81\!\cdots\!53$$ \beta_{5} - $$12\!\cdots\!37$$ \beta_{6} + $$38\!\cdots\!40$$ \beta_{7} - $$18\!\cdots\!16$$ \beta_{8}) q^{39} +(-$$51\!\cdots\!50$$ - $$42\!\cdots\!30$$ \beta_{1} + $$96\!\cdots\!50$$ \beta_{2} - $$22\!\cdots\!50$$ \beta_{3} - $$20\!\cdots\!40$$ \beta_{4} + $$33\!\cdots\!80$$ \beta_{5} + $$86\!\cdots\!10$$ \beta_{6} - $$16\!\cdots\!90$$ \beta_{7} + $$88\!\cdots\!80$$ \beta_{8}) q^{40} +(-$$16\!\cdots\!62$$ - $$41\!\cdots\!22$$ \beta_{1} + $$51\!\cdots\!10$$ \beta_{2} + $$35\!\cdots\!34$$ \beta_{3} + $$27\!\cdots\!74$$ \beta_{4} + $$35\!\cdots\!64$$ \beta_{5} - $$10\!\cdots\!00$$ \beta_{6} + $$51\!\cdots\!70$$ \beta_{7} - $$29\!\cdots\!50$$ \beta_{8}) q^{41} +($$12\!\cdots\!80$$ - $$21\!\cdots\!96$$ \beta_{1} - $$12\!\cdots\!40$$ \beta_{2} + $$12\!\cdots\!68$$ \beta_{3} - $$22\!\cdots\!60$$ \beta_{4} - $$65\!\cdots\!84$$ \beta_{5} - $$14\!\cdots\!04$$ \beta_{6} - $$81\!\cdots\!60$$ \beta_{7} + $$58\!\cdots\!60$$ \beta_{8}) q^{42} +($$65\!\cdots\!03$$ + $$28\!\cdots\!76$$ \beta_{1} - $$27\!\cdots\!41$$ \beta_{2} + $$89\!\cdots\!76$$ \beta_{3} - $$26\!\cdots\!80$$ \beta_{4} + $$16\!\cdots\!40$$ \beta_{5} + $$11\!\cdots\!60$$ \beta_{6} - $$18\!\cdots\!76$$ \beta_{7} + $$34\!\cdots\!00$$ \beta_{8}) q^{43} +(-$$68\!\cdots\!44$$ + $$47\!\cdots\!44$$ \beta_{1} + $$36\!\cdots\!44$$ \beta_{2} + $$78\!\cdots\!08$$ \beta_{3} + $$29\!\cdots\!68$$ \beta_{4} + $$37\!\cdots\!60$$ \beta_{5} - $$44\!\cdots\!08$$ \beta_{6} + $$19\!\cdots\!00$$ \beta_{7} - $$90\!\cdots\!44$$ \beta_{8}) q^{44} +(-$$35\!\cdots\!67$$ + $$11\!\cdots\!63$$ \beta_{1} + $$46\!\cdots\!44$$ \beta_{2} - $$11\!\cdots\!20$$ \beta_{3} + $$54\!\cdots\!07$$ \beta_{4} - $$33\!\cdots\!40$$ \beta_{5} + $$10\!\cdots\!20$$ \beta_{6} - $$81\!\cdots\!30$$ \beta_{7} + $$44\!\cdots\!10$$ \beta_{8}) q^{45} +(-$$20\!\cdots\!62$$ + $$53\!\cdots\!18$$ \beta_{1} + $$92\!\cdots\!90$$ \beta_{2} + $$14\!\cdots\!44$$ \beta_{3} - $$13\!\cdots\!66$$ \beta_{4} + $$78\!\cdots\!34$$ \beta_{5} - $$11\!\cdots\!40$$ \beta_{6} + $$19\!\cdots\!80$$ \beta_{7} - $$12\!\cdots\!20$$ \beta_{8}) q^{46} +($$45\!\cdots\!20$$ + $$64\!\cdots\!90$$ \beta_{1} - $$18\!\cdots\!22$$ \beta_{2} + $$60\!\cdots\!72$$ \beta_{3} - $$27\!\cdots\!80$$ \beta_{4} + $$10\!\cdots\!50$$ \beta_{5} - $$12\!\cdots\!90$$ \beta_{6} - $$12\!\cdots\!32$$ \beta_{7} + $$15\!\cdots\!20$$ \beta_{8}) q^{47} +($$15\!\cdots\!00$$ - $$36\!\cdots\!64$$ \beta_{1} - $$11\!\cdots\!52$$ \beta_{2} - $$25\!\cdots\!28$$ \beta_{3} + $$13\!\cdots\!00$$ \beta_{4} - $$20\!\cdots\!76$$ \beta_{5} + $$34\!\cdots\!04$$ \beta_{6} - $$11\!\cdots\!20$$ \beta_{7} + $$56\!\cdots\!00$$ \beta_{8}) q^{48} +(-$$55\!\cdots\!51$$ - $$45\!\cdots\!52$$ \beta_{1} - $$21\!\cdots\!20$$ \beta_{2} - $$37\!\cdots\!16$$ \beta_{3} + $$16\!\cdots\!64$$ \beta_{4} - $$16\!\cdots\!56$$ \beta_{5} + $$15\!\cdots\!40$$ \beta_{6} + $$59\!\cdots\!80$$ \beta_{7} - $$42\!\cdots\!80$$ \beta_{8}) q^{49} +(-$$36\!\cdots\!25$$ - $$48\!\cdots\!75$$ \beta_{1} + $$70\!\cdots\!00$$ \beta_{2} - $$15\!\cdots\!00$$ \beta_{3} - $$12\!\cdots\!00$$ \beta_{4} + $$12\!\cdots\!00$$ \beta_{5} - $$51\!\cdots\!00$$ \beta_{6} - $$14\!\cdots\!00$$ \beta_{7} + $$14\!\cdots\!00$$ \beta_{8}) q^{50} +($$47\!\cdots\!38$$ - $$10\!\cdots\!50$$ \beta_{1} + $$18\!\cdots\!92$$ \beta_{2} + $$63\!\cdots\!88$$ \beta_{3} - $$39\!\cdots\!72$$ \beta_{4} - $$32\!\cdots\!86$$ \beta_{5} - $$22\!\cdots\!54$$ \beta_{6} + $$10\!\cdots\!40$$ \beta_{7} - $$22\!\cdots\!72$$ \beta_{8}) q^{51} +($$65\!\cdots\!08$$ - $$43\!\cdots\!24$$ \beta_{1} + $$62\!\cdots\!50$$ \beta_{2} + $$10\!\cdots\!42$$ \beta_{3} + $$98\!\cdots\!40$$ \beta_{4} + $$44\!\cdots\!80$$ \beta_{5} + $$20\!\cdots\!20$$ \beta_{6} + $$52\!\cdots\!08$$ \beta_{7} - $$32\!\cdots\!00$$ \beta_{8}) q^{52} +(-$$59\!\cdots\!91$$ + $$22\!\cdots\!71$$ \beta_{1} - $$16\!\cdots\!60$$ \beta_{2} - $$31\!\cdots\!68$$ \beta_{3} + $$10\!\cdots\!75$$ \beta_{4} - $$90\!\cdots\!88$$ \beta_{5} - $$66\!\cdots\!08$$ \beta_{6} - $$22\!\cdots\!46$$ \beta_{7} + $$33\!\cdots\!70$$ \beta_{8}) q^{53} +(-$$17\!\cdots\!66$$ + $$14\!\cdots\!14$$ \beta_{1} - $$52\!\cdots\!30$$ \beta_{2} - $$11\!\cdots\!20$$ \beta_{3} - $$89\!\cdots\!22$$ \beta_{4} + $$55\!\cdots\!10$$ \beta_{5} + $$81\!\cdots\!32$$ \beta_{6} + $$38\!\cdots\!40$$ \beta_{7} - $$10\!\cdots\!24$$ \beta_{8}) q^{54} +($$16\!\cdots\!48$$ + $$59\!\cdots\!63$$ \beta_{1} - $$77\!\cdots\!61$$ \beta_{2} + $$12\!\cdots\!80$$ \beta_{3} + $$12\!\cdots\!72$$ \beta_{4} - $$18\!\cdots\!75$$ \beta_{5} + $$18\!\cdots\!25$$ \beta_{6} + $$18\!\cdots\!00$$ \beta_{7} + $$15\!\cdots\!00$$ \beta_{8}) q^{55} +($$82\!\cdots\!64$$ + $$12\!\cdots\!08$$ \beta_{1} + $$25\!\cdots\!64$$ \beta_{2} + $$10\!\cdots\!48$$ \beta_{3} + $$12\!\cdots\!36$$ \beta_{4} + $$14\!\cdots\!60$$ \beta_{5} - $$10\!\cdots\!76$$ \beta_{6} - $$11\!\cdots\!20$$ \beta_{7} + $$26\!\cdots\!32$$ \beta_{8}) q^{56} +(-$$10\!\cdots\!98$$ + $$96\!\cdots\!49$$ \beta_{1} + $$13\!\cdots\!91$$ \beta_{2} - $$11\!\cdots\!99$$ \beta_{3} + $$83\!\cdots\!85$$ \beta_{4} + $$80\!\cdots\!50$$ \beta_{5} + $$20\!\cdots\!80$$ \beta_{6} - $$13\!\cdots\!81$$ \beta_{7} - $$22\!\cdots\!15$$ \beta_{8}) q^{57} +(-$$16\!\cdots\!70$$ - $$78\!\cdots\!94$$ \beta_{1} + $$96\!\cdots\!08$$ \beta_{2} - $$77\!\cdots\!60$$ \beta_{3} - $$60\!\cdots\!80$$ \beta_{4} - $$25\!\cdots\!16$$ \beta_{5} - $$14\!\cdots\!36$$ \beta_{6} + $$10\!\cdots\!92$$ \beta_{7} + $$61\!\cdots\!40$$ \beta_{8}) q^{58} +(-$$19\!\cdots\!21$$ - $$18\!\cdots\!36$$ \beta_{1} - $$46\!\cdots\!17$$ \beta_{2} + $$54\!\cdots\!20$$ \beta_{3} + $$98\!\cdots\!56$$ \beta_{4} + $$18\!\cdots\!64$$ \beta_{5} - $$52\!\cdots\!32$$ \beta_{6} - $$12\!\cdots\!80$$ \beta_{7} - $$58\!\cdots\!76$$ \beta_{8}) q^{59} +($$13\!\cdots\!16$$ - $$61\!\cdots\!28$$ \beta_{1} - $$19\!\cdots\!32$$ \beta_{2} + $$29\!\cdots\!60$$ \beta_{3} - $$12\!\cdots\!28$$ \beta_{4} + $$63\!\cdots\!64$$ \beta_{5} - $$41\!\cdots\!32$$ \beta_{6} - $$10\!\cdots\!92$$ \beta_{7} - $$23\!\cdots\!56$$ \beta_{8}) q^{60} +($$11\!\cdots\!97$$ + $$15\!\cdots\!03$$ \beta_{1} + $$77\!\cdots\!84$$ \beta_{2} + $$66\!\cdots\!56$$ \beta_{3} + $$35\!\cdots\!07$$ \beta_{4} + $$22\!\cdots\!28$$ \beta_{5} + $$38\!\cdots\!16$$ \beta_{6} + $$50\!\cdots\!50$$ \beta_{7} + $$11\!\cdots\!38$$ \beta_{8}) q^{61} +(-$$20\!\cdots\!56$$ - $$11\!\cdots\!48$$ \beta_{1} + $$52\!\cdots\!16$$ \beta_{2} - $$21\!\cdots\!92$$ \beta_{3} - $$21\!\cdots\!20$$ \beta_{4} - $$95\!\cdots\!44$$ \beta_{5} + $$11\!\cdots\!96$$ \beta_{6} - $$83\!\cdots\!40$$ \beta_{7} - $$24\!\cdots\!80$$ \beta_{8}) q^{62} +(-$$58\!\cdots\!92$$ + $$79\!\cdots\!67$$ \beta_{1} + $$24\!\cdots\!71$$ \beta_{2} - $$22\!\cdots\!76$$ \beta_{3} + $$34\!\cdots\!60$$ \beta_{4} + $$80\!\cdots\!09$$ \beta_{5} - $$54\!\cdots\!91$$ \beta_{6} - $$11\!\cdots\!12$$ \beta_{7} + $$13\!\cdots\!80$$ \beta_{8}) q^{63} +($$98\!\cdots\!04$$ + $$24\!\cdots\!56$$ \beta_{1} + $$12\!\cdots\!76$$ \beta_{2} + $$20\!\cdots\!80$$ \beta_{3} - $$14\!\cdots\!32$$ \beta_{4} + $$29\!\cdots\!68$$ \beta_{5} + $$79\!\cdots\!20$$ \beta_{6} + $$94\!\cdots\!60$$ \beta_{7} + $$14\!\cdots\!60$$ \beta_{8}) q^{64} +($$37\!\cdots\!12$$ + $$10\!\cdots\!34$$ \beta_{1} - $$14\!\cdots\!74$$ \beta_{2} + $$19\!\cdots\!70$$ \beta_{3} + $$15\!\cdots\!94$$ \beta_{4} - $$61\!\cdots\!32$$ \beta_{5} + $$12\!\cdots\!16$$ \beta_{6} - $$20\!\cdots\!54$$ \beta_{7} - $$45\!\cdots\!22$$ \beta_{8}) q^{65} +($$64\!\cdots\!12$$ - $$33\!\cdots\!40$$ \beta_{1} - $$31\!\cdots\!44$$ \beta_{2} + $$15\!\cdots\!04$$ \beta_{3} - $$62\!\cdots\!36$$ \beta_{4} - $$37\!\cdots\!28$$ \beta_{5} - $$66\!\cdots\!12$$ \beta_{6} + $$38\!\cdots\!20$$ \beta_{7} + $$51\!\cdots\!84$$ \beta_{8}) q^{66} +($$74\!\cdots\!65$$ - $$59\!\cdots\!54$$ \beta_{1} + $$26\!\cdots\!79$$ \beta_{2} - $$10\!\cdots\!92$$ \beta_{3} + $$33\!\cdots\!40$$ \beta_{4} - $$33\!\cdots\!66$$ \beta_{5} + $$71\!\cdots\!14$$ \beta_{6} + $$97\!\cdots\!24$$ \beta_{7} + $$10\!\cdots\!80$$ \beta_{8}) q^{67} +(-$$58\!\cdots\!72$$ - $$52\!\cdots\!72$$ \beta_{1} + $$15\!\cdots\!18$$ \beta_{2} - $$94\!\cdots\!02$$ \beta_{3} + $$18\!\cdots\!20$$ \beta_{4} + $$13\!\cdots\!68$$ \beta_{5} + $$14\!\cdots\!68$$ \beta_{6} - $$26\!\cdots\!24$$ \beta_{7} - $$57\!\cdots\!40$$ \beta_{8}) q^{68} +($$13\!\cdots\!76$$ - $$60\!\cdots\!94$$ \beta_{1} + $$17\!\cdots\!34$$ \beta_{2} - $$43\!\cdots\!02$$ \beta_{3} - $$15\!\cdots\!98$$ \beta_{4} - $$33\!\cdots\!00$$ \beta_{5} - $$35\!\cdots\!52$$ \beta_{6} + $$21\!\cdots\!30$$ \beta_{7} + $$10\!\cdots\!14$$ \beta_{8}) q^{69} +($$33\!\cdots\!64$$ + $$21\!\cdots\!44$$ \beta_{1} - $$27\!\cdots\!48$$ \beta_{2} + $$19\!\cdots\!40$$ \beta_{3} - $$39\!\cdots\!24$$ \beta_{4} - $$67\!\cdots\!60$$ \beta_{5} - $$57\!\cdots\!20$$ \beta_{6} + $$53\!\cdots\!80$$ \beta_{7} + $$89\!\cdots\!40$$ \beta_{8}) q^{70} +($$61\!\cdots\!12$$ + $$59\!\cdots\!79$$ \beta_{1} - $$10\!\cdots\!93$$ \beta_{2} - $$22\!\cdots\!52$$ \beta_{3} - $$10\!\cdots\!84$$ \beta_{4} + $$14\!\cdots\!89$$ \beta_{5} + $$28\!\cdots\!33$$ \beta_{6} - $$17\!\cdots\!20$$ \beta_{7} - $$52\!\cdots\!56$$ \beta_{8}) q^{71} +($$18\!\cdots\!19$$ + $$13\!\cdots\!31$$ \beta_{1} - $$16\!\cdots\!17$$ \beta_{2} + $$35\!\cdots\!41$$ \beta_{3} + $$57\!\cdots\!20$$ \beta_{4} - $$17\!\cdots\!38$$ \beta_{5} - $$81\!\cdots\!93$$ \beta_{6} + $$18\!\cdots\!55$$ \beta_{7} + $$13\!\cdots\!80$$ \beta_{8}) q^{72} +($$14\!\cdots\!56$$ + $$12\!\cdots\!33$$ \beta_{1} + $$26\!\cdots\!15$$ \beta_{2} - $$14\!\cdots\!07$$ \beta_{3} - $$55\!\cdots\!75$$ \beta_{4} - $$52\!\cdots\!06$$ \beta_{5} - $$14\!\cdots\!36$$ \beta_{6} + $$42\!\cdots\!79$$ \beta_{7} - $$86\!\cdots\!55$$ \beta_{8}) q^{73} +($$80\!\cdots\!30$$ + $$93\!\cdots\!34$$ \beta_{1} + $$88\!\cdots\!16$$ \beta_{2} + $$33\!\cdots\!00$$ \beta_{3} + $$35\!\cdots\!44$$ \beta_{4} - $$27\!\cdots\!52$$ \beta_{5} + $$30\!\cdots\!24$$ \beta_{6} + $$15\!\cdots\!40$$ \beta_{7} - $$34\!\cdots\!68$$ \beta_{8}) q^{74} +($$20\!\cdots\!75$$ - $$37\!\cdots\!00$$ \beta_{1} + $$25\!\cdots\!75$$ \beta_{2} - $$34\!\cdots\!00$$ \beta_{3} - $$36\!\cdots\!00$$ \beta_{4} + $$17\!\cdots\!00$$ \beta_{5} - $$35\!\cdots\!00$$ \beta_{6} - $$24\!\cdots\!00$$ \beta_{7} + $$11\!\cdots\!00$$ \beta_{8}) q^{75} +($$41\!\cdots\!68$$ - $$14\!\cdots\!04$$ \beta_{1} - $$19\!\cdots\!12$$ \beta_{2} + $$15\!\cdots\!36$$ \beta_{3} + $$59\!\cdots\!52$$ \beta_{4} - $$57\!\cdots\!40$$ \beta_{5} - $$72\!\cdots\!92$$ \beta_{6} - $$27\!\cdots\!80$$ \beta_{7} - $$12\!\cdots\!56$$ \beta_{8}) q^{76} +($$43\!\cdots\!92$$ - $$19\!\cdots\!50$$ \beta_{1} - $$70\!\cdots\!54$$ \beta_{2} - $$31\!\cdots\!18$$ \beta_{3} + $$96\!\cdots\!50$$ \beta_{4} - $$65\!\cdots\!36$$ \beta_{5} + $$78\!\cdots\!44$$ \beta_{6} + $$13\!\cdots\!90$$ \beta_{7} - $$14\!\cdots\!50$$ \beta_{8}) q^{77} +($$18\!\cdots\!66$$ - $$46\!\cdots\!86$$ \beta_{1} - $$16\!\cdots\!74$$ \beta_{2} - $$37\!\cdots\!64$$ \beta_{3} - $$25\!\cdots\!10$$ \beta_{4} + $$65\!\cdots\!58$$ \beta_{5} - $$29\!\cdots\!92$$ \beta_{6} - $$18\!\cdots\!92$$ \beta_{7} + $$75\!\cdots\!00$$ \beta_{8}) q^{78} +($$19\!\cdots\!28$$ + $$83\!\cdots\!14$$ \beta_{1} + $$61\!\cdots\!46$$ \beta_{2} - $$58\!\cdots\!68$$ \beta_{3} + $$46\!\cdots\!68$$ \beta_{4} + $$89\!\cdots\!30$$ \beta_{5} + $$50\!\cdots\!62$$ \beta_{6} - $$34\!\cdots\!00$$ \beta_{7} - $$10\!\cdots\!84$$ \beta_{8}) q^{79} +($$74\!\cdots\!44$$ + $$29\!\cdots\!24$$ \beta_{1} + $$37\!\cdots\!92$$ \beta_{2} + $$37\!\cdots\!40$$ \beta_{3} - $$77\!\cdots\!04$$ \beta_{4} + $$25\!\cdots\!40$$ \beta_{5} - $$15\!\cdots\!20$$ \beta_{6} + $$16\!\cdots\!80$$ \beta_{7} - $$17\!\cdots\!60$$ \beta_{8}) q^{80} +($$95\!\cdots\!79$$ + $$50\!\cdots\!19$$ \beta_{1} + $$15\!\cdots\!29$$ \beta_{2} - $$23\!\cdots\!13$$ \beta_{3} + $$32\!\cdots\!71$$ \beta_{4} - $$15\!\cdots\!06$$ \beta_{5} - $$19\!\cdots\!92$$ \beta_{6} - $$15\!\cdots\!15$$ \beta_{7} + $$31\!\cdots\!19$$ \beta_{8}) q^{81} +($$95\!\cdots\!02$$ + $$17\!\cdots\!38$$ \beta_{1} + $$40\!\cdots\!32$$ \beta_{2} - $$38\!\cdots\!04$$ \beta_{3} - $$13\!\cdots\!00$$ \beta_{4} + $$10\!\cdots\!36$$ \beta_{5} + $$86\!\cdots\!76$$ \beta_{6} - $$39\!\cdots\!88$$ \beta_{7} - $$50\!\cdots\!40$$ \beta_{8}) q^{82} +($$11\!\cdots\!81$$ + $$49\!\cdots\!52$$ \beta_{1} - $$32\!\cdots\!47$$ \beta_{2} - $$79\!\cdots\!20$$ \beta_{3} - $$58\!\cdots\!80$$ \beta_{4} + $$44\!\cdots\!80$$ \beta_{5} - $$14\!\cdots\!40$$ \beta_{6} + $$11\!\cdots\!80$$ \beta_{7} + $$25\!\cdots\!80$$ \beta_{8}) q^{83} +($$32\!\cdots\!20$$ - $$26\!\cdots\!24$$ \beta_{1} - $$13\!\cdots\!12$$ \beta_{2} + $$13\!\cdots\!24$$ \beta_{3} - $$50\!\cdots\!72$$ \beta_{4} - $$67\!\cdots\!72$$ \beta_{5} - $$48\!\cdots\!80$$ \beta_{6} + $$70\!\cdots\!80$$ \beta_{7} + $$19\!\cdots\!60$$ \beta_{8}) q^{84} +($$31\!\cdots\!06$$ - $$78\!\cdots\!88$$ \beta_{1} - $$14\!\cdots\!62$$ \beta_{2} + $$16\!\cdots\!10$$ \beta_{3} + $$21\!\cdots\!32$$ \beta_{4} - $$20\!\cdots\!36$$ \beta_{5} + $$32\!\cdots\!68$$ \beta_{6} - $$42\!\cdots\!42$$ \beta_{7} - $$31\!\cdots\!06$$ \beta_{8}) q^{85} +(-$$62\!\cdots\!33$$ - $$32\!\cdots\!33$$ \beta_{1} + $$23\!\cdots\!07$$ \beta_{2} + $$36\!\cdots\!54$$ \beta_{3} + $$45\!\cdots\!39$$ \beta_{4} - $$97\!\cdots\!45$$ \beta_{5} + $$50\!\cdots\!16$$ \beta_{6} + $$44\!\cdots\!60$$ \beta_{7} + $$19\!\cdots\!88$$ \beta_{8}) q^{86} +(-$$26\!\cdots\!60$$ - $$14\!\cdots\!51$$ \beta_{1} + $$76\!\cdots\!49$$ \beta_{2} - $$79\!\cdots\!16$$ \beta_{3} - $$12\!\cdots\!20$$ \beta_{4} + $$64\!\cdots\!31$$ \beta_{5} - $$64\!\cdots\!89$$ \beta_{6} + $$12\!\cdots\!44$$ \beta_{7} - $$93\!\cdots\!60$$ \beta_{8}) q^{87} +(-$$72\!\cdots\!36$$ - $$31\!\cdots\!56$$ \beta_{1} + $$25\!\cdots\!40$$ \beta_{2} - $$47\!\cdots\!08$$ \beta_{3} + $$10\!\cdots\!80$$ \beta_{4} - $$37\!\cdots\!76$$ \beta_{5} - $$18\!\cdots\!76$$ \beta_{6} - $$34\!\cdots\!60$$ \beta_{7} + $$17\!\cdots\!20$$ \beta_{8}) q^{88} +(-$$98\!\cdots\!68$$ + $$59\!\cdots\!65$$ \beta_{1} - $$10\!\cdots\!05$$ \beta_{2} + $$23\!\cdots\!41$$ \beta_{3} - $$41\!\cdots\!43$$ \beta_{4} - $$18\!\cdots\!34$$ \beta_{5} + $$74\!\cdots\!24$$ \beta_{6} + $$42\!\cdots\!35$$ \beta_{7} + $$63\!\cdots\!57$$ \beta_{8}) q^{89} +(-$$26\!\cdots\!18$$ + $$21\!\cdots\!14$$ \beta_{1} - $$45\!\cdots\!64$$ \beta_{2} + $$89\!\cdots\!20$$ \beta_{3} + $$58\!\cdots\!04$$ \beta_{4} + $$30\!\cdots\!08$$ \beta_{5} - $$38\!\cdots\!04$$ \beta_{6} + $$11\!\cdots\!76$$ \beta_{7} - $$86\!\cdots\!32$$ \beta_{8}) q^{90} +(-$$22\!\cdots\!24$$ - $$26\!\cdots\!72$$ \beta_{1} - $$38\!\cdots\!36$$ \beta_{2} + $$16\!\cdots\!48$$ \beta_{3} - $$15\!\cdots\!64$$ \beta_{4} - $$16\!\cdots\!00$$ \beta_{5} - $$20\!\cdots\!36$$ \beta_{6} - $$18\!\cdots\!00$$ \beta_{7} + $$14\!\cdots\!52$$ \beta_{8}) q^{91} +(-$$10\!\cdots\!88$$ - $$22\!\cdots\!72$$ \beta_{1} + $$53\!\cdots\!60$$ \beta_{2} - $$52\!\cdots\!04$$ \beta_{3} - $$12\!\cdots\!40$$ \beta_{4} + $$67\!\cdots\!96$$ \beta_{5} + $$43\!\cdots\!76$$ \beta_{6} - $$53\!\cdots\!48$$ \beta_{7} + $$12\!\cdots\!00$$ \beta_{8}) q^{92} +(-$$11\!\cdots\!56$$ - $$35\!\cdots\!64$$ \beta_{1} + $$29\!\cdots\!80$$ \beta_{2} + $$17\!\cdots\!84$$ \beta_{3} - $$43\!\cdots\!60$$ \beta_{4} - $$12\!\cdots\!48$$ \beta_{5} - $$20\!\cdots\!28$$ \beta_{6} + $$46\!\cdots\!32$$ \beta_{7} - $$40\!\cdots\!80$$ \beta_{8}) q^{93} +(-$$14\!\cdots\!80$$ - $$15\!\cdots\!96$$ \beta_{1} + $$38\!\cdots\!60$$ \beta_{2} - $$77\!\cdots\!40$$ \beta_{3} - $$40\!\cdots\!72$$ \beta_{4} - $$19\!\cdots\!60$$ \beta_{5} - $$34\!\cdots\!88$$ \beta_{6} - $$39\!\cdots\!20$$ \beta_{7} + $$52\!\cdots\!16$$ \beta_{8}) q^{94} +($$21\!\cdots\!00$$ - $$46\!\cdots\!95$$ \beta_{1} + $$54\!\cdots\!25$$ \beta_{2} + $$17\!\cdots\!00$$ \beta_{3} + $$14\!\cdots\!40$$ \beta_{4} + $$68\!\cdots\!95$$ \beta_{5} + $$89\!\cdots\!15$$ \beta_{6} - $$13\!\cdots\!60$$ \beta_{7} + $$12\!\cdots\!20$$ \beta_{8}) q^{95} +($$90\!\cdots\!12$$ - $$11\!\cdots\!84$$ \beta_{1} - $$25\!\cdots\!28$$ \beta_{2} + $$18\!\cdots\!72$$ \beta_{3} - $$16\!\cdots\!92$$ \beta_{4} - $$21\!\cdots\!92$$ \beta_{5} - $$20\!\cdots\!80$$ \beta_{6} - $$52\!\cdots\!20$$ \beta_{7} - $$49\!\cdots\!40$$ \beta_{8}) q^{96} +($$20\!\cdots\!52$$ + $$52\!\cdots\!41$$ \beta_{1} + $$93\!\cdots\!15$$ \beta_{2} - $$15\!\cdots\!47$$ \beta_{3} - $$10\!\cdots\!55$$ \beta_{4} - $$49\!\cdots\!82$$ \beta_{5} + $$22\!\cdots\!68$$ \beta_{6} + $$21\!\cdots\!71$$ \beta_{7} - $$10\!\cdots\!15$$ \beta_{8}) q^{97} +($$10\!\cdots\!89$$ + $$10\!\cdots\!67$$ \beta_{1} - $$38\!\cdots\!28$$ \beta_{2} - $$36\!\cdots\!64$$ \beta_{3} - $$13\!\cdots\!20$$ \beta_{4} - $$27\!\cdots\!84$$ \beta_{5} + $$50\!\cdots\!76$$ \beta_{6} + $$18\!\cdots\!92$$ \beta_{7} + $$11\!\cdots\!80$$ \beta_{8}) q^{98} +($$23\!\cdots\!75$$ + $$97\!\cdots\!92$$ \beta_{1} + $$12\!\cdots\!91$$ \beta_{2} - $$20\!\cdots\!88$$ \beta_{3} + $$46\!\cdots\!44$$ \beta_{4} + $$29\!\cdots\!60$$ \beta_{5} - $$13\!\cdots\!84$$ \beta_{6} - $$15\!\cdots\!20$$ \beta_{7} + $$85\!\cdots\!88$$ \beta_{8}) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9q + 5465779613435496q^{2} + $$15\!\cdots\!12$$q^{3} + $$64\!\cdots\!92$$q^{4} + $$24\!\cdots\!50$$q^{5} - $$12\!\cdots\!32$$q^{6} - $$93\!\cdots\!44$$q^{7} - $$10\!\cdots\!40$$q^{8} + $$36\!\cdots\!13$$q^{9} + O(q^{10}) $	$ 9q + 5465779613435496q^{2} + $$15\!\cdots\!12$$q^{3} + $$64\!\cdots\!92$$q^{4} + $$24\!\cdots\!50$$q^{5} - $$12\!\cdots\!32$$q^{6} - $$93\!\cdots\!44$$q^{7} - $$10\!\cdots\!40$$q^{8} + $$36\!\cdots\!13$$q^{9} - $$10\!\cdots\!00$$q^{10} + $$75\!\cdots\!48$$q^{11} - $$79\!\cdots\!44$$q^{12} + $$38\!\cdots\!82$$q^{13} - $$74\!\cdots\!16$$q^{14} + $$11\!\cdots\!00$$q^{15} + $$29\!\cdots\!64$$q^{16} + $$55\!\cdots\!26$$q^{17} + $$23\!\cdots\!72$$q^{18} + $$85\!\cdots\!60$$q^{19} + $$13\!\cdots\!00$$q^{20} + $$98\!\cdots\!48$$q^{21} - $$21\!\cdots\!88$$q^{22} - $$10\!\cdots\!48$$q^{23} - $$29\!\cdots\!20$$q^{24} + $$20\!\cdots\!75$$q^{25} + $$29\!\cdots\!48$$q^{26} + $$12\!\cdots\!40$$q^{27} - $$77\!\cdots\!72$$q^{28} + $$35\!\cdots\!90$$q^{29} + $$17\!\cdots\!00$$q^{30} + $$80\!\cdots\!08$$q^{31} - $$14\!\cdots\!64$$q^{32} + $$39\!\cdots\!64$$q^{33} + $$11\!\cdots\!64$$q^{34} + $$49\!\cdots\!00$$q^{35} - $$47\!\cdots\!56$$q^{36} - $$21\!\cdots\!34$$q^{37} + $$82\!\cdots\!40$$q^{38} + $$12\!\cdots\!56$$q^{39} - $$46\!\cdots\!00$$q^{40} - $$15\!\cdots\!62$$q^{41} + $$11\!\cdots\!12$$q^{42} + $$58\!\cdots\!92$$q^{43} - $$61\!\cdots\!76$$q^{44} - $$31\!\cdots\!50$$q^{45} - $$18\!\cdots\!72$$q^{46} + $$41\!\cdots\!36$$q^{47} + $$14\!\cdots\!52$$q^{48} - $$49\!\cdots\!63$$q^{49} - $$32\!\cdots\!00$$q^{50} + $$42\!\cdots\!08$$q^{51} + $$59\!\cdots\!16$$q^{52} - $$53\!\cdots\!38$$q^{53} - $$15\!\cdots\!40$$q^{54} + $$14\!\cdots\!00$$q^{55} + $$74\!\cdots\!40$$q^{56} - $$93\!\cdots\!20$$q^{57} - $$14\!\cdots\!40$$q^{58} - $$17\!\cdots\!20$$q^{59} + $$11\!\cdots\!00$$q^{60} + $$99\!\cdots\!98$$q^{61} - $$18\!\cdots\!48$$q^{62} - $$52\!\cdots\!08$$q^{63} + $$88\!\cdots\!12$$q^{64} + $$33\!\cdots\!00$$q^{65} + $$57\!\cdots\!96$$q^{66} + $$67\!\cdots\!76$$q^{67} - $$52\!\cdots\!12$$q^{68} + $$12\!\cdots\!16$$q^{69} + $$30\!\cdots\!00$$q^{70} + $$55\!\cdots\!28$$q^{71} + $$16\!\cdots\!20$$q^{72} + $$13\!\cdots\!02$$q^{73} + $$72\!\cdots\!24$$q^{74} + $$18\!\cdots\!00$$q^{75} + $$36\!\cdots\!80$$q^{76} + $$38\!\cdots\!32$$q^{77} + $$16\!\cdots\!64$$q^{78} + $$17\!\cdots\!40$$q^{79} + $$67\!\cdots\!00$$q^{80} + $$85\!\cdots\!89$$q^{81} + $$86\!\cdots\!72$$q^{82} + $$10\!\cdots\!72$$q^{83} + $$29\!\cdots\!24$$q^{84} + $$28\!\cdots\!00$$q^{85} - $$56\!\cdots\!12$$q^{86} - $$23\!\cdots\!80$$q^{87} - $$64\!\cdots\!80$$q^{88} - $$88\!\cdots\!30$$q^{89} - $$23\!\cdots\!00$$q^{90} - $$20\!\cdots\!72$$q^{91} - $$96\!\cdots\!24$$q^{92} - $$10\!\cdots\!56$$q^{93} - $$13\!\cdots\!96$$q^{94} + $$19\!\cdots\!00$$q^{95} + $$81\!\cdots\!88$$q^{96} + $$18\!\cdots\!86$$q^{97} + $$91\!\cdots\!28$$q^{98} + $$21\!\cdots\!36$$q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $x^{9} - 4 x^{8} - $$18\!\cdots\!64$$ x^{7} - $$73\!\cdots\!04$$ x^{6} + $$10\!\cdots\!46$$ x^{5} + $$37\!\cdots\!92$$ x^{4} - $$21\!\cdots\!48$$ x^{3} - $$45\!\cdots\!88$$ x^{2} + $$13\!\cdots\!17$$ x + $$27\!\cdots\!52$:

$\beta_{0}$	$=$	$ 1 $
$\beta_{1}$	$=$	$ 24 \nu - 11 $
$\beta_{2}$	$=$	$($$-$$12\!\cdots\!79$$ \nu^{8} - $$17\!\cdots\!29$$ \nu^{7} + $$22\!\cdots\!45$$ \nu^{6} + $$27\!\cdots\!23$$ \nu^{5} - $$10\!\cdots\!49$$ \nu^{4} - $$11\!\cdots\!03$$ \nu^{3} + $$11\!\cdots\!11$$ \nu^{2} + $$11\!\cdots\!21$$ \nu + $$21\!\cdots\!68$$)/ $$18\!\cdots\!24$
$\beta_{3}$	$=$	$($$-$$47\!\cdots\!19$$ \nu^{8} - $$66\!\cdots\!69$$ \nu^{7} + $$86\!\cdots\!45$$ \nu^{6} + $$10\!\cdots\!03$$ \nu^{5} - $$40\!\cdots\!89$$ \nu^{4} - $$44\!\cdots\!83$$ \nu^{3} + $$10\!\cdots\!43$$ \nu^{2} + $$42\!\cdots\!73$$ \nu - $$24\!\cdots\!20$$)/ $$10\!\cdots\!72$
$\beta_{4}$	$=$	$($$-$$32\!\cdots\!89$$ \nu^{8} + $$90\!\cdots\!09$$ \nu^{7} + $$58\!\cdots\!03$$ \nu^{6} - $$12\!\cdots\!75$$ \nu^{5} - $$29\!\cdots\!19$$ \nu^{4} + $$41\!\cdots\!75$$ \nu^{3} + $$28\!\cdots\!97$$ \nu^{2} - $$53\!\cdots\!37$$ \nu + $$16\!\cdots\!36$$)/ $$96\!\cdots\!00$
$\beta_{5}$	$=$	$($$59\!\cdots\!89$$ \nu^{8} + $$35\!\cdots\!91$$ \nu^{7} - $$91\!\cdots\!03$$ \nu^{6} - $$13\!\cdots\!25$$ \nu^{5} + $$38\!\cdots\!19$$ \nu^{4} + $$73\!\cdots\!25$$ \nu^{3} - $$40\!\cdots\!97$$ \nu^{2} - $$91\!\cdots\!63$$ \nu - $$93\!\cdots\!36$$)/ $$25\!\cdots\!00$
$\beta_{6}$	$=$	$($$-$$50\!\cdots\!53$$ \nu^{8} + $$54\!\cdots\!93$$ \nu^{7} + $$89\!\cdots\!31$$ \nu^{6} - $$76\!\cdots\!75$$ \nu^{5} - $$47\!\cdots\!63$$ \nu^{4} + $$50\!\cdots\!75$$ \nu^{3} + $$84\!\cdots\!69$$ \nu^{2} - $$88\!\cdots\!49$$ \nu - $$31\!\cdots\!28$$)/ $$11\!\cdots\!00$
$\beta_{7}$	$=$	$($$-$$26\!\cdots\!07$$ \nu^{8} + $$47\!\cdots\!67$$ \nu^{7} + $$44\!\cdots\!89$$ \nu^{6} - $$57\!\cdots\!25$$ \nu^{5} - $$21\!\cdots\!97$$ \nu^{4} + $$41\!\cdots\!25$$ \nu^{3} + $$28\!\cdots\!11$$ \nu^{2} - $$13\!\cdots\!31$$ \nu + $$53\!\cdots\!68$$)/ $$11\!\cdots\!00$
$\beta_{8}$	$=$	$($$-$$45\!\cdots\!37$$ \nu^{8} + $$14\!\cdots\!97$$ \nu^{7} + $$10\!\cdots\!99$$ \nu^{6} + $$31\!\cdots\!25$$ \nu^{5} - $$90\!\cdots\!27$$ \nu^{4} - $$18\!\cdots\!25$$ \nu^{3} + $$26\!\cdots\!01$$ \nu^{2} + $$43\!\cdots\!79$$ \nu - $$14\!\cdots\!12$$)/ $$76\!\cdots\!00$

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

$1$	$=$	$\beta_0$
$\nu$	$=$	$($$\beta_{1} + 11$$)/24$
$\nu^{2}$	$=$	$($$\beta_{3} - 664037 \beta_{2} + 1454496237795986 \beta_{1} + 233205805914080463879155578823890$$)/576$
$\nu^{3}$	$=$	$($$\beta_{7} - 287 \beta_{6} + 326098 \beta_{5} - 11536093720 \beta_{4} + 6080152289344487 \beta_{3} - 1729190174695774540615 \beta_{2} + 389982933498588803529772202388069 \beta_{1} + 339196967390022887798427489622681198989944874045$$)/13824$
$\nu^{4}$	$=$	$($$-399171419712 \beta_{8} + 1032326586055479 \beta_{7} - 1073780444029653737 \beta_{6} + 398942241277981412382 \beta_{5} + 88256816768613430614080024 \beta_{4} + 73419319872788947466478796229417 \beta_{3} - 40830610558608177706493963427534012393 \beta_{2} + 229589003907921368619341708370163101878428065411 \beta_{1} + 11368285537427622943708687074161312670632101143477642887442636587$$)/41472$
$\nu^{5}$	$=$	$($$1040276766472715976193601280 \beta_{8} + 5386861631393577261729528541249 \beta_{7} - 2503564908944438757999586187538911 \beta_{6} + 965686401281995211115857145364280914 \beta_{5} - 13058856798979542643874920464718010944280 \beta_{4} + 45718502092925457823634876203711469669762023863 \beta_{3} - 16942881902956622105520393094251233220884436358727511 \beta_{2} + 1438922850832020154332150699622892850112530279258047393493043109 \beta_{1} + 3346343043049345609121544223011877028625201201516347548008693867606683570380605$$)/62208$
$\nu^{6}$	$=$	$($$-$$69\!\cdots\!00$$ \beta_{8} + $$88\!\cdots\!81$$ \beta_{7} - $$73\!\cdots\!19$$ \beta_{6} + $$58\!\cdots\!94$$ \beta_{5} + $$61\!\cdots\!88$$ \beta_{4} + $$35\!\cdots\!43$$ \beta_{3} - $$16\!\cdots\!35$$ \beta_{2} + $$17\!\cdots\!85$$ \beta_{1} + $$46\!\cdots\!93$$)/20736$
$\nu^{7}$	$=$	$($$10\!\cdots\!80$$ \beta_{8} + $$37\!\cdots\!81$$ \beta_{7} - $$21\!\cdots\!15$$ \beta_{6} + $$41\!\cdots\!14$$ \beta_{5} + $$20\!\cdots\!24$$ \beta_{4} + $$39\!\cdots\!51$$ \beta_{3} - $$20\!\cdots\!27$$ \beta_{2} + $$86\!\cdots\!33$$ \beta_{1} + $$33\!\cdots\!45$$)/41472$
$\nu^{8}$	$=$	$($$42\!\cdots\!96$$ \beta_{8} + $$69\!\cdots\!03$$ \beta_{7} - $$51\!\cdots\!17$$ \beta_{6} + $$51\!\cdots\!14$$ \beta_{5} + $$40\!\cdots\!76$$ \beta_{4} + $$21\!\cdots\!81$$ \beta_{3} - $$85\!\cdots\!13$$ \beta_{2} + $$13\!\cdots\!47$$ \beta_{1} + $$25\!\cdots\!07$$)/124416$

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

1.03210e15
6.73733e14
4.82720e14
3.47594e14
−2.01415e13
−3.22655e14
−5.32911e14
−7.86617e14
−8.73826e14

−2.41632e16

−3.68177e24

4.21601e32

3.92322e37

8.89632e40

−1.24740e45

−6.26652e48

−1.11358e51

−9.47976e53

1.2

−1.55623e16

1.35366e25

7.99254e31

−3.05298e37

−2.10660e41

6.30256e44

1.28130e48

−9.43892e50

4.75113e53

1.3

−1.09780e16

−5.69800e25

−4.17435e31

2.81503e36

6.25524e41

3.42403e44

2.23954e48

2.11959e51

−3.09033e52

1.4

−7.73495e15

6.57221e25

−1.02430e32

3.43777e37

−5.08357e41

9.22106e44

2.04736e48

3.19227e51

−2.65909e53

1.5

1.09070e15

6.73736e23

−1.61070e32

1.05761e37

7.34847e38

−2.03867e45

−3.52656e47

−1.12668e51

1.15354e52

1.6

8.35103e15

−1.73817e25

−9.25196e31

3.17576e36

−1.45155e41

2.61780e45

−2.12767e48

−8.25006e50

2.65209e52

1.7

1.33972e16

4.78035e25

1.72251e31

−4.75138e37

6.40432e41

−7.19244e44

−1.94305e48

1.15805e51

−6.36552e53

1.8

1.94861e16

−5.50693e25

2.17450e32

−2.12470e37

−1.07309e42

−2.04209e45

1.07545e48

1.90550e51

−4.14022e53

1.9

2.15791e16

2.13607e25

3.03400e32

3.38441e37

4.60946e41

5.98905e44

3.04570e48

−6.70851e50

7.30327e53

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	1.108.a.a	✓	9

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.108.a.a	✓	9	1.a	even	1	1	trivial

Hecke kernels

This newform subspace is the entire newspace $S_{108}^{\mathrm{new}}(\Gamma_0(1))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ -$$16\!\cdots\!56$$ + $$14\!\cdots\!36$$ T + $$95\!\cdots\!64$$ T^{2} - $$39\!\cdots\!84$$ T^{3} - $$13\!\cdots\!56$$ T^{4} + $$33\!\cdots\!36$$ T^{5} + $$54\!\cdots\!44$$ T^{6} - $$10\!\cdots\!64$$ T^{7} - 5465779613435496 T^{8} + T^{9} $
$3$	$ -$$12\!\cdots\!52$$ + $$15\!\cdots\!96$$ T + $$40\!\cdots\!32$$ T^{2} - $$37\!\cdots\!36$$ T^{3} - $$16\!\cdots\!92$$ T^{4} + $$12\!\cdots\!16$$ T^{5} + $$93\!\cdots\!12$$ T^{6} - $$67\!\cdots\!76$$ T^{7} - $$15\!\cdots\!12$$ T^{8} + T^{9} $
$5$	$13\!\cdots\!00$$ - $$99\!\cdots\!00$$ T + $$21\!\cdots\!00$$ T^{2} - $$72\!\cdots\!00$$ T^{3} - $$88\!\cdots\!00$$ T^{4} + $$32\!\cdots\!00$$ T^{5} + $$93\!\cdots\!00$$ T^{6} - $$35\!\cdots\!00$$ T^{7} - $$24\!\cdots\!50$$ T^{8} + T^{9} $
$7$	$ -$$11\!\cdots\!96$$ + $$52\!\cdots\!16$$ T - $$10\!\cdots\!56$$ T^{2} - $$17\!\cdots\!24$$ T^{3} + $$10\!\cdots\!44$$ T^{4} + $$19\!\cdots\!76$$ T^{5} - $$87\!\cdots\!36$$ T^{6} - $$90\!\cdots\!44$$ T^{7} + $$93\!\cdots\!44$$ T^{8} + T^{9} $
$11$	$ -$$20\!\cdots\!92$$ + $$27\!\cdots\!24$$ T + $$42\!\cdots\!32$$ T^{2} - $$57\!\cdots\!64$$ T^{3} - $$29\!\cdots\!32$$ T^{4} + $$39\!\cdots\!56$$ T^{5} + $$82\!\cdots\!68$$ T^{6} - $$11\!\cdots\!76$$ T^{7} - $$75\!\cdots\!48$$ T^{8} + T^{9} $
$13$	$ -$$74\!\cdots\!32$$ + $$56\!\cdots\!76$$ T + $$41\!\cdots\!72$$ T^{2} - $$20\!\cdots\!96$$ T^{3} - $$57\!\cdots\!92$$ T^{4} + $$21\!\cdots\!56$$ T^{5} + $$28\!\cdots\!72$$ T^{6} - $$84\!\cdots\!96$$ T^{7} - $$38\!\cdots\!82$$ T^{8} + T^{9} $
$17$	$28\!\cdots\!24$$ - $$23\!\cdots\!24$$ T + $$36\!\cdots\!04$$ T^{2} - $$15\!\cdots\!04$$ T^{3} - $$13\!\cdots\!56$$ T^{4} + $$12\!\cdots\!56$$ T^{5} + $$62\!\cdots\!04$$ T^{6} - $$20\!\cdots\!04$$ T^{7} - $$55\!\cdots\!26$$ T^{8} + T^{9} $
$19$	$ -$$14\!\cdots\!00$$ + $$16\!\cdots\!00$$ T - $$51\!\cdots\!00$$ T^{2} + $$51\!\cdots\!00$$ T^{3} + $$38\!\cdots\!00$$ T^{4} - $$28\!\cdots\!00$$ T^{5} + $$11\!\cdots\!00$$ T^{6} + $$67\!\cdots\!00$$ T^{7} - $$85\!\cdots\!60$$ T^{8} + T^{9} $
$23$	$25\!\cdots\!88$$ + $$16\!\cdots\!56$$ T - $$32\!\cdots\!88$$ T^{2} - $$29\!\cdots\!56$$ T^{3} + $$12\!\cdots\!08$$ T^{4} + $$12\!\cdots\!96$$ T^{5} - $$18\!\cdots\!68$$ T^{6} - $$18\!\cdots\!16$$ T^{7} + $$10\!\cdots\!48$$ T^{8} + T^{9} $
$29$	$ -$$11\!\cdots\!00$$ - $$88\!\cdots\!00$$ T + $$13\!\cdots\!00$$ T^{2} + $$51\!\cdots\!00$$ T^{3} - $$13\!\cdots\!00$$ T^{4} + $$91\!\cdots\!00$$ T^{5} + $$38\!\cdots\!00$$ T^{6} - $$81\!\cdots\!00$$ T^{7} - $$35\!\cdots\!90$$ T^{8} + T^{9} $
$31$	$48\!\cdots\!48$$ - $$17\!\cdots\!36$$ T + $$51\!\cdots\!12$$ T^{2} + $$21\!\cdots\!56$$ T^{3} - $$17\!\cdots\!32$$ T^{4} + $$20\!\cdots\!36$$ T^{5} + $$10\!\cdots\!48$$ T^{6} - $$13\!\cdots\!16$$ T^{7} - $$80\!\cdots\!08$$ T^{8} + T^{9} $
$37$	$ -$$77\!\cdots\!36$$ + $$17\!\cdots\!96$$ T - $$24\!\cdots\!76$$ T^{2} - $$88\!\cdots\!64$$ T^{3} + $$24\!\cdots\!44$$ T^{4} + $$29\!\cdots\!16$$ T^{5} - $$42\!\cdots\!16$$ T^{6} - $$30\!\cdots\!24$$ T^{7} + $$21\!\cdots\!34$$ T^{8} + T^{9} $
$41$	$ -$$24\!\cdots\!32$$ + $$14\!\cdots\!84$$ T - $$16\!\cdots\!48$$ T^{2} - $$44\!\cdots\!84$$ T^{3} + $$12\!\cdots\!68$$ T^{4} + $$39\!\cdots\!76$$ T^{5} - $$10\!\cdots\!12$$ T^{6} - $$11\!\cdots\!36$$ T^{7} + $$15\!\cdots\!62$$ T^{8} + T^{9} $
$43$	$29\!\cdots\!28$$ + $$32\!\cdots\!16$$ T + $$68\!\cdots\!92$$ T^{2} - $$12\!\cdots\!76$$ T^{3} - $$11\!\cdots\!92$$ T^{4} + $$12\!\cdots\!76$$ T^{5} + $$16\!\cdots\!52$$ T^{6} - $$21\!\cdots\!56$$ T^{7} - $$58\!\cdots\!92$$ T^{8} + T^{9} $
$47$	$ -$$14\!\cdots\!16$$ + $$45\!\cdots\!56$$ T + $$10\!\cdots\!84$$ T^{2} - $$48\!\cdots\!44$$ T^{3} - $$10\!\cdots\!56$$ T^{4} + $$26\!\cdots\!96$$ T^{5} + $$37\!\cdots\!24$$ T^{6} - $$81\!\cdots\!84$$ T^{7} - $$41\!\cdots\!36$$ T^{8} + T^{9} $
$53$	$12\!\cdots\!48$$ + $$12\!\cdots\!96$$ T - $$38\!\cdots\!68$$ T^{2} - $$42\!\cdots\!36$$ T^{3} + $$26\!\cdots\!08$$ T^{4} + $$34\!\cdots\!16$$ T^{5} - $$67\!\cdots\!88$$ T^{6} - $$10\!\cdots\!76$$ T^{7} + $$53\!\cdots\!38$$ T^{8} + T^{9} $
$59$	$ -$$12\!\cdots\!00$$ + $$11\!\cdots\!00$$ T + $$56\!\cdots\!00$$ T^{2} - $$26\!\cdots\!00$$ T^{3} + $$17\!\cdots\!00$$ T^{4} + $$23\!\cdots\!00$$ T^{5} - $$14\!\cdots\!00$$ T^{6} - $$90\!\cdots\!00$$ T^{7} + $$17\!\cdots\!20$$ T^{8} + T^{9} $
$61$	$12\!\cdots\!08$$ - $$58\!\cdots\!76$$ T - $$22\!\cdots\!68$$ T^{2} + $$21\!\cdots\!36$$ T^{3} - $$15\!\cdots\!32$$ T^{4} - $$57\!\cdots\!44$$ T^{5} + $$18\!\cdots\!68$$ T^{6} + $$82\!\cdots\!24$$ T^{7} - $$99\!\cdots\!98$$ T^{8} + T^{9} $
$67$	$11\!\cdots\!24$$ + $$10\!\cdots\!76$$ T - $$35\!\cdots\!96$$ T^{2} - $$52\!\cdots\!04$$ T^{3} - $$13\!\cdots\!56$$ T^{4} + $$75\!\cdots\!56$$ T^{5} + $$23\!\cdots\!04$$ T^{6} - $$47\!\cdots\!04$$ T^{7} - $$67\!\cdots\!76$$ T^{8} + T^{9} $
$71$	$ -$$13\!\cdots\!72$$ + $$92\!\cdots\!44$$ T + $$11\!\cdots\!72$$ T^{2} - $$15\!\cdots\!04$$ T^{3} - $$11\!\cdots\!32$$ T^{4} + $$19\!\cdots\!96$$ T^{5} + $$44\!\cdots\!08$$ T^{6} - $$79\!\cdots\!96$$ T^{7} - $$55\!\cdots\!28$$ T^{8} + T^{9} $
$73$	$ -$$75\!\cdots\!12$$ - $$31\!\cdots\!44$$ T - $$12\!\cdots\!88$$ T^{2} + $$40\!\cdots\!44$$ T^{3} + $$67\!\cdots\!08$$ T^{4} - $$19\!\cdots\!04$$ T^{5} + $$55\!\cdots\!32$$ T^{6} - $$18\!\cdots\!16$$ T^{7} - $$13\!\cdots\!02$$ T^{8} + T^{9} $
$79$	$ -$$51\!\cdots\!00$$ + $$17\!\cdots\!00$$ T - $$18\!\cdots\!00$$ T^{2} - $$18\!\cdots\!00$$ T^{3} + $$11\!\cdots\!00$$ T^{4} + $$15\!\cdots\!00$$ T^{5} - $$33\!\cdots\!00$$ T^{6} + $$11\!\cdots\!00$$ T^{7} - $$17\!\cdots\!40$$ T^{8} + T^{9} $
$83$	$ -$$26\!\cdots\!92$$ - $$10\!\cdots\!64$$ T + $$11\!\cdots\!52$$ T^{2} + $$19\!\cdots\!84$$ T^{3} - $$17\!\cdots\!92$$ T^{4} + $$17\!\cdots\!36$$ T^{5} + $$79\!\cdots\!92$$ T^{6} - $$54\!\cdots\!36$$ T^{7} - $$10\!\cdots\!72$$ T^{8} + T^{9} $
$89$	$ -$$19\!\cdots\!00$$ - $$34\!\cdots\!00$$ T - $$42\!\cdots\!00$$ T^{2} + $$53\!\cdots\!00$$ T^{3} - $$32\!\cdots\!00$$ T^{4} - $$67\!\cdots\!00$$ T^{5} - $$10\!\cdots\!00$$ T^{6} + $$20\!\cdots\!00$$ T^{7} + $$88\!\cdots\!30$$ T^{8} + T^{9} $
$97$	$ -$$76\!\cdots\!16$$ + $$17\!\cdots\!56$$ T - $$14\!\cdots\!16$$ T^{2} + $$55\!\cdots\!56$$ T^{3} - $$83\!\cdots\!56$$ T^{4} - $$60\!\cdots\!04$$ T^{5} + $$91\!\cdots\!24$$ T^{6} - $$38\!\cdots\!84$$ T^{7} - $$18\!\cdots\!86$$ T^{8} + T^{9} $
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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:	\( k \)	\(=\)	\( 108 \)
Character orbit:	\([\chi]\)	\(=\)	1.a (trivial)

\(f(q)\)	\(=\)	\( q +(607308845937277 - \beta_{1}) q^{2} +(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3} +(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +O(q^{10})\)	\( q +(607308845937277 - \beta_{1}) q^{2} +(\)\(17\!\cdots\!33\)\( + 62113396 \beta_{1} + \beta_{2}) q^{3} +(\)\(71\!\cdots\!70\)\( + 239878545921410 \beta_{1} - 664037 \beta_{2} + \beta_{3}) q^{4} +(\)\(27\!\cdots\!09\)\( + \)\(50\!\cdots\!29\)\( \beta_{1} + 72785241362 \beta_{2} + 37490 \beta_{3} + \beta_{4}) q^{5} +(-\)\(13\!\cdots\!31\)\( + \)\(25\!\cdots\!77\)\( \beta_{1} + 2153888516744697 \beta_{2} + 14282730 \beta_{3} - 459 \beta_{4} + \beta_{5}) q^{6} +(-\)\(10\!\cdots\!24\)\( + \)\(32\!\cdots\!71\)\( \beta_{1} + 7204540145655150787 \beta_{2} - 2014591938312 \beta_{3} + 9986300 \beta_{4} + 381 \beta_{5} + \beta_{6}) q^{7} +(-\)\(11\!\cdots\!77\)\( - \)\(63\!\cdots\!41\)\( \beta_{1} + \)\(51\!\cdots\!47\)\( \beta_{2} - 4258225751532623 \beta_{3} + 11536093720 \beta_{4} - 326098 \beta_{5} + 287 \beta_{6} - \beta_{7}) q^{8} +(\)\(41\!\cdots\!23\)\( - \)\(10\!\cdots\!03\)\( \beta_{1} + \)\(17\!\cdots\!19\)\( \beta_{2} - 2240805579461315103 \beta_{3} - 954963839103 \beta_{4} - 675589406 \beta_{5} - 63368 \beta_{6} - 245 \beta_{7} + \beta_{8}) q^{9} +(-\)\(11\!\cdots\!14\)\( - \)\(85\!\cdots\!98\)\( \beta_{1} + \)\(34\!\cdots\!28\)\( \beta_{2} - \)\(42\!\cdots\!40\)\( \beta_{3} + 3035644428618332 \beta_{4} + 137787538404 \beta_{5} + 147169648 \beta_{6} - 120312 \beta_{7} - 216 \beta_{8}) q^{10} +(\)\(83\!\cdots\!59\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(26\!\cdots\!77\)\( \beta_{2} - \)\(30\!\cdots\!24\)\( \beta_{3} + 39655237059879084 \beta_{4} + 18943495258882 \beta_{5} + 20412816178 \beta_{6} + 12038260 \beta_{7} + 23004 \beta_{8}) q^{11} +(-\)\(88\!\cdots\!96\)\( + \)\(84\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!32\)\( \beta_{3} + 17417687351454383040 \beta_{4} + 8277620636103552 \beta_{5} + 1126938004992 \beta_{6} - 91849536 \beta_{7} - 1609920 \beta_{8}) q^{12} +(\)\(42\!\cdots\!13\)\( - \)\(29\!\cdots\!41\)\( \beta_{1} + \)\(84\!\cdots\!80\)\( \beta_{2} + \)\(12\!\cdots\!40\)\( \beta_{3} - 1292007267852162005 \beta_{4} + 688583074305208092 \beta_{5} - 165811929566128 \beta_{6} - 46020832614 \beta_{7} + 83255310 \beta_{8}) q^{13} +(-\)\(82\!\cdots\!78\)\( + \)\(43\!\cdots\!78\)\( \beta_{1} + \)\(78\!\cdots\!18\)\( \beta_{2} + \)\(22\!\cdots\!84\)\( \beta_{3} + \)\(34\!\cdots\!22\)\( \beta_{4} + 17161982422390714858 \beta_{5} + 4494593484494656 \beta_{6} + 3819461803360 \beta_{7} - 3391893792 \beta_{8}) q^{14} +(\)\(12\!\cdots\!72\)\( - \)\(49\!\cdots\!31\)\( \beta_{1} - \)\(43\!\cdots\!19\)\( \beta_{2} + \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(28\!\cdots\!84\)\( \beta_{4} - \)\(14\!\cdots\!57\)\( \beta_{5} + 71321094534085191 \beta_{6} - 177279761520504 \beta_{7} + 113342680728 \beta_{8}) q^{15} +(\)\(32\!\cdots\!36\)\( + \)\(77\!\cdots\!76\)\( \beta_{1} - \)\(67\!\cdots\!24\)\( \beta_{2} + \)\(88\!\cdots\!92\)\( \beta_{3} + \)\(73\!\cdots\!32\)\( \beta_{4} + \)\(23\!\cdots\!60\)\( \beta_{5} - 7893052997101223272 \beta_{6} + 5829377304694680 \beta_{7} - 3193371357696 \beta_{8}) q^{16} +(\)\(61\!\cdots\!04\)\( - \)\(51\!\cdots\!51\)\( \beta_{1} + \)\(30\!\cdots\!87\)\( \beta_{2} - \)\(44\!\cdots\!23\)\( \beta_{3} + \)\(45\!\cdots\!85\)\( \beta_{4} + \)\(86\!\cdots\!14\)\( \beta_{5} + \)\(25\!\cdots\!84\)\( \beta_{6} - 147577032261572145 \beta_{7} + 77391112194765 \beta_{8}) q^{17} +(\)\(26\!\cdots\!33\)\( - \)\(54\!\cdots\!45\)\( \beta_{1} - \)\(12\!\cdots\!88\)\( \beta_{2} + \)\(56\!\cdots\!08\)\( \beta_{3} - \)\(44\!\cdots\!20\)\( \beta_{4} - \)\(14\!\cdots\!04\)\( \beta_{5} - \)\(45\!\cdots\!64\)\( \beta_{6} + 2995387893492815280 \beta_{7} - 1637786543560080 \beta_{8}) q^{18} +(\)\(95\!\cdots\!49\)\( - \)\(36\!\cdots\!30\)\( \beta_{1} - \)\(75\!\cdots\!37\)\( \beta_{2} + \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!68\)\( \beta_{4} - \)\(67\!\cdots\!94\)\( \beta_{5} + \)\(45\!\cdots\!14\)\( \beta_{6} - 49760167465115702820 \beta_{7} + 30620433423116052 \beta_{8}) q^{19} +(\)\(14\!\cdots\!52\)\( + \)\(10\!\cdots\!32\)\( \beta_{1} + \)\(59\!\cdots\!86\)\( \beta_{2} + \)\(26\!\cdots\!70\)\( \beta_{3} + \)\(90\!\cdots\!88\)\( \beta_{4} + \)\(51\!\cdots\!80\)\( \beta_{5} + \)\(96\!\cdots\!60\)\( \beta_{6} + \)\(68\!\cdots\!60\)\( \beta_{7} - 510445792877080320 \beta_{8}) q^{20} +(\)\(10\!\cdots\!08\)\( - \)\(50\!\cdots\!94\)\( \beta_{1} - \)\(12\!\cdots\!98\)\( \beta_{2} + \)\(67\!\cdots\!70\)\( \beta_{3} + \)\(16\!\cdots\!94\)\( \beta_{4} - \)\(69\!\cdots\!64\)\( \beta_{5} - \)\(89\!\cdots\!68\)\( \beta_{6} - \)\(76\!\cdots\!30\)\( \beta_{7} + 7643011086349644726 \beta_{8}) q^{21} +(-\)\(23\!\cdots\!17\)\( - \)\(35\!\cdots\!21\)\( \beta_{1} + \)\(45\!\cdots\!75\)\( \beta_{2} - \)\(25\!\cdots\!70\)\( \beta_{3} - \)\(13\!\cdots\!85\)\( \beta_{4} + \)\(23\!\cdots\!47\)\( \beta_{5} + \)\(17\!\cdots\!32\)\( \beta_{6} + \)\(66\!\cdots\!56\)\( \beta_{7} - \)\(10\!\cdots\!60\)\( \beta_{8}) q^{22} +(-\)\(11\!\cdots\!80\)\( + \)\(57\!\cdots\!65\)\( \beta_{1} + \)\(88\!\cdots\!29\)\( \beta_{2} - \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!95\)\( \beta_{5} - \)\(20\!\cdots\!45\)\( \beta_{6} - \)\(39\!\cdots\!00\)\( \beta_{7} + \)\(12\!\cdots\!60\)\( \beta_{8}) q^{23} +(-\)\(32\!\cdots\!60\)\( + \)\(54\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!08\)\( \beta_{2} + \)\(46\!\cdots\!44\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4} - \)\(63\!\cdots\!52\)\( \beta_{5} + \)\(16\!\cdots\!08\)\( \beta_{6} + \)\(18\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!56\)\( \beta_{8}) q^{24} +(\)\(23\!\cdots\!75\)\( + \)\(76\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3} - \)\(96\!\cdots\!50\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5} - \)\(67\!\cdots\!00\)\( \beta_{6} + \)\(32\!\cdots\!50\)\( \beta_{7} + \)\(14\!\cdots\!50\)\( \beta_{8}) q^{25} +(\)\(33\!\cdots\!98\)\( - \)\(58\!\cdots\!06\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(65\!\cdots\!56\)\( \beta_{3} - \)\(55\!\cdots\!60\)\( \beta_{4} + \)\(29\!\cdots\!96\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(53\!\cdots\!60\)\( \beta_{7} - \)\(13\!\cdots\!72\)\( \beta_{8}) q^{26} +(\)\(13\!\cdots\!82\)\( + \)\(74\!\cdots\!46\)\( \beta_{1} + \)\(91\!\cdots\!08\)\( \beta_{2} - \)\(84\!\cdots\!72\)\( \beta_{3} + \)\(70\!\cdots\!40\)\( \beta_{4} - \)\(42\!\cdots\!42\)\( \beta_{5} + \)\(48\!\cdots\!38\)\( \beta_{6} + \)\(58\!\cdots\!36\)\( \beta_{7} + \)\(11\!\cdots\!40\)\( \beta_{8}) q^{27} +(-\)\(85\!\cdots\!36\)\( - \)\(32\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2} - \)\(63\!\cdots\!32\)\( \beta_{3} + \)\(65\!\cdots\!00\)\( \beta_{4} + \)\(21\!\cdots\!48\)\( \beta_{5} - \)\(38\!\cdots\!32\)\( \beta_{6} - \)\(48\!\cdots\!24\)\( \beta_{7} - \)\(91\!\cdots\!20\)\( \beta_{8}) q^{28} +(\)\(39\!\cdots\!69\)\( + \)\(80\!\cdots\!49\)\( \beta_{1} - \)\(17\!\cdots\!98\)\( \beta_{2} + \)\(19\!\cdots\!10\)\( \beta_{3} - \)\(36\!\cdots\!15\)\( \beta_{4} + \)\(12\!\cdots\!56\)\( \beta_{5} + \)\(11\!\cdots\!36\)\( \beta_{6} + \)\(33\!\cdots\!20\)\( \beta_{7} + \)\(66\!\cdots\!48\)\( \beta_{8}) q^{29} +(\)\(19\!\cdots\!58\)\( - \)\(14\!\cdots\!22\)\( \beta_{1} - \)\(25\!\cdots\!06\)\( \beta_{2} + \)\(33\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!02\)\( \beta_{4} - \)\(79\!\cdots\!30\)\( \beta_{5} + \)\(90\!\cdots\!40\)\( \beta_{6} - \)\(18\!\cdots\!60\)\( \beta_{7} - \)\(43\!\cdots\!80\)\( \beta_{8}) q^{30} +(\)\(88\!\cdots\!68\)\( + \)\(88\!\cdots\!24\)\( \beta_{1} - \)\(76\!\cdots\!52\)\( \beta_{2} - \)\(10\!\cdots\!04\)\( \beta_{3} + \)\(56\!\cdots\!48\)\( \beta_{4} + \)\(36\!\cdots\!20\)\( \beta_{5} - \)\(14\!\cdots\!28\)\( \beta_{6} + \)\(75\!\cdots\!40\)\( \beta_{7} + \)\(26\!\cdots\!96\)\( \beta_{8}) q^{31} +(-\)\(16\!\cdots\!28\)\( - \)\(69\!\cdots\!40\)\( \beta_{1} + \)\(84\!\cdots\!44\)\( \beta_{2} - \)\(13\!\cdots\!08\)\( \beta_{3} - \)\(36\!\cdots\!20\)\( \beta_{4} + \)\(96\!\cdots\!04\)\( \beta_{5} + \)\(10\!\cdots\!04\)\( \beta_{6} - \)\(19\!\cdots\!20\)\( \beta_{7} - \)\(14\!\cdots\!80\)\( \beta_{8}) q^{32} +(\)\(43\!\cdots\!10\)\( - \)\(26\!\cdots\!29\)\( \beta_{1} + \)\(21\!\cdots\!69\)\( \beta_{2} + \)\(86\!\cdots\!91\)\( \beta_{3} - \)\(15\!\cdots\!25\)\( \beta_{4} - \)\(19\!\cdots\!86\)\( \beta_{5} - \)\(49\!\cdots\!96\)\( \beta_{6} - \)\(20\!\cdots\!79\)\( \beta_{7} + \)\(69\!\cdots\!55\)\( \beta_{8}) q^{33} +(\)\(12\!\cdots\!46\)\( - \)\(37\!\cdots\!70\)\( \beta_{1} + \)\(15\!\cdots\!12\)\( \beta_{2} + \)\(36\!\cdots\!80\)\( \beta_{3} + \)\(16\!\cdots\!92\)\( \beta_{4} + \)\(10\!\cdots\!96\)\( \beta_{5} + \)\(11\!\cdots\!44\)\( \beta_{6} + \)\(52\!\cdots\!60\)\( \beta_{7} - \)\(29\!\cdots\!08\)\( \beta_{8}) q^{34} +(\)\(54\!\cdots\!76\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} - \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(11\!\cdots\!68\)\( \beta_{4} - \)\(11\!\cdots\!76\)\( \beta_{5} + \)\(30\!\cdots\!88\)\( \beta_{6} - \)\(27\!\cdots\!72\)\( \beta_{7} + \)\(10\!\cdots\!04\)\( \beta_{8}) q^{35} +(-\)\(52\!\cdots\!14\)\( - \)\(10\!\cdots\!58\)\( \beta_{1} - \)\(29\!\cdots\!21\)\( \beta_{2} - \)\(51\!\cdots\!11\)\( \beta_{3} - \)\(32\!\cdots\!04\)\( \beta_{4} - \)\(13\!\cdots\!44\)\( \beta_{5} - \)\(39\!\cdots\!00\)\( \beta_{6} + \)\(41\!\cdots\!80\)\( \beta_{7} - \)\(29\!\cdots\!00\)\( \beta_{8}) q^{36} +(-\)\(23\!\cdots\!87\)\( - \)\(34\!\cdots\!45\)\( \beta_{1} - \)\(84\!\cdots\!56\)\( \beta_{2} - \)\(51\!\cdots\!04\)\( \beta_{3} + \)\(94\!\cdots\!35\)\( \beta_{4} + \)\(75\!\cdots\!56\)\( \beta_{5} + \)\(17\!\cdots\!36\)\( \beta_{6} + \)\(57\!\cdots\!82\)\( \beta_{7} + \)\(39\!\cdots\!50\)\( \beta_{8}) q^{37} +(\)\(91\!\cdots\!89\)\( - \)\(23\!\cdots\!71\)\( \beta_{1} - \)\(13\!\cdots\!27\)\( \beta_{2} + \)\(81\!\cdots\!98\)\( \beta_{3} + \)\(38\!\cdots\!25\)\( \beta_{4} - \)\(78\!\cdots\!31\)\( \beta_{5} - \)\(26\!\cdots\!36\)\( \beta_{6} - \)\(63\!\cdots\!76\)\( \beta_{7} + \)\(18\!\cdots\!20\)\( \beta_{8}) q^{38} +(\)\(13\!\cdots\!32\)\( - \)\(79\!\cdots\!55\)\( \beta_{1} + \)\(11\!\cdots\!41\)\( \beta_{2} + \)\(29\!\cdots\!84\)\( \beta_{3} - \)\(24\!\cdots\!36\)\( \beta_{4} - \)\(81\!\cdots\!53\)\( \beta_{5} - \)\(12\!\cdots\!37\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(18\!\cdots\!16\)\( \beta_{8}) q^{39} +(-\)\(51\!\cdots\!50\)\( - \)\(42\!\cdots\!30\)\( \beta_{1} + \)\(96\!\cdots\!50\)\( \beta_{2} - \)\(22\!\cdots\!50\)\( \beta_{3} - \)\(20\!\cdots\!40\)\( \beta_{4} + \)\(33\!\cdots\!80\)\( \beta_{5} + \)\(86\!\cdots\!10\)\( \beta_{6} - \)\(16\!\cdots\!90\)\( \beta_{7} + \)\(88\!\cdots\!80\)\( \beta_{8}) q^{40} +(-\)\(16\!\cdots\!62\)\( - \)\(41\!\cdots\!22\)\( \beta_{1} + \)\(51\!\cdots\!10\)\( \beta_{2} + \)\(35\!\cdots\!34\)\( \beta_{3} + \)\(27\!\cdots\!74\)\( \beta_{4} + \)\(35\!\cdots\!64\)\( \beta_{5} - \)\(10\!\cdots\!00\)\( \beta_{6} + \)\(51\!\cdots\!70\)\( \beta_{7} - \)\(29\!\cdots\!50\)\( \beta_{8}) q^{41} +(\)\(12\!\cdots\!80\)\( - \)\(21\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{3} - \)\(22\!\cdots\!60\)\( \beta_{4} - \)\(65\!\cdots\!84\)\( \beta_{5} - \)\(14\!\cdots\!04\)\( \beta_{6} - \)\(81\!\cdots\!60\)\( \beta_{7} + \)\(58\!\cdots\!60\)\( \beta_{8}) q^{42} +(\)\(65\!\cdots\!03\)\( + \)\(28\!\cdots\!76\)\( \beta_{1} - \)\(27\!\cdots\!41\)\( \beta_{2} + \)\(89\!\cdots\!76\)\( \beta_{3} - \)\(26\!\cdots\!80\)\( \beta_{4} + \)\(16\!\cdots\!40\)\( \beta_{5} + \)\(11\!\cdots\!60\)\( \beta_{6} - \)\(18\!\cdots\!76\)\( \beta_{7} + \)\(34\!\cdots\!00\)\( \beta_{8}) q^{43} +(-\)\(68\!\cdots\!44\)\( + \)\(47\!\cdots\!44\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} + \)\(78\!\cdots\!08\)\( \beta_{3} + \)\(29\!\cdots\!68\)\( \beta_{4} + \)\(37\!\cdots\!60\)\( \beta_{5} - \)\(44\!\cdots\!08\)\( \beta_{6} + \)\(19\!\cdots\!00\)\( \beta_{7} - \)\(90\!\cdots\!44\)\( \beta_{8}) q^{44} +(-\)\(35\!\cdots\!67\)\( + \)\(11\!\cdots\!63\)\( \beta_{1} + \)\(46\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} + \)\(54\!\cdots\!07\)\( \beta_{4} - \)\(33\!\cdots\!40\)\( \beta_{5} + \)\(10\!\cdots\!20\)\( \beta_{6} - \)\(81\!\cdots\!30\)\( \beta_{7} + \)\(44\!\cdots\!10\)\( \beta_{8}) q^{45} +(-\)\(20\!\cdots\!62\)\( + \)\(53\!\cdots\!18\)\( \beta_{1} + \)\(92\!\cdots\!90\)\( \beta_{2} + \)\(14\!\cdots\!44\)\( \beta_{3} - \)\(13\!\cdots\!66\)\( \beta_{4} + \)\(78\!\cdots\!34\)\( \beta_{5} - \)\(11\!\cdots\!40\)\( \beta_{6} + \)\(19\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!20\)\( \beta_{8}) q^{46} +(\)\(45\!\cdots\!20\)\( + \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!22\)\( \beta_{2} + \)\(60\!\cdots\!72\)\( \beta_{3} - \)\(27\!\cdots\!80\)\( \beta_{4} + \)\(10\!\cdots\!50\)\( \beta_{5} - \)\(12\!\cdots\!90\)\( \beta_{6} - \)\(12\!\cdots\!32\)\( \beta_{7} + \)\(15\!\cdots\!20\)\( \beta_{8}) q^{47} +(\)\(15\!\cdots\!00\)\( - \)\(36\!\cdots\!64\)\( \beta_{1} - \)\(11\!\cdots\!52\)\( \beta_{2} - \)\(25\!\cdots\!28\)\( \beta_{3} + \)\(13\!\cdots\!00\)\( \beta_{4} - \)\(20\!\cdots\!76\)\( \beta_{5} + \)\(34\!\cdots\!04\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(56\!\cdots\!00\)\( \beta_{8}) q^{48} +(-\)\(55\!\cdots\!51\)\( - \)\(45\!\cdots\!52\)\( \beta_{1} - \)\(21\!\cdots\!20\)\( \beta_{2} - \)\(37\!\cdots\!16\)\( \beta_{3} + \)\(16\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!56\)\( \beta_{5} + \)\(15\!\cdots\!40\)\( \beta_{6} + \)\(59\!\cdots\!80\)\( \beta_{7} - \)\(42\!\cdots\!80\)\( \beta_{8}) q^{49} +(-\)\(36\!\cdots\!25\)\( - \)\(48\!\cdots\!75\)\( \beta_{1} + \)\(70\!\cdots\!00\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(12\!\cdots\!00\)\( \beta_{4} + \)\(12\!\cdots\!00\)\( \beta_{5} - \)\(51\!\cdots\!00\)\( \beta_{6} - \)\(14\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!00\)\( \beta_{8}) q^{50} +(\)\(47\!\cdots\!38\)\( - \)\(10\!\cdots\!50\)\( \beta_{1} + \)\(18\!\cdots\!92\)\( \beta_{2} + \)\(63\!\cdots\!88\)\( \beta_{3} - \)\(39\!\cdots\!72\)\( \beta_{4} - \)\(32\!\cdots\!86\)\( \beta_{5} - \)\(22\!\cdots\!54\)\( \beta_{6} + \)\(10\!\cdots\!40\)\( \beta_{7} - \)\(22\!\cdots\!72\)\( \beta_{8}) q^{51} +(\)\(65\!\cdots\!08\)\( - \)\(43\!\cdots\!24\)\( \beta_{1} + \)\(62\!\cdots\!50\)\( \beta_{2} + \)\(10\!\cdots\!42\)\( \beta_{3} + \)\(98\!\cdots\!40\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} + \)\(20\!\cdots\!20\)\( \beta_{6} + \)\(52\!\cdots\!08\)\( \beta_{7} - \)\(32\!\cdots\!00\)\( \beta_{8}) q^{52} +(-\)\(59\!\cdots\!91\)\( + \)\(22\!\cdots\!71\)\( \beta_{1} - \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(31\!\cdots\!68\)\( \beta_{3} + \)\(10\!\cdots\!75\)\( \beta_{4} - \)\(90\!\cdots\!88\)\( \beta_{5} - \)\(66\!\cdots\!08\)\( \beta_{6} - \)\(22\!\cdots\!46\)\( \beta_{7} + \)\(33\!\cdots\!70\)\( \beta_{8}) q^{53} +(-\)\(17\!\cdots\!66\)\( + \)\(14\!\cdots\!14\)\( \beta_{1} - \)\(52\!\cdots\!30\)\( \beta_{2} - \)\(11\!\cdots\!20\)\( \beta_{3} - \)\(89\!\cdots\!22\)\( \beta_{4} + \)\(55\!\cdots\!10\)\( \beta_{5} + \)\(81\!\cdots\!32\)\( \beta_{6} + \)\(38\!\cdots\!40\)\( \beta_{7} - \)\(10\!\cdots\!24\)\( \beta_{8}) q^{54} +(\)\(16\!\cdots\!48\)\( + \)\(59\!\cdots\!63\)\( \beta_{1} - \)\(77\!\cdots\!61\)\( \beta_{2} + \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(12\!\cdots\!72\)\( \beta_{4} - \)\(18\!\cdots\!75\)\( \beta_{5} + \)\(18\!\cdots\!25\)\( \beta_{6} + \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(15\!\cdots\!00\)\( \beta_{8}) q^{55} +(\)\(82\!\cdots\!64\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(25\!\cdots\!64\)\( \beta_{2} + \)\(10\!\cdots\!48\)\( \beta_{3} + \)\(12\!\cdots\!36\)\( \beta_{4} + \)\(14\!\cdots\!60\)\( \beta_{5} - \)\(10\!\cdots\!76\)\( \beta_{6} - \)\(11\!\cdots\!20\)\( \beta_{7} + \)\(26\!\cdots\!32\)\( \beta_{8}) q^{56} +(-\)\(10\!\cdots\!98\)\( + \)\(96\!\cdots\!49\)\( \beta_{1} + \)\(13\!\cdots\!91\)\( \beta_{2} - \)\(11\!\cdots\!99\)\( \beta_{3} + \)\(83\!\cdots\!85\)\( \beta_{4} + \)\(80\!\cdots\!50\)\( \beta_{5} + \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(13\!\cdots\!81\)\( \beta_{7} - \)\(22\!\cdots\!15\)\( \beta_{8}) q^{57} +(-\)\(16\!\cdots\!70\)\( - \)\(78\!\cdots\!94\)\( \beta_{1} + \)\(96\!\cdots\!08\)\( \beta_{2} - \)\(77\!\cdots\!60\)\( \beta_{3} - \)\(60\!\cdots\!80\)\( \beta_{4} - \)\(25\!\cdots\!16\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(10\!\cdots\!92\)\( \beta_{7} + \)\(61\!\cdots\!40\)\( \beta_{8}) q^{58} +(-\)\(19\!\cdots\!21\)\( - \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(46\!\cdots\!17\)\( \beta_{2} + \)\(54\!\cdots\!20\)\( \beta_{3} + \)\(98\!\cdots\!56\)\( \beta_{4} + \)\(18\!\cdots\!64\)\( \beta_{5} - \)\(52\!\cdots\!32\)\( \beta_{6} - \)\(12\!\cdots\!80\)\( \beta_{7} - \)\(58\!\cdots\!76\)\( \beta_{8}) q^{59} +(\)\(13\!\cdots\!16\)\( - \)\(61\!\cdots\!28\)\( \beta_{1} - \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(29\!\cdots\!60\)\( \beta_{3} - \)\(12\!\cdots\!28\)\( \beta_{4} + \)\(63\!\cdots\!64\)\( \beta_{5} - \)\(41\!\cdots\!32\)\( \beta_{6} - \)\(10\!\cdots\!92\)\( \beta_{7} - \)\(23\!\cdots\!56\)\( \beta_{8}) q^{60} +(\)\(11\!\cdots\!97\)\( + \)\(15\!\cdots\!03\)\( \beta_{1} + \)\(77\!\cdots\!84\)\( \beta_{2} + \)\(66\!\cdots\!56\)\( \beta_{3} + \)\(35\!\cdots\!07\)\( \beta_{4} + \)\(22\!\cdots\!28\)\( \beta_{5} + \)\(38\!\cdots\!16\)\( \beta_{6} + \)\(50\!\cdots\!50\)\( \beta_{7} + \)\(11\!\cdots\!38\)\( \beta_{8}) q^{61} +(-\)\(20\!\cdots\!56\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(52\!\cdots\!16\)\( \beta_{2} - \)\(21\!\cdots\!92\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(95\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!96\)\( \beta_{6} - \)\(83\!\cdots\!40\)\( \beta_{7} - \)\(24\!\cdots\!80\)\( \beta_{8}) q^{62} +(-\)\(58\!\cdots\!92\)\( + \)\(79\!\cdots\!67\)\( \beta_{1} + \)\(24\!\cdots\!71\)\( \beta_{2} - \)\(22\!\cdots\!76\)\( \beta_{3} + \)\(34\!\cdots\!60\)\( \beta_{4} + \)\(80\!\cdots\!09\)\( \beta_{5} - \)\(54\!\cdots\!91\)\( \beta_{6} - \)\(11\!\cdots\!12\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{63} +(\)\(98\!\cdots\!04\)\( + \)\(24\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!76\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!32\)\( \beta_{4} + \)\(29\!\cdots\!68\)\( \beta_{5} + \)\(79\!\cdots\!20\)\( \beta_{6} + \)\(94\!\cdots\!60\)\( \beta_{7} + \)\(14\!\cdots\!60\)\( \beta_{8}) q^{64} +(\)\(37\!\cdots\!12\)\( + \)\(10\!\cdots\!34\)\( \beta_{1} - \)\(14\!\cdots\!74\)\( \beta_{2} + \)\(19\!\cdots\!70\)\( \beta_{3} + \)\(15\!\cdots\!94\)\( \beta_{4} - \)\(61\!\cdots\!32\)\( \beta_{5} + \)\(12\!\cdots\!16\)\( \beta_{6} - \)\(20\!\cdots\!54\)\( \beta_{7} - \)\(45\!\cdots\!22\)\( \beta_{8}) q^{65} +(\)\(64\!\cdots\!12\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(31\!\cdots\!44\)\( \beta_{2} + \)\(15\!\cdots\!04\)\( \beta_{3} - \)\(62\!\cdots\!36\)\( \beta_{4} - \)\(37\!\cdots\!28\)\( \beta_{5} - \)\(66\!\cdots\!12\)\( \beta_{6} + \)\(38\!\cdots\!20\)\( \beta_{7} + \)\(51\!\cdots\!84\)\( \beta_{8}) q^{66} +(\)\(74\!\cdots\!65\)\( - \)\(59\!\cdots\!54\)\( \beta_{1} + \)\(26\!\cdots\!79\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4} - \)\(33\!\cdots\!66\)\( \beta_{5} + \)\(71\!\cdots\!14\)\( \beta_{6} + \)\(97\!\cdots\!24\)\( \beta_{7} + \)\(10\!\cdots\!80\)\( \beta_{8}) q^{67} +(-\)\(58\!\cdots\!72\)\( - \)\(52\!\cdots\!72\)\( \beta_{1} + \)\(15\!\cdots\!18\)\( \beta_{2} - \)\(94\!\cdots\!02\)\( \beta_{3} + \)\(18\!\cdots\!20\)\( \beta_{4} + \)\(13\!\cdots\!68\)\( \beta_{5} + \)\(14\!\cdots\!68\)\( \beta_{6} - \)\(26\!\cdots\!24\)\( \beta_{7} - \)\(57\!\cdots\!40\)\( \beta_{8}) q^{68} +(\)\(13\!\cdots\!76\)\( - \)\(60\!\cdots\!94\)\( \beta_{1} + \)\(17\!\cdots\!34\)\( \beta_{2} - \)\(43\!\cdots\!02\)\( \beta_{3} - \)\(15\!\cdots\!98\)\( \beta_{4} - \)\(33\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!52\)\( \beta_{6} + \)\(21\!\cdots\!30\)\( \beta_{7} + \)\(10\!\cdots\!14\)\( \beta_{8}) q^{69} +(\)\(33\!\cdots\!64\)\( + \)\(21\!\cdots\!44\)\( \beta_{1} - \)\(27\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!40\)\( \beta_{3} - \)\(39\!\cdots\!24\)\( \beta_{4} - \)\(67\!\cdots\!60\)\( \beta_{5} - \)\(57\!\cdots\!20\)\( \beta_{6} + \)\(53\!\cdots\!80\)\( \beta_{7} + \)\(89\!\cdots\!40\)\( \beta_{8}) q^{70} +(\)\(61\!\cdots\!12\)\( + \)\(59\!\cdots\!79\)\( \beta_{1} - \)\(10\!\cdots\!93\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3} - \)\(10\!\cdots\!84\)\( \beta_{4} + \)\(14\!\cdots\!89\)\( \beta_{5} + \)\(28\!\cdots\!33\)\( \beta_{6} - \)\(17\!\cdots\!20\)\( \beta_{7} - \)\(52\!\cdots\!56\)\( \beta_{8}) q^{71} +(\)\(18\!\cdots\!19\)\( + \)\(13\!\cdots\!31\)\( \beta_{1} - \)\(16\!\cdots\!17\)\( \beta_{2} + \)\(35\!\cdots\!41\)\( \beta_{3} + \)\(57\!\cdots\!20\)\( \beta_{4} - \)\(17\!\cdots\!38\)\( \beta_{5} - \)\(81\!\cdots\!93\)\( \beta_{6} + \)\(18\!\cdots\!55\)\( \beta_{7} + \)\(13\!\cdots\!80\)\( \beta_{8}) q^{72} +(\)\(14\!\cdots\!56\)\( + \)\(12\!\cdots\!33\)\( \beta_{1} + \)\(26\!\cdots\!15\)\( \beta_{2} - \)\(14\!\cdots\!07\)\( \beta_{3} - \)\(55\!\cdots\!75\)\( \beta_{4} - \)\(52\!\cdots\!06\)\( \beta_{5} - \)\(14\!\cdots\!36\)\( \beta_{6} + \)\(42\!\cdots\!79\)\( \beta_{7} - \)\(86\!\cdots\!55\)\( \beta_{8}) q^{73} +(\)\(80\!\cdots\!30\)\( + \)\(93\!\cdots\!34\)\( \beta_{1} + \)\(88\!\cdots\!16\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(35\!\cdots\!44\)\( \beta_{4} - \)\(27\!\cdots\!52\)\( \beta_{5} + \)\(30\!\cdots\!24\)\( \beta_{6} + \)\(15\!\cdots\!40\)\( \beta_{7} - \)\(34\!\cdots\!68\)\( \beta_{8}) q^{74} +(\)\(20\!\cdots\!75\)\( - \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(25\!\cdots\!75\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3} - \)\(36\!\cdots\!00\)\( \beta_{4} + \)\(17\!\cdots\!00\)\( \beta_{5} - \)\(35\!\cdots\!00\)\( \beta_{6} - \)\(24\!\cdots\!00\)\( \beta_{7} + \)\(11\!\cdots\!00\)\( \beta_{8}) q^{75} +(\)\(41\!\cdots\!68\)\( - \)\(14\!\cdots\!04\)\( \beta_{1} - \)\(19\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!36\)\( \beta_{3} + \)\(59\!\cdots\!52\)\( \beta_{4} - \)\(57\!\cdots\!40\)\( \beta_{5} - \)\(72\!\cdots\!92\)\( \beta_{6} - \)\(27\!\cdots\!80\)\( \beta_{7} - \)\(12\!\cdots\!56\)\( \beta_{8}) q^{76} +(\)\(43\!\cdots\!92\)\( - \)\(19\!\cdots\!50\)\( \beta_{1} - \)\(70\!\cdots\!54\)\( \beta_{2} - \)\(31\!\cdots\!18\)\( \beta_{3} + \)\(96\!\cdots\!50\)\( \beta_{4} - \)\(65\!\cdots\!36\)\( \beta_{5} + \)\(78\!\cdots\!44\)\( \beta_{6} + \)\(13\!\cdots\!90\)\( \beta_{7} - \)\(14\!\cdots\!50\)\( \beta_{8}) q^{77} +(\)\(18\!\cdots\!66\)\( - \)\(46\!\cdots\!86\)\( \beta_{1} - \)\(16\!\cdots\!74\)\( \beta_{2} - \)\(37\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!10\)\( \beta_{4} + \)\(65\!\cdots\!58\)\( \beta_{5} - \)\(29\!\cdots\!92\)\( \beta_{6} - \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(75\!\cdots\!00\)\( \beta_{8}) q^{78} +(\)\(19\!\cdots\!28\)\( + \)\(83\!\cdots\!14\)\( \beta_{1} + \)\(61\!\cdots\!46\)\( \beta_{2} - \)\(58\!\cdots\!68\)\( \beta_{3} + \)\(46\!\cdots\!68\)\( \beta_{4} + \)\(89\!\cdots\!30\)\( \beta_{5} + \)\(50\!\cdots\!62\)\( \beta_{6} - \)\(34\!\cdots\!00\)\( \beta_{7} - \)\(10\!\cdots\!84\)\( \beta_{8}) q^{79} +(\)\(74\!\cdots\!44\)\( + \)\(29\!\cdots\!24\)\( \beta_{1} + \)\(37\!\cdots\!92\)\( \beta_{2} + \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(77\!\cdots\!04\)\( \beta_{4} + \)\(25\!\cdots\!40\)\( \beta_{5} - \)\(15\!\cdots\!20\)\( \beta_{6} + \)\(16\!\cdots\!80\)\( \beta_{7} - \)\(17\!\cdots\!60\)\( \beta_{8}) q^{80} +(\)\(95\!\cdots\!79\)\( + \)\(50\!\cdots\!19\)\( \beta_{1} + \)\(15\!\cdots\!29\)\( \beta_{2} - \)\(23\!\cdots\!13\)\( \beta_{3} + \)\(32\!\cdots\!71\)\( \beta_{4} - \)\(15\!\cdots\!06\)\( \beta_{5} - \)\(19\!\cdots\!92\)\( \beta_{6} - \)\(15\!\cdots\!15\)\( \beta_{7} + \)\(31\!\cdots\!19\)\( \beta_{8}) q^{81} +(\)\(95\!\cdots\!02\)\( + \)\(17\!\cdots\!38\)\( \beta_{1} + \)\(40\!\cdots\!32\)\( \beta_{2} - \)\(38\!\cdots\!04\)\( \beta_{3} - \)\(13\!\cdots\!00\)\( \beta_{4} + \)\(10\!\cdots\!36\)\( \beta_{5} + \)\(86\!\cdots\!76\)\( \beta_{6} - \)\(39\!\cdots\!88\)\( \beta_{7} - \)\(50\!\cdots\!40\)\( \beta_{8}) q^{82} +(\)\(11\!\cdots\!81\)\( + \)\(49\!\cdots\!52\)\( \beta_{1} - \)\(32\!\cdots\!47\)\( \beta_{2} - \)\(79\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!80\)\( \beta_{4} + \)\(44\!\cdots\!80\)\( \beta_{5} - \)\(14\!\cdots\!40\)\( \beta_{6} + \)\(11\!\cdots\!80\)\( \beta_{7} + \)\(25\!\cdots\!80\)\( \beta_{8}) q^{83} +(\)\(32\!\cdots\!20\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} - \)\(13\!\cdots\!12\)\( \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3} - \)\(50\!\cdots\!72\)\( \beta_{4} - \)\(67\!\cdots\!72\)\( \beta_{5} - \)\(48\!\cdots\!80\)\( \beta_{6} + \)\(70\!\cdots\!80\)\( \beta_{7} + \)\(19\!\cdots\!60\)\( \beta_{8}) q^{84} +(\)\(31\!\cdots\!06\)\( - \)\(78\!\cdots\!88\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} + \)\(16\!\cdots\!10\)\( \beta_{3} + \)\(21\!\cdots\!32\)\( \beta_{4} - \)\(20\!\cdots\!36\)\( \beta_{5} + \)\(32\!\cdots\!68\)\( \beta_{6} - \)\(42\!\cdots\!42\)\( \beta_{7} - \)\(31\!\cdots\!06\)\( \beta_{8}) q^{85} +(-\)\(62\!\cdots\!33\)\( - \)\(32\!\cdots\!33\)\( \beta_{1} + \)\(23\!\cdots\!07\)\( \beta_{2} + \)\(36\!\cdots\!54\)\( \beta_{3} + \)\(45\!\cdots\!39\)\( \beta_{4} - \)\(97\!\cdots\!45\)\( \beta_{5} + \)\(50\!\cdots\!16\)\( \beta_{6} + \)\(44\!\cdots\!60\)\( \beta_{7} + \)\(19\!\cdots\!88\)\( \beta_{8}) q^{86} +(-\)\(26\!\cdots\!60\)\( - \)\(14\!\cdots\!51\)\( \beta_{1} + \)\(76\!\cdots\!49\)\( \beta_{2} - \)\(79\!\cdots\!16\)\( \beta_{3} - \)\(12\!\cdots\!20\)\( \beta_{4} + \)\(64\!\cdots\!31\)\( \beta_{5} - \)\(64\!\cdots\!89\)\( \beta_{6} + \)\(12\!\cdots\!44\)\( \beta_{7} - \)\(93\!\cdots\!60\)\( \beta_{8}) q^{87} +(-\)\(72\!\cdots\!36\)\( - \)\(31\!\cdots\!56\)\( \beta_{1} + \)\(25\!\cdots\!40\)\( \beta_{2} - \)\(47\!\cdots\!08\)\( \beta_{3} + \)\(10\!\cdots\!80\)\( \beta_{4} - \)\(37\!\cdots\!76\)\( \beta_{5} - \)\(18\!\cdots\!76\)\( \beta_{6} - \)\(34\!\cdots\!60\)\( \beta_{7} + \)\(17\!\cdots\!20\)\( \beta_{8}) q^{88} +(-\)\(98\!\cdots\!68\)\( + \)\(59\!\cdots\!65\)\( \beta_{1} - \)\(10\!\cdots\!05\)\( \beta_{2} + \)\(23\!\cdots\!41\)\( \beta_{3} - \)\(41\!\cdots\!43\)\( \beta_{4} - \)\(18\!\cdots\!34\)\( \beta_{5} + \)\(74\!\cdots\!24\)\( \beta_{6} + \)\(42\!\cdots\!35\)\( \beta_{7} + \)\(63\!\cdots\!57\)\( \beta_{8}) q^{89} +(-\)\(26\!\cdots\!18\)\( + \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(45\!\cdots\!64\)\( \beta_{2} + \)\(89\!\cdots\!20\)\( \beta_{3} + \)\(58\!\cdots\!04\)\( \beta_{4} + \)\(30\!\cdots\!08\)\( \beta_{5} - \)\(38\!\cdots\!04\)\( \beta_{6} + \)\(11\!\cdots\!76\)\( \beta_{7} - \)\(86\!\cdots\!32\)\( \beta_{8}) q^{90} +(-\)\(22\!\cdots\!24\)\( - \)\(26\!\cdots\!72\)\( \beta_{1} - \)\(38\!\cdots\!36\)\( \beta_{2} + \)\(16\!\cdots\!48\)\( \beta_{3} - \)\(15\!\cdots\!64\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - \)\(20\!\cdots\!36\)\( \beta_{6} - \)\(18\!\cdots\!00\)\( \beta_{7} + \)\(14\!\cdots\!52\)\( \beta_{8}) q^{91} +(-\)\(10\!\cdots\!88\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} + \)\(53\!\cdots\!60\)\( \beta_{2} - \)\(52\!\cdots\!04\)\( \beta_{3} - \)\(12\!\cdots\!40\)\( \beta_{4} + \)\(67\!\cdots\!96\)\( \beta_{5} + \)\(43\!\cdots\!76\)\( \beta_{6} - \)\(53\!\cdots\!48\)\( \beta_{7} + \)\(12\!\cdots\!00\)\( \beta_{8}) q^{92} +(-\)\(11\!\cdots\!56\)\( - \)\(35\!\cdots\!64\)\( \beta_{1} + \)\(29\!\cdots\!80\)\( \beta_{2} + \)\(17\!\cdots\!84\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4} - \)\(12\!\cdots\!48\)\( \beta_{5} - \)\(20\!\cdots\!28\)\( \beta_{6} + \)\(46\!\cdots\!32\)\( \beta_{7} - \)\(40\!\cdots\!80\)\( \beta_{8}) q^{93} +(-\)\(14\!\cdots\!80\)\( - \)\(15\!\cdots\!96\)\( \beta_{1} + \)\(38\!\cdots\!60\)\( \beta_{2} - \)\(77\!\cdots\!40\)\( \beta_{3} - \)\(40\!\cdots\!72\)\( \beta_{4} - \)\(19\!\cdots\!60\)\( \beta_{5} - \)\(34\!\cdots\!88\)\( \beta_{6} - \)\(39\!\cdots\!20\)\( \beta_{7} + \)\(52\!\cdots\!16\)\( \beta_{8}) q^{94} +(\)\(21\!\cdots\!00\)\( - \)\(46\!\cdots\!95\)\( \beta_{1} + \)\(54\!\cdots\!25\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!40\)\( \beta_{4} + \)\(68\!\cdots\!95\)\( \beta_{5} + \)\(89\!\cdots\!15\)\( \beta_{6} - \)\(13\!\cdots\!60\)\( \beta_{7} + \)\(12\!\cdots\!20\)\( \beta_{8}) q^{95} +(\)\(90\!\cdots\!12\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!28\)\( \beta_{2} + \)\(18\!\cdots\!72\)\( \beta_{3} - \)\(16\!\cdots\!92\)\( \beta_{4} - \)\(21\!\cdots\!92\)\( \beta_{5} - \)\(20\!\cdots\!80\)\( \beta_{6} - \)\(52\!\cdots\!20\)\( \beta_{7} - \)\(49\!\cdots\!40\)\( \beta_{8}) q^{96} +(\)\(20\!\cdots\!52\)\( + \)\(52\!\cdots\!41\)\( \beta_{1} + \)\(93\!\cdots\!15\)\( \beta_{2} - \)\(15\!\cdots\!47\)\( \beta_{3} - \)\(10\!\cdots\!55\)\( \beta_{4} - \)\(49\!\cdots\!82\)\( \beta_{5} + \)\(22\!\cdots\!68\)\( \beta_{6} + \)\(21\!\cdots\!71\)\( \beta_{7} - \)\(10\!\cdots\!15\)\( \beta_{8}) q^{97} +(\)\(10\!\cdots\!89\)\( + \)\(10\!\cdots\!67\)\( \beta_{1} - \)\(38\!\cdots\!28\)\( \beta_{2} - \)\(36\!\cdots\!64\)\( \beta_{3} - \)\(13\!\cdots\!20\)\( \beta_{4} - \)\(27\!\cdots\!84\)\( \beta_{5} + \)\(50\!\cdots\!76\)\( \beta_{6} + \)\(18\!\cdots\!92\)\( \beta_{7} + \)\(11\!\cdots\!80\)\( \beta_{8}) q^{98} +(\)\(23\!\cdots\!75\)\( + \)\(97\!\cdots\!92\)\( \beta_{1} + \)\(12\!\cdots\!91\)\( \beta_{2} - \)\(20\!\cdots\!88\)\( \beta_{3} + \)\(46\!\cdots\!44\)\( \beta_{4} + \)\(29\!\cdots\!60\)\( \beta_{5} - \)\(13\!\cdots\!84\)\( \beta_{6} - \)\(15\!\cdots\!20\)\( \beta_{7} + \)\(85\!\cdots\!88\)\( \beta_{8}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9q + 5465779613435496q^{2} + \)\(15\!\cdots\!12\)\(q^{3} + \)\(64\!\cdots\!92\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(12\!\cdots\!32\)\(q^{6} - \)\(93\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!40\)\(q^{8} + \)\(36\!\cdots\!13\)\(q^{9} + O(q^{10}) \)	\( 9q + 5465779613435496q^{2} + \)\(15\!\cdots\!12\)\(q^{3} + \)\(64\!\cdots\!92\)\(q^{4} + \)\(24\!\cdots\!50\)\(q^{5} - \)\(12\!\cdots\!32\)\(q^{6} - \)\(93\!\cdots\!44\)\(q^{7} - \)\(10\!\cdots\!40\)\(q^{8} + \)\(36\!\cdots\!13\)\(q^{9} - \)\(10\!\cdots\!00\)\(q^{10} + \)\(75\!\cdots\!48\)\(q^{11} - \)\(79\!\cdots\!44\)\(q^{12} + \)\(38\!\cdots\!82\)\(q^{13} - \)\(74\!\cdots\!16\)\(q^{14} + \)\(11\!\cdots\!00\)\(q^{15} + \)\(29\!\cdots\!64\)\(q^{16} + \)\(55\!\cdots\!26\)\(q^{17} + \)\(23\!\cdots\!72\)\(q^{18} + \)\(85\!\cdots\!60\)\(q^{19} + \)\(13\!\cdots\!00\)\(q^{20} + \)\(98\!\cdots\!48\)\(q^{21} - \)\(21\!\cdots\!88\)\(q^{22} - \)\(10\!\cdots\!48\)\(q^{23} - \)\(29\!\cdots\!20\)\(q^{24} + \)\(20\!\cdots\!75\)\(q^{25} + \)\(29\!\cdots\!48\)\(q^{26} + \)\(12\!\cdots\!40\)\(q^{27} - \)\(77\!\cdots\!72\)\(q^{28} + \)\(35\!\cdots\!90\)\(q^{29} + \)\(17\!\cdots\!00\)\(q^{30} + \)\(80\!\cdots\!08\)\(q^{31} - \)\(14\!\cdots\!64\)\(q^{32} + \)\(39\!\cdots\!64\)\(q^{33} + \)\(11\!\cdots\!64\)\(q^{34} + \)\(49\!\cdots\!00\)\(q^{35} - \)\(47\!\cdots\!56\)\(q^{36} - \)\(21\!\cdots\!34\)\(q^{37} + \)\(82\!\cdots\!40\)\(q^{38} + \)\(12\!\cdots\!56\)\(q^{39} - \)\(46\!\cdots\!00\)\(q^{40} - \)\(15\!\cdots\!62\)\(q^{41} + \)\(11\!\cdots\!12\)\(q^{42} + \)\(58\!\cdots\!92\)\(q^{43} - \)\(61\!\cdots\!76\)\(q^{44} - \)\(31\!\cdots\!50\)\(q^{45} - \)\(18\!\cdots\!72\)\(q^{46} + \)\(41\!\cdots\!36\)\(q^{47} + \)\(14\!\cdots\!52\)\(q^{48} - \)\(49\!\cdots\!63\)\(q^{49} - \)\(32\!\cdots\!00\)\(q^{50} + \)\(42\!\cdots\!08\)\(q^{51} + \)\(59\!\cdots\!16\)\(q^{52} - \)\(53\!\cdots\!38\)\(q^{53} - \)\(15\!\cdots\!40\)\(q^{54} + \)\(14\!\cdots\!00\)\(q^{55} + \)\(74\!\cdots\!40\)\(q^{56} - \)\(93\!\cdots\!20\)\(q^{57} - \)\(14\!\cdots\!40\)\(q^{58} - \)\(17\!\cdots\!20\)\(q^{59} + \)\(11\!\cdots\!00\)\(q^{60} + \)\(99\!\cdots\!98\)\(q^{61} - \)\(18\!\cdots\!48\)\(q^{62} - \)\(52\!\cdots\!08\)\(q^{63} + \)\(88\!\cdots\!12\)\(q^{64} + \)\(33\!\cdots\!00\)\(q^{65} + \)\(57\!\cdots\!96\)\(q^{66} + \)\(67\!\cdots\!76\)\(q^{67} - \)\(52\!\cdots\!12\)\(q^{68} + \)\(12\!\cdots\!16\)\(q^{69} + \)\(30\!\cdots\!00\)\(q^{70} + \)\(55\!\cdots\!28\)\(q^{71} + \)\(16\!\cdots\!20\)\(q^{72} + \)\(13\!\cdots\!02\)\(q^{73} + \)\(72\!\cdots\!24\)\(q^{74} + \)\(18\!\cdots\!00\)\(q^{75} + \)\(36\!\cdots\!80\)\(q^{76} + \)\(38\!\cdots\!32\)\(q^{77} + \)\(16\!\cdots\!64\)\(q^{78} + \)\(17\!\cdots\!40\)\(q^{79} + \)\(67\!\cdots\!00\)\(q^{80} + \)\(85\!\cdots\!89\)\(q^{81} + \)\(86\!\cdots\!72\)\(q^{82} + \)\(10\!\cdots\!72\)\(q^{83} + \)\(29\!\cdots\!24\)\(q^{84} + \)\(28\!\cdots\!00\)\(q^{85} - \)\(56\!\cdots\!12\)\(q^{86} - \)\(23\!\cdots\!80\)\(q^{87} - \)\(64\!\cdots\!80\)\(q^{88} - \)\(88\!\cdots\!30\)\(q^{89} - \)\(23\!\cdots\!00\)\(q^{90} - \)\(20\!\cdots\!72\)\(q^{91} - \)\(96\!\cdots\!24\)\(q^{92} - \)\(10\!\cdots\!56\)\(q^{93} - \)\(13\!\cdots\!96\)\(q^{94} + \)\(19\!\cdots\!00\)\(q^{95} + \)\(81\!\cdots\!88\)\(q^{96} + \)\(18\!\cdots\!86\)\(q^{97} + \)\(91\!\cdots\!28\)\(q^{98} + \)\(21\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

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