LMFDB - Newform 1.68.a.a

Newspace parameters

Level:	$ N $	=	$ 1 $
Weight:	$ k $	=	$ 68 $
Character orbit:	$[\chi]$	=	1.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$28.429035193$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 $
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +($$66\!\cdots\!25$$ + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +($$29\!\cdots\!36$$ + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +($$67\!\cdots\!98$$ - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +($$65\!\cdots\!36$$ - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +($$55\!\cdots\!99$$ + $$96\!\cdots\!02$$ \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +O(q^{10})$	$ q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +($$66\!\cdots\!25$$ + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +($$29\!\cdots\!36$$ + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +($$67\!\cdots\!98$$ - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +($$65\!\cdots\!36$$ - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +($$55\!\cdots\!99$$ + $$96\!\cdots\!02$$ \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +(-$$10\!\cdots\!50$$ - $$36\!\cdots\!46$$ \beta_{1} + 387747904434926976 \beta_{2} - 10724715572992 \beta_{3} + 6662909664 \beta_{4}) q^{10} +($$40\!\cdots\!97$$ - $$14\!\cdots\!25$$ \beta_{1} + 4432914057561345675 \beta_{2} + 337416804371680 \beta_{3} - 154884117080 \beta_{4}) q^{11} +(-$$41\!\cdots\!16$$ - $$10\!\cdots\!44$$ \beta_{1} - 12007852213004039576 \beta_{2} - 2856419893635300 \beta_{3} + 2653671352320 \beta_{4}) q^{12} +($$35\!\cdots\!77$$ - $$19\!\cdots\!87$$ \beta_{1} - $$15\!\cdots\!60$$ \beta_{2} - 29875083592222700 \beta_{3} - 34622077787505 \beta_{4}) q^{13} +($$58\!\cdots\!44$$ - $$18\!\cdots\!92$$ \beta_{1} + 33318747846508840128 \beta_{2} + 964772473179561088 \beta_{3} + 344998573654000 \beta_{4}) q^{14} +(-$$18\!\cdots\!50$$ - $$17\!\cdots\!22$$ \beta_{1} + $$22\!\cdots\!82$$ \beta_{2} - 10229182088434956144 \beta_{3} - 2522507816770452 \beta_{4}) q^{15} +($$36\!\cdots\!04$$ - $$18\!\cdots\!52$$ \beta_{1} - $$10\!\cdots\!32$$ \beta_{2} + 43878715027197261888 \beta_{3} + 11266877309667840 \beta_{4}) q^{16} +($$15\!\cdots\!32$$ - $$71\!\cdots\!02$$ \beta_{1} - $$95\!\cdots\!12$$ \beta_{2} + $$18\!\cdots\!00$$ \beta_{3} + 8396275243314890 \beta_{4}) q^{17} +(-$$20\!\cdots\!21$$ + $$30\!\cdots\!91$$ \beta_{1} + $$80\!\cdots\!76$$ \beta_{2} - $$39\!\cdots\!00$$ \beta_{3} - 700896140455976640 \beta_{4}) q^{18} +($$79\!\cdots\!79$$ + $$29\!\cdots\!09$$ \beta_{1} + $$63\!\cdots\!69$$ \beta_{2} + $$24\!\cdots\!64$$ \beta_{3} + 7385036747071057560 \beta_{4}) q^{19} +(-$$30\!\cdots\!50$$ + $$24\!\cdots\!02$$ \beta_{1} - $$20\!\cdots\!12$$ \beta_{2} - $$48\!\cdots\!46$$ \beta_{3} - 47581246658342739968 \beta_{4}) q^{20} +(-$$35\!\cdots\!00$$ - $$27\!\cdots\!32$$ \beta_{1} + $$16\!\cdots\!88$$ \beta_{2} - $$36\!\cdots\!52$$ \beta_{3} + $$19\!\cdots\!00$$ \beta_{4}) q^{21} +($$36\!\cdots\!72$$ - $$71\!\cdots\!32$$ \beta_{1} + $$35\!\cdots\!60$$ \beta_{2} + $$32\!\cdots\!00$$ \beta_{3} - $$20\!\cdots\!80$$ \beta_{4}) q^{22} +(-$$83\!\cdots\!90$$ - $$73\!\cdots\!10$$ \beta_{1} - $$14\!\cdots\!86$$ \beta_{2} - $$10\!\cdots\!00$$ \beta_{3} - $$38\!\cdots\!00$$ \beta_{4}) q^{23} +(-$$26\!\cdots\!48$$ + $$45\!\cdots\!92$$ \beta_{1} - $$78\!\cdots\!28$$ \beta_{2} - $$48\!\cdots\!48$$ \beta_{3} + $$36\!\cdots\!60$$ \beta_{4}) q^{24} +($$35\!\cdots\!75$$ + $$38\!\cdots\!00$$ \beta_{1} + $$94\!\cdots\!00$$ \beta_{2} + $$13\!\cdots\!00$$ \beta_{3} - $$20\!\cdots\!00$$ \beta_{4}) q^{25} +($$42\!\cdots\!94$$ - $$62\!\cdots\!18$$ \beta_{1} + $$12\!\cdots\!12$$ \beta_{2} - $$20\!\cdots\!68$$ \beta_{3} + $$74\!\cdots\!20$$ \beta_{4}) q^{26} +($$37\!\cdots\!34$$ - $$15\!\cdots\!74$$ \beta_{1} - $$25\!\cdots\!18$$ \beta_{2} - $$23\!\cdots\!00$$ \beta_{3} - $$17\!\cdots\!20$$ \beta_{4}) q^{27} +($$29\!\cdots\!20$$ - $$17\!\cdots\!60$$ \beta_{1} - $$18\!\cdots\!24$$ \beta_{2} + $$15\!\cdots\!00$$ \beta_{3} + $$48\!\cdots\!80$$ \beta_{4}) q^{28} +($$37\!\cdots\!01$$ + $$75\!\cdots\!21$$ \beta_{1} + $$75\!\cdots\!36$$ \beta_{2} - $$43\!\cdots\!64$$ \beta_{3} + $$15\!\cdots\!95$$ \beta_{4}) q^{29} +($$36\!\cdots\!00$$ + $$15\!\cdots\!08$$ \beta_{1} + $$17\!\cdots\!52$$ \beta_{2} + $$45\!\cdots\!16$$ \beta_{3} - $$76\!\cdots\!72$$ \beta_{4}) q^{30} +($$73\!\cdots\!12$$ + $$24\!\cdots\!00$$ \beta_{1} - $$37\!\cdots\!00$$ \beta_{2} - $$46\!\cdots\!60$$ \beta_{3} + $$18\!\cdots\!60$$ \beta_{4}) q^{31} +($$31\!\cdots\!16$$ - $$66\!\cdots\!36$$ \beta_{1} - $$91\!\cdots\!80$$ \beta_{2} + $$55\!\cdots\!00$$ \beta_{3} - $$13\!\cdots\!60$$ \beta_{4}) q^{32} +($$48\!\cdots\!38$$ - $$23\!\cdots\!58$$ \beta_{1} + $$63\!\cdots\!92$$ \beta_{2} - $$38\!\cdots\!00$$ \beta_{3} - $$80\!\cdots\!10$$ \beta_{4}) q^{33} +($$15\!\cdots\!18$$ - $$77\!\cdots\!18$$ \beta_{1} - $$51\!\cdots\!88$$ \beta_{2} + $$98\!\cdots\!92$$ \beta_{3} + $$35\!\cdots\!60$$ \beta_{4}) q^{34} +($$90\!\cdots\!00$$ + $$24\!\cdots\!84$$ \beta_{1} - $$16\!\cdots\!04$$ \beta_{2} - $$28\!\cdots\!32$$ \beta_{3} - $$63\!\cdots\!56$$ \beta_{4}) q^{35} +(-$$76\!\cdots\!51$$ + $$78\!\cdots\!23$$ \beta_{1} - $$18\!\cdots\!82$$ \beta_{2} - $$49\!\cdots\!87$$ \beta_{3} + $$26\!\cdots\!40$$ \beta_{4}) q^{36} +(-$$11\!\cdots\!15$$ - $$90\!\cdots\!15$$ \beta_{1} + $$19\!\cdots\!32$$ \beta_{2} + $$11\!\cdots\!00$$ \beta_{3} + $$24\!\cdots\!35$$ \beta_{4}) q^{37} +(-$$62\!\cdots\!36$$ - $$27\!\cdots\!84$$ \beta_{1} - $$15\!\cdots\!48$$ \beta_{2} - $$56\!\cdots\!00$$ \beta_{3} - $$43\!\cdots\!60$$ \beta_{4}) q^{38} +(-$$14\!\cdots\!18$$ - $$64\!\cdots\!62$$ \beta_{1} - $$43\!\cdots\!42$$ \beta_{2} + $$20\!\cdots\!48$$ \beta_{3} - $$81\!\cdots\!80$$ \beta_{4}) q^{39} +(-$$37\!\cdots\!00$$ + $$14\!\cdots\!60$$ \beta_{1} - $$24\!\cdots\!60$$ \beta_{2} - $$12\!\cdots\!80$$ \beta_{3} + $$28\!\cdots\!60$$ \beta_{4}) q^{40} +($$22\!\cdots\!82$$ + $$16\!\cdots\!00$$ \beta_{1} + $$78\!\cdots\!00$$ \beta_{2} + $$38\!\cdots\!40$$ \beta_{3} + $$62\!\cdots\!60$$ \beta_{4}) q^{41} +($$56\!\cdots\!20$$ + $$75\!\cdots\!00$$ \beta_{1} - $$72\!\cdots\!56$$ \beta_{2} + $$21\!\cdots\!00$$ \beta_{3} - $$22\!\cdots\!40$$ \beta_{4}) q^{42} +($$13\!\cdots\!81$$ - $$39\!\cdots\!81$$ \beta_{1} + $$21\!\cdots\!47$$ \beta_{2} - $$23\!\cdots\!00$$ \beta_{3} + $$67\!\cdots\!00$$ \beta_{4}) q^{43} +($$98\!\cdots\!28$$ - $$59\!\cdots\!48$$ \beta_{1} - $$11\!\cdots\!68$$ \beta_{2} + $$45\!\cdots\!12$$ \beta_{3} + $$15\!\cdots\!60$$ \beta_{4}) q^{44} +($$19\!\cdots\!25$$ - $$33\!\cdots\!87$$ \beta_{1} + $$14\!\cdots\!72$$ \beta_{2} - $$91\!\cdots\!24$$ \beta_{3} - $$43\!\cdots\!17$$ \beta_{4}) q^{45} +($$15\!\cdots\!12$$ + $$10\!\cdots\!20$$ \beta_{1} + $$12\!\cdots\!20$$ \beta_{2} - $$28\!\cdots\!40$$ \beta_{3} + $$18\!\cdots\!60$$ \beta_{4}) q^{46} +(-$$25\!\cdots\!96$$ + $$44\!\cdots\!96$$ \beta_{1} - $$19\!\cdots\!16$$ \beta_{2} - $$14\!\cdots\!00$$ \beta_{3} + $$15\!\cdots\!00$$ \beta_{4}) q^{47} +(-$$39\!\cdots\!20$$ + $$36\!\cdots\!00$$ \beta_{1} - $$83\!\cdots\!56$$ \beta_{2} + $$16\!\cdots\!00$$ \beta_{3} - $$35\!\cdots\!60$$ \beta_{4}) q^{48} +(-$$15\!\cdots\!47$$ - $$12\!\cdots\!40$$ \beta_{1} - $$17\!\cdots\!40$$ \beta_{2} + $$12\!\cdots\!00$$ \beta_{3} - $$26\!\cdots\!40$$ \beta_{4}) q^{49} +(-$$78\!\cdots\!75$$ - $$39\!\cdots\!75$$ \beta_{1} + $$89\!\cdots\!00$$ \beta_{2} - $$35\!\cdots\!00$$ \beta_{3} + $$11\!\cdots\!00$$ \beta_{4}) q^{50} +(-$$67\!\cdots\!42$$ + $$36\!\cdots\!26$$ \beta_{1} - $$51\!\cdots\!34$$ \beta_{2} + $$12\!\cdots\!96$$ \beta_{3} - $$57\!\cdots\!60$$ \beta_{4}) q^{51} +($$13\!\cdots\!18$$ + $$54\!\cdots\!82$$ \beta_{1} - $$32\!\cdots\!04$$ \beta_{2} + $$48\!\cdots\!50$$ \beta_{3} - $$46\!\cdots\!00$$ \beta_{4}) q^{52} +($$20\!\cdots\!69$$ + $$31\!\cdots\!21$$ \beta_{1} - $$35\!\cdots\!44$$ \beta_{2} + $$20\!\cdots\!00$$ \beta_{3} + $$66\!\cdots\!95$$ \beta_{4}) q^{53} +($$38\!\cdots\!44$$ - $$20\!\cdots\!76$$ \beta_{1} + $$58\!\cdots\!84$$ \beta_{2} - $$87\!\cdots\!96$$ \beta_{3} + $$17\!\cdots\!60$$ \beta_{4}) q^{54} +($$13\!\cdots\!50$$ + $$32\!\cdots\!58$$ \beta_{1} + $$96\!\cdots\!02$$ \beta_{2} - $$67\!\cdots\!84$$ \beta_{3} - $$64\!\cdots\!72$$ \beta_{4}) q^{55} +($$32\!\cdots\!16$$ - $$30\!\cdots\!64$$ \beta_{1} - $$17\!\cdots\!24$$ \beta_{2} + $$19\!\cdots\!76$$ \beta_{3} + $$30\!\cdots\!20$$ \beta_{4}) q^{56} +(-$$18\!\cdots\!22$$ + $$51\!\cdots\!22$$ \beta_{1} - $$62\!\cdots\!36$$ \beta_{2} - $$33\!\cdots\!00$$ \beta_{3} + $$19\!\cdots\!50$$ \beta_{4}) q^{57} +(-$$12\!\cdots\!34$$ + $$34\!\cdots\!14$$ \beta_{1} - $$37\!\cdots\!72$$ \beta_{2} - $$47\!\cdots\!00$$ \beta_{3} - $$42\!\cdots\!60$$ \beta_{4}) q^{58} +(-$$61\!\cdots\!03$$ + $$11\!\cdots\!87$$ \beta_{1} + $$12\!\cdots\!67$$ \beta_{2} + $$49\!\cdots\!32$$ \beta_{3} + $$78\!\cdots\!00$$ \beta_{4}) q^{59} +(-$$26\!\cdots\!00$$ - $$12\!\cdots\!96$$ \beta_{1} + $$14\!\cdots\!76$$ \beta_{2} + $$14\!\cdots\!08$$ \beta_{3} + $$85\!\cdots\!64$$ \beta_{4}) q^{60} +(-$$22\!\cdots\!03$$ + $$14\!\cdots\!25$$ \beta_{1} - $$12\!\cdots\!00$$ \beta_{2} + $$81\!\cdots\!00$$ \beta_{3} - $$11\!\cdots\!25$$ \beta_{4}) q^{61} +(-$$43\!\cdots\!88$$ - $$76\!\cdots\!92$$ \beta_{1} - $$51\!\cdots\!20$$ \beta_{2} - $$12\!\cdots\!00$$ \beta_{3} + $$11\!\cdots\!60$$ \beta_{4}) q^{62} +(-$$41\!\cdots\!02$$ + $$62\!\cdots\!82$$ \beta_{1} + $$27\!\cdots\!98$$ \beta_{2} - $$43\!\cdots\!00$$ \beta_{3} - $$50\!\cdots\!60$$ \beta_{4}) q^{63} +($$12\!\cdots\!20$$ - $$14\!\cdots\!88$$ \beta_{1} + $$14\!\cdots\!92$$ \beta_{2} + $$55\!\cdots\!12$$ \beta_{3} + $$31\!\cdots\!20$$ \beta_{4}) q^{64} +($$34\!\cdots\!00$$ + $$33\!\cdots\!68$$ \beta_{1} - $$94\!\cdots\!08$$ \beta_{2} + $$17\!\cdots\!36$$ \beta_{3} + $$16\!\cdots\!88$$ \beta_{4}) q^{65} +($$55\!\cdots\!92$$ + $$68\!\cdots\!88$$ \beta_{1} + $$26\!\cdots\!08$$ \beta_{2} - $$20\!\cdots\!52$$ \beta_{3} - $$21\!\cdots\!80$$ \beta_{4}) q^{66} +(-$$23\!\cdots\!73$$ + $$58\!\cdots\!53$$ \beta_{1} - $$60\!\cdots\!27$$ \beta_{2} - $$73\!\cdots\!00$$ \beta_{3} + $$33\!\cdots\!40$$ \beta_{4}) q^{67} +($$16\!\cdots\!02$$ - $$19\!\cdots\!42$$ \beta_{1} - $$62\!\cdots\!28$$ \beta_{2} + $$15\!\cdots\!50$$ \beta_{3} + $$50\!\cdots\!80$$ \beta_{4}) q^{68} +($$71\!\cdots\!48$$ - $$43\!\cdots\!56$$ \beta_{1} + $$76\!\cdots\!04$$ \beta_{2} + $$26\!\cdots\!44$$ \beta_{3} - $$70\!\cdots\!60$$ \beta_{4}) q^{69} +(-$$52\!\cdots\!00$$ - $$43\!\cdots\!76$$ \beta_{1} + $$38\!\cdots\!56$$ \beta_{2} - $$27\!\cdots\!52$$ \beta_{3} + $$96\!\cdots\!84$$ \beta_{4}) q^{70} +(-$$22\!\cdots\!58$$ + $$16\!\cdots\!50$$ \beta_{1} - $$23\!\cdots\!50$$ \beta_{2} + $$41\!\cdots\!00$$ \beta_{3} - $$10\!\cdots\!00$$ \beta_{4}) q^{71} +(-$$14\!\cdots\!68$$ + $$10\!\cdots\!08$$ \beta_{1} - $$56\!\cdots\!24$$ \beta_{2} - $$62\!\cdots\!00$$ \beta_{3} + $$11\!\cdots\!20$$ \beta_{4}) q^{72} +(-$$60\!\cdots\!00$$ - $$28\!\cdots\!70$$ \beta_{1} + $$14\!\cdots\!04$$ \beta_{2} + $$33\!\cdots\!00$$ \beta_{3} + $$14\!\cdots\!90$$ \beta_{4}) q^{73} +($$18\!\cdots\!26$$ + $$42\!\cdots\!90$$ \beta_{1} - $$17\!\cdots\!60$$ \beta_{2} + $$18\!\cdots\!00$$ \beta_{3} - $$72\!\cdots\!60$$ \beta_{4}) q^{74} +($$43\!\cdots\!25$$ - $$41\!\cdots\!25$$ \beta_{1} + $$68\!\cdots\!75$$ \beta_{2} - $$29\!\cdots\!00$$ \beta_{3} + $$83\!\cdots\!00$$ \beta_{4}) q^{75} +($$41\!\cdots\!12$$ + $$15\!\cdots\!52$$ \beta_{1} - $$89\!\cdots\!68$$ \beta_{2} - $$13\!\cdots\!68$$ \beta_{3} - $$29\!\cdots\!60$$ \beta_{4}) q^{76} +($$10\!\cdots\!56$$ - $$47\!\cdots\!76$$ \beta_{1} - $$81\!\cdots\!28$$ \beta_{2} + $$29\!\cdots\!00$$ \beta_{3} + $$54\!\cdots\!40$$ \beta_{4}) q^{77} +($$12\!\cdots\!52$$ + $$12\!\cdots\!08$$ \beta_{1} - $$36\!\cdots\!36$$ \beta_{2} + $$66\!\cdots\!00$$ \beta_{3} + $$23\!\cdots\!80$$ \beta_{4}) q^{78} +($$59\!\cdots\!36$$ + $$85\!\cdots\!56$$ \beta_{1} - $$81\!\cdots\!04$$ \beta_{2} - $$63\!\cdots\!44$$ \beta_{3} - $$77\!\cdots\!40$$ \beta_{4}) q^{79} +(-$$30\!\cdots\!00$$ + $$12\!\cdots\!04$$ \beta_{1} + $$42\!\cdots\!76$$ \beta_{2} - $$14\!\cdots\!92$$ \beta_{3} + $$14\!\cdots\!64$$ \beta_{4}) q^{80} +(-$$22\!\cdots\!25$$ - $$39\!\cdots\!26$$ \beta_{1} + $$13\!\cdots\!84$$ \beta_{2} + $$12\!\cdots\!64$$ \beta_{3} + $$24\!\cdots\!50$$ \beta_{4}) q^{81} +(-$$35\!\cdots\!18$$ - $$22\!\cdots\!62$$ \beta_{1} - $$23\!\cdots\!20$$ \beta_{2} + $$10\!\cdots\!00$$ \beta_{3} - $$28\!\cdots\!40$$ \beta_{4}) q^{82} +(-$$94\!\cdots\!57$$ - $$35\!\cdots\!43$$ \beta_{1} - $$14\!\cdots\!27$$ \beta_{2} + $$14\!\cdots\!00$$ \beta_{3} + $$10\!\cdots\!00$$ \beta_{4}) q^{83} +(-$$10\!\cdots\!68$$ - $$19\!\cdots\!24$$ \beta_{1} + $$99\!\cdots\!16$$ \beta_{2} - $$20\!\cdots\!84$$ \beta_{3} + $$12\!\cdots\!20$$ \beta_{4}) q^{84} +(-$$78\!\cdots\!50$$ + $$15\!\cdots\!94$$ \beta_{1} + $$65\!\cdots\!36$$ \beta_{2} - $$53\!\cdots\!12$$ \beta_{3} + $$64\!\cdots\!54$$ \beta_{4}) q^{85} +($$99\!\cdots\!28$$ + $$18\!\cdots\!76$$ \beta_{1} + $$23\!\cdots\!16$$ \beta_{2} - $$18\!\cdots\!44$$ \beta_{3} - $$10\!\cdots\!20$$ \beta_{4}) q^{86} +($$71\!\cdots\!82$$ + $$24\!\cdots\!38$$ \beta_{1} + $$71\!\cdots\!46$$ \beta_{2} + $$20\!\cdots\!00$$ \beta_{3} - $$15\!\cdots\!40$$ \beta_{4}) q^{87} +($$86\!\cdots\!92$$ - $$88\!\cdots\!12$$ \beta_{1} - $$15\!\cdots\!84$$ \beta_{2} + $$69\!\cdots\!00$$ \beta_{3} + $$38\!\cdots\!40$$ \beta_{4}) q^{88} +(-$$23\!\cdots\!12$$ + $$45\!\cdots\!98$$ \beta_{1} - $$86\!\cdots\!32$$ \beta_{2} - $$12\!\cdots\!32$$ \beta_{3} - $$74\!\cdots\!90$$ \beta_{4}) q^{89} +($$95\!\cdots\!50$$ - $$15\!\cdots\!82$$ \beta_{1} + $$13\!\cdots\!92$$ \beta_{2} - $$84\!\cdots\!64$$ \beta_{3} - $$25\!\cdots\!12$$ \beta_{4}) q^{90} +($$46\!\cdots\!08$$ + $$82\!\cdots\!96$$ \beta_{1} - $$38\!\cdots\!64$$ \beta_{2} + $$60\!\cdots\!36$$ \beta_{3} - $$37\!\cdots\!80$$ \beta_{4}) q^{91} +(-$$19\!\cdots\!68$$ + $$38\!\cdots\!88$$ \beta_{1} + $$26\!\cdots\!48$$ \beta_{2} + $$14\!\cdots\!00$$ \beta_{3} - $$25\!\cdots\!40$$ \beta_{4}) q^{92} +(-$$36\!\cdots\!52$$ + $$11\!\cdots\!92$$ \beta_{1} + $$34\!\cdots\!72$$ \beta_{2} + $$27\!\cdots\!00$$ \beta_{3} + $$21\!\cdots\!20$$ \beta_{4}) q^{93} +(-$$98\!\cdots\!40$$ + $$51\!\cdots\!24$$ \beta_{1} - $$24\!\cdots\!16$$ \beta_{2} - $$49\!\cdots\!96$$ \beta_{3} - $$11\!\cdots\!40$$ \beta_{4}) q^{94} +(-$$40\!\cdots\!50$$ + $$11\!\cdots\!90$$ \beta_{1} - $$18\!\cdots\!90$$ \beta_{2} + $$52\!\cdots\!80$$ \beta_{3} - $$30\!\cdots\!60$$ \beta_{4}) q^{95} +(-$$44\!\cdots\!24$$ - $$60\!\cdots\!36$$ \beta_{1} + $$25\!\cdots\!24$$ \beta_{2} - $$19\!\cdots\!16$$ \beta_{3} + $$50\!\cdots\!20$$ \beta_{4}) q^{96} +($$57\!\cdots\!24$$ - $$21\!\cdots\!14$$ \beta_{1} - $$76\!\cdots\!16$$ \beta_{2} + $$27\!\cdots\!00$$ \beta_{3} + $$50\!\cdots\!30$$ \beta_{4}) q^{97} +($$24\!\cdots\!03$$ - $$68\!\cdots\!33$$ \beta_{1} - $$10\!\cdots\!40$$ \beta_{2} + $$25\!\cdots\!00$$ \beta_{3} + $$79\!\cdots\!60$$ \beta_{4}) q^{98} +($$32\!\cdots\!53$$ + $$51\!\cdots\!19$$ \beta_{1} + $$62\!\cdots\!79$$ \beta_{2} - $$36\!\cdots\!96$$ \beta_{3} - $$18\!\cdots\!20$$ \beta_{4}) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5q + 5554901256q^{2} + 3443360269119372q^{3} + $$35\!\cdots\!40$$q^{4} + $$33\!\cdots\!50$$q^{5} + $$14\!\cdots\!60$$q^{6} + $$33\!\cdots\!56$$q^{7} + $$32\!\cdots\!80$$q^{8} + $$27\!\cdots\!85$$q^{9} + O(q^{10}) $	$ 5q + 5554901256q^{2} + 3443360269119372q^{3} + $$35\!\cdots\!40$$q^{4} + $$33\!\cdots\!50$$q^{5} + $$14\!\cdots\!60$$q^{6} + $$33\!\cdots\!56$$q^{7} + $$32\!\cdots\!80$$q^{8} + $$27\!\cdots\!85$$q^{9} - $$52\!\cdots\!00$$q^{10} + $$20\!\cdots\!60$$q^{11} - $$20\!\cdots\!24$$q^{12} + $$17\!\cdots\!02$$q^{13} + $$29\!\cdots\!80$$q^{14} - $$93\!\cdots\!00$$q^{15} + $$18\!\cdots\!80$$q^{16} + $$75\!\cdots\!06$$q^{17} - $$10\!\cdots\!68$$q^{18} + $$39\!\cdots\!00$$q^{19} - $$15\!\cdots\!00$$q^{20} - $$17\!\cdots\!40$$q^{21} + $$18\!\cdots\!32$$q^{22} - $$41\!\cdots\!68$$q^{23} - $$13\!\cdots\!00$$q^{24} + $$17\!\cdots\!75$$q^{25} + $$21\!\cdots\!60$$q^{26} + $$18\!\cdots\!20$$q^{27} + $$14\!\cdots\!48$$q^{28} + $$18\!\cdots\!50$$q^{29} + $$18\!\cdots\!00$$q^{30} + $$36\!\cdots\!60$$q^{31} + $$15\!\cdots\!96$$q^{32} + $$24\!\cdots\!84$$q^{33} + $$78\!\cdots\!80$$q^{34} + $$45\!\cdots\!00$$q^{35} - $$38\!\cdots\!20$$q^{36} - $$56\!\cdots\!94$$q^{37} - $$31\!\cdots\!80$$q^{38} - $$71\!\cdots\!80$$q^{39} - $$18\!\cdots\!00$$q^{40} + $$11\!\cdots\!10$$q^{41} + $$28\!\cdots\!92$$q^{42} + $$65\!\cdots\!92$$q^{43} + $$49\!\cdots\!80$$q^{44} + $$99\!\cdots\!50$$q^{45} + $$79\!\cdots\!60$$q^{46} - $$12\!\cdots\!44$$q^{47} - $$19\!\cdots\!68$$q^{48} - $$77\!\cdots\!35$$q^{49} - $$39\!\cdots\!00$$q^{50} - $$33\!\cdots\!40$$q^{51} + $$67\!\cdots\!16$$q^{52} + $$10\!\cdots\!22$$q^{53} + $$19\!\cdots\!00$$q^{54} + $$66\!\cdots\!00$$q^{55} + $$16\!\cdots\!00$$q^{56} - $$92\!\cdots\!60$$q^{57} - $$61\!\cdots\!20$$q^{58} - $$30\!\cdots\!00$$q^{59} - $$13\!\cdots\!00$$q^{60} - $$11\!\cdots\!90$$q^{61} - $$21\!\cdots\!28$$q^{62} - $$20\!\cdots\!68$$q^{63} + $$63\!\cdots\!40$$q^{64} + $$17\!\cdots\!00$$q^{65} + $$27\!\cdots\!20$$q^{66} - $$11\!\cdots\!44$$q^{67} + $$81\!\cdots\!48$$q^{68} + $$35\!\cdots\!20$$q^{69} - $$26\!\cdots\!00$$q^{70} - $$11\!\cdots\!40$$q^{71} - $$74\!\cdots\!40$$q^{72} - $$30\!\cdots\!18$$q^{73} + $$92\!\cdots\!80$$q^{74} + $$21\!\cdots\!00$$q^{75} + $$20\!\cdots\!00$$q^{76} + $$50\!\cdots\!32$$q^{77} + $$61\!\cdots\!64$$q^{78} + $$29\!\cdots\!00$$q^{79} - $$15\!\cdots\!00$$q^{80} - $$11\!\cdots\!95$$q^{81} - $$17\!\cdots\!08$$q^{82} - $$47\!\cdots\!88$$q^{83} - $$52\!\cdots\!20$$q^{84} - $$39\!\cdots\!00$$q^{85} + $$49\!\cdots\!60$$q^{86} + $$35\!\cdots\!60$$q^{87} + $$43\!\cdots\!60$$q^{88} - $$11\!\cdots\!50$$q^{89} + $$47\!\cdots\!00$$q^{90} + $$23\!\cdots\!60$$q^{91} - $$98\!\cdots\!44$$q^{92} - $$18\!\cdots\!36$$q^{93} - $$49\!\cdots\!20$$q^{94} - $$20\!\cdots\!00$$q^{95} - $$22\!\cdots\!40$$q^{96} + $$28\!\cdots\!06$$q^{97} + $$12\!\cdots\!08$$q^{98} + $$16\!\cdots\!20$$q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000$:

$\beta_{0}$	$=$	$ 1 $
$\beta_{1}$	$=$	$ 24 \nu - 5 $
$\beta_{2}$	$=$	$($$-1463 \nu^{4} + 61316688179 \nu^{3} + 1134677465176013610916 \nu^{2} - 175871625402178878144147591024 \nu - 92322819705534579210406721528027067456$$)/ $$37\!\cdots\!28$
$\beta_{3}$	$=$	$($$-1461537 \nu^{4} + 61255371490821 \nu^{3} + 1493224099349187733125372 \nu^{2} - 157864696669790310577560760925712 \nu - 227382046305944113582206155994463766078912$$)/ $$62\!\cdots\!88$
$\beta_{4}$	$=$	$($$4738030953 \nu^{4} - 6783896933686506093 \nu^{3} - 6265292406150844870725632988 \nu^{2} + 4428965898625782506113044897501160080 \nu + 1149735030546567532337013572577831183558493120$$)/ $$15\!\cdots\!20$

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

$1$	$=$	$\beta_0$
$\nu$	$=$	$($$\beta_{1} + 5$$)/24$
$\nu^{2}$	$=$	$($$\beta_{3} - 5994 \beta_{2} - 1189790397 \beta_{1} + 216434076347341357501$$)/576$
$\nu^{3}$	$=$	$($$-409640 \beta_{4} - 1180151429 \beta_{3} + 3889870865010 \beta_{2} + 43600466505262789881 \beta_{1} - 32188957769729085286587521881$$)/1728$
$\nu^{4}$	$=$	$($$-17168672690120 \beta_{4} + 2277286000240633339 \beta_{3} - 18208831024879688328462 \beta_{2} - 9596312969973284759959813095 \beta_{1} + 393192791921864844386864518484726790407$$)/1728$

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

7.63188e8
6.06038e8
−1.64845e8
−3.05046e8
−8.99335e8

−1.72055e10

−1.09955e16

1.48456e20

4.93311e23

1.89183e26

1.79351e28

−1.51842e28

2.81916e31

−8.48767e33

1.2

−1.34339e10

7.87958e15

3.28965e19

−3.01383e23

−1.05854e26

−4.20742e26

1.54057e30

−3.06216e31

4.04876e33

1.3

5.06726e9

−7.82458e15

−1.21897e20

−8.33551e22

−3.96491e25

−1.15070e28

−1.36548e30

−3.14854e31

−4.22381e32

1.4

8.43209e9

1.56682e16

−7.64738e19

3.83370e23

1.32115e26

−1.75533e27

−1.88919e30

1.52782e32

3.23261e33

1.5

2.26950e10

−1.28429e15

3.67490e20

−1.61198e23

−2.91471e25

2.93806e28

4.99099e30

−9.10601e31

−3.65838e33

Inner twists

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

Hecke kernels

There are no other newforms in $S_{68}^{\mathrm{new}}(\Gamma_0(1))$.

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 1 \)
Weight:	\( k \)	=	\( 68 \)
Character orbit:	\([\chi]\)	=	1.a (trivial)

\(f(q)\)	\(=\)	\( q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +O(q^{10})\)	\( q +(1110980251 - \beta_{1}) q^{2} +(688672053797479 - 131979 \beta_{1} + \beta_{2}) q^{3} +(70094400875776967549 - 3411750909 \beta_{1} - 5994 \beta_{2} + \beta_{3}) q^{4} +(\)\(66\!\cdots\!25\)\( + 5225164572589 \beta_{1} - 881784 \beta_{2} - 172 \beta_{3} - \beta_{4}) q^{5} +(\)\(29\!\cdots\!36\)\( + 1611846772269804 \beta_{1} - 781391136 \beta_{2} + 16704 \beta_{3} - 360 \beta_{4}) q^{6} +(\)\(67\!\cdots\!98\)\( - 234147910821976578 \beta_{1} - 494133901254 \beta_{2} + 83062000 \beta_{3} - 11340 \beta_{4}) q^{7} +(\)\(65\!\cdots\!36\)\( - 61324159169951174696 \beta_{1} - 51096613883472 \beta_{2} + 12774152200 \beta_{3} + 3277120 \beta_{4}) q^{8} +(\)\(55\!\cdots\!99\)\( + \)\(96\!\cdots\!02\)\( \beta_{1} + 4943811769157232 \beta_{2} - 181017634248 \beta_{3} - 196006230 \beta_{4}) q^{9} +(-\)\(10\!\cdots\!50\)\( - \)\(36\!\cdots\!46\)\( \beta_{1} + 387747904434926976 \beta_{2} - 10724715572992 \beta_{3} + 6662909664 \beta_{4}) q^{10} +(\)\(40\!\cdots\!97\)\( - \)\(14\!\cdots\!25\)\( \beta_{1} + 4432914057561345675 \beta_{2} + 337416804371680 \beta_{3} - 154884117080 \beta_{4}) q^{11} +(-\)\(41\!\cdots\!16\)\( - \)\(10\!\cdots\!44\)\( \beta_{1} - 12007852213004039576 \beta_{2} - 2856419893635300 \beta_{3} + 2653671352320 \beta_{4}) q^{12} +(\)\(35\!\cdots\!77\)\( - \)\(19\!\cdots\!87\)\( \beta_{1} - \)\(15\!\cdots\!60\)\( \beta_{2} - 29875083592222700 \beta_{3} - 34622077787505 \beta_{4}) q^{13} +(\)\(58\!\cdots\!44\)\( - \)\(18\!\cdots\!92\)\( \beta_{1} + 33318747846508840128 \beta_{2} + 964772473179561088 \beta_{3} + 344998573654000 \beta_{4}) q^{14} +(-\)\(18\!\cdots\!50\)\( - \)\(17\!\cdots\!22\)\( \beta_{1} + \)\(22\!\cdots\!82\)\( \beta_{2} - 10229182088434956144 \beta_{3} - 2522507816770452 \beta_{4}) q^{15} +(\)\(36\!\cdots\!04\)\( - \)\(18\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} + 43878715027197261888 \beta_{3} + 11266877309667840 \beta_{4}) q^{16} +(\)\(15\!\cdots\!32\)\( - \)\(71\!\cdots\!02\)\( \beta_{1} - \)\(95\!\cdots\!12\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} + 8396275243314890 \beta_{4}) q^{17} +(-\)\(20\!\cdots\!21\)\( + \)\(30\!\cdots\!91\)\( \beta_{1} + \)\(80\!\cdots\!76\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} - 700896140455976640 \beta_{4}) q^{18} +(\)\(79\!\cdots\!79\)\( + \)\(29\!\cdots\!09\)\( \beta_{1} + \)\(63\!\cdots\!69\)\( \beta_{2} + \)\(24\!\cdots\!64\)\( \beta_{3} + 7385036747071057560 \beta_{4}) q^{19} +(-\)\(30\!\cdots\!50\)\( + \)\(24\!\cdots\!02\)\( \beta_{1} - \)\(20\!\cdots\!12\)\( \beta_{2} - \)\(48\!\cdots\!46\)\( \beta_{3} - 47581246658342739968 \beta_{4}) q^{20} +(-\)\(35\!\cdots\!00\)\( - \)\(27\!\cdots\!32\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(36\!\cdots\!52\)\( \beta_{3} + \)\(19\!\cdots\!00\)\( \beta_{4}) q^{21} +(\)\(36\!\cdots\!72\)\( - \)\(71\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!60\)\( \beta_{2} + \)\(32\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!80\)\( \beta_{4}) q^{22} +(-\)\(83\!\cdots\!90\)\( - \)\(73\!\cdots\!10\)\( \beta_{1} - \)\(14\!\cdots\!86\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(38\!\cdots\!00\)\( \beta_{4}) q^{23} +(-\)\(26\!\cdots\!48\)\( + \)\(45\!\cdots\!92\)\( \beta_{1} - \)\(78\!\cdots\!28\)\( \beta_{2} - \)\(48\!\cdots\!48\)\( \beta_{3} + \)\(36\!\cdots\!60\)\( \beta_{4}) q^{24} +(\)\(35\!\cdots\!75\)\( + \)\(38\!\cdots\!00\)\( \beta_{1} + \)\(94\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} - \)\(20\!\cdots\!00\)\( \beta_{4}) q^{25} +(\)\(42\!\cdots\!94\)\( - \)\(62\!\cdots\!18\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} + \)\(74\!\cdots\!20\)\( \beta_{4}) q^{26} +(\)\(37\!\cdots\!34\)\( - \)\(15\!\cdots\!74\)\( \beta_{1} - \)\(25\!\cdots\!18\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} - \)\(17\!\cdots\!20\)\( \beta_{4}) q^{27} +(\)\(29\!\cdots\!20\)\( - \)\(17\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} + \)\(48\!\cdots\!80\)\( \beta_{4}) q^{28} +(\)\(37\!\cdots\!01\)\( + \)\(75\!\cdots\!21\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2} - \)\(43\!\cdots\!64\)\( \beta_{3} + \)\(15\!\cdots\!95\)\( \beta_{4}) q^{29} +(\)\(36\!\cdots\!00\)\( + \)\(15\!\cdots\!08\)\( \beta_{1} + \)\(17\!\cdots\!52\)\( \beta_{2} + \)\(45\!\cdots\!16\)\( \beta_{3} - \)\(76\!\cdots\!72\)\( \beta_{4}) q^{30} +(\)\(73\!\cdots\!12\)\( + \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!00\)\( \beta_{2} - \)\(46\!\cdots\!60\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{31} +(\)\(31\!\cdots\!16\)\( - \)\(66\!\cdots\!36\)\( \beta_{1} - \)\(91\!\cdots\!80\)\( \beta_{2} + \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(13\!\cdots\!60\)\( \beta_{4}) q^{32} +(\)\(48\!\cdots\!38\)\( - \)\(23\!\cdots\!58\)\( \beta_{1} + \)\(63\!\cdots\!92\)\( \beta_{2} - \)\(38\!\cdots\!00\)\( \beta_{3} - \)\(80\!\cdots\!10\)\( \beta_{4}) q^{33} +(\)\(15\!\cdots\!18\)\( - \)\(77\!\cdots\!18\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} + \)\(98\!\cdots\!92\)\( \beta_{3} + \)\(35\!\cdots\!60\)\( \beta_{4}) q^{34} +(\)\(90\!\cdots\!00\)\( + \)\(24\!\cdots\!84\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3} - \)\(63\!\cdots\!56\)\( \beta_{4}) q^{35} +(-\)\(76\!\cdots\!51\)\( + \)\(78\!\cdots\!23\)\( \beta_{1} - \)\(18\!\cdots\!82\)\( \beta_{2} - \)\(49\!\cdots\!87\)\( \beta_{3} + \)\(26\!\cdots\!40\)\( \beta_{4}) q^{36} +(-\)\(11\!\cdots\!15\)\( - \)\(90\!\cdots\!15\)\( \beta_{1} + \)\(19\!\cdots\!32\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!35\)\( \beta_{4}) q^{37} +(-\)\(62\!\cdots\!36\)\( - \)\(27\!\cdots\!84\)\( \beta_{1} - \)\(15\!\cdots\!48\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4}) q^{38} +(-\)\(14\!\cdots\!18\)\( - \)\(64\!\cdots\!62\)\( \beta_{1} - \)\(43\!\cdots\!42\)\( \beta_{2} + \)\(20\!\cdots\!48\)\( \beta_{3} - \)\(81\!\cdots\!80\)\( \beta_{4}) q^{39} +(-\)\(37\!\cdots\!00\)\( + \)\(14\!\cdots\!60\)\( \beta_{1} - \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3} + \)\(28\!\cdots\!60\)\( \beta_{4}) q^{40} +(\)\(22\!\cdots\!82\)\( + \)\(16\!\cdots\!00\)\( \beta_{1} + \)\(78\!\cdots\!00\)\( \beta_{2} + \)\(38\!\cdots\!40\)\( \beta_{3} + \)\(62\!\cdots\!60\)\( \beta_{4}) q^{41} +(\)\(56\!\cdots\!20\)\( + \)\(75\!\cdots\!00\)\( \beta_{1} - \)\(72\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!40\)\( \beta_{4}) q^{42} +(\)\(13\!\cdots\!81\)\( - \)\(39\!\cdots\!81\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} - \)\(23\!\cdots\!00\)\( \beta_{3} + \)\(67\!\cdots\!00\)\( \beta_{4}) q^{43} +(\)\(98\!\cdots\!28\)\( - \)\(59\!\cdots\!48\)\( \beta_{1} - \)\(11\!\cdots\!68\)\( \beta_{2} + \)\(45\!\cdots\!12\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{44} +(\)\(19\!\cdots\!25\)\( - \)\(33\!\cdots\!87\)\( \beta_{1} + \)\(14\!\cdots\!72\)\( \beta_{2} - \)\(91\!\cdots\!24\)\( \beta_{3} - \)\(43\!\cdots\!17\)\( \beta_{4}) q^{45} +(\)\(15\!\cdots\!12\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} - \)\(28\!\cdots\!40\)\( \beta_{3} + \)\(18\!\cdots\!60\)\( \beta_{4}) q^{46} +(-\)\(25\!\cdots\!96\)\( + \)\(44\!\cdots\!96\)\( \beta_{1} - \)\(19\!\cdots\!16\)\( \beta_{2} - \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!00\)\( \beta_{4}) q^{47} +(-\)\(39\!\cdots\!20\)\( + \)\(36\!\cdots\!00\)\( \beta_{1} - \)\(83\!\cdots\!56\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} - \)\(35\!\cdots\!60\)\( \beta_{4}) q^{48} +(-\)\(15\!\cdots\!47\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!40\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} - \)\(26\!\cdots\!40\)\( \beta_{4}) q^{49} +(-\)\(78\!\cdots\!75\)\( - \)\(39\!\cdots\!75\)\( \beta_{1} + \)\(89\!\cdots\!00\)\( \beta_{2} - \)\(35\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4}) q^{50} +(-\)\(67\!\cdots\!42\)\( + \)\(36\!\cdots\!26\)\( \beta_{1} - \)\(51\!\cdots\!34\)\( \beta_{2} + \)\(12\!\cdots\!96\)\( \beta_{3} - \)\(57\!\cdots\!60\)\( \beta_{4}) q^{51} +(\)\(13\!\cdots\!18\)\( + \)\(54\!\cdots\!82\)\( \beta_{1} - \)\(32\!\cdots\!04\)\( \beta_{2} + \)\(48\!\cdots\!50\)\( \beta_{3} - \)\(46\!\cdots\!00\)\( \beta_{4}) q^{52} +(\)\(20\!\cdots\!69\)\( + \)\(31\!\cdots\!21\)\( \beta_{1} - \)\(35\!\cdots\!44\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(66\!\cdots\!95\)\( \beta_{4}) q^{53} +(\)\(38\!\cdots\!44\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(58\!\cdots\!84\)\( \beta_{2} - \)\(87\!\cdots\!96\)\( \beta_{3} + \)\(17\!\cdots\!60\)\( \beta_{4}) q^{54} +(\)\(13\!\cdots\!50\)\( + \)\(32\!\cdots\!58\)\( \beta_{1} + \)\(96\!\cdots\!02\)\( \beta_{2} - \)\(67\!\cdots\!84\)\( \beta_{3} - \)\(64\!\cdots\!72\)\( \beta_{4}) q^{55} +(\)\(32\!\cdots\!16\)\( - \)\(30\!\cdots\!64\)\( \beta_{1} - \)\(17\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!76\)\( \beta_{3} + \)\(30\!\cdots\!20\)\( \beta_{4}) q^{56} +(-\)\(18\!\cdots\!22\)\( + \)\(51\!\cdots\!22\)\( \beta_{1} - \)\(62\!\cdots\!36\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57} +(-\)\(12\!\cdots\!34\)\( + \)\(34\!\cdots\!14\)\( \beta_{1} - \)\(37\!\cdots\!72\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3} - \)\(42\!\cdots\!60\)\( \beta_{4}) q^{58} +(-\)\(61\!\cdots\!03\)\( + \)\(11\!\cdots\!87\)\( \beta_{1} + \)\(12\!\cdots\!67\)\( \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + \)\(78\!\cdots\!00\)\( \beta_{4}) q^{59} +(-\)\(26\!\cdots\!00\)\( - \)\(12\!\cdots\!96\)\( \beta_{1} + \)\(14\!\cdots\!76\)\( \beta_{2} + \)\(14\!\cdots\!08\)\( \beta_{3} + \)\(85\!\cdots\!64\)\( \beta_{4}) q^{60} +(-\)\(22\!\cdots\!03\)\( + \)\(14\!\cdots\!25\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} - \)\(11\!\cdots\!25\)\( \beta_{4}) q^{61} +(-\)\(43\!\cdots\!88\)\( - \)\(76\!\cdots\!92\)\( \beta_{1} - \)\(51\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!60\)\( \beta_{4}) q^{62} +(-\)\(41\!\cdots\!02\)\( + \)\(62\!\cdots\!82\)\( \beta_{1} + \)\(27\!\cdots\!98\)\( \beta_{2} - \)\(43\!\cdots\!00\)\( \beta_{3} - \)\(50\!\cdots\!60\)\( \beta_{4}) q^{63} +(\)\(12\!\cdots\!20\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!92\)\( \beta_{2} + \)\(55\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!20\)\( \beta_{4}) q^{64} +(\)\(34\!\cdots\!00\)\( + \)\(33\!\cdots\!68\)\( \beta_{1} - \)\(94\!\cdots\!08\)\( \beta_{2} + \)\(17\!\cdots\!36\)\( \beta_{3} + \)\(16\!\cdots\!88\)\( \beta_{4}) q^{65} +(\)\(55\!\cdots\!92\)\( + \)\(68\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2} - \)\(20\!\cdots\!52\)\( \beta_{3} - \)\(21\!\cdots\!80\)\( \beta_{4}) q^{66} +(-\)\(23\!\cdots\!73\)\( + \)\(58\!\cdots\!53\)\( \beta_{1} - \)\(60\!\cdots\!27\)\( \beta_{2} - \)\(73\!\cdots\!00\)\( \beta_{3} + \)\(33\!\cdots\!40\)\( \beta_{4}) q^{67} +(\)\(16\!\cdots\!02\)\( - \)\(19\!\cdots\!42\)\( \beta_{1} - \)\(62\!\cdots\!28\)\( \beta_{2} + \)\(15\!\cdots\!50\)\( \beta_{3} + \)\(50\!\cdots\!80\)\( \beta_{4}) q^{68} +(\)\(71\!\cdots\!48\)\( - \)\(43\!\cdots\!56\)\( \beta_{1} + \)\(76\!\cdots\!04\)\( \beta_{2} + \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(70\!\cdots\!60\)\( \beta_{4}) q^{69} +(-\)\(52\!\cdots\!00\)\( - \)\(43\!\cdots\!76\)\( \beta_{1} + \)\(38\!\cdots\!56\)\( \beta_{2} - \)\(27\!\cdots\!52\)\( \beta_{3} + \)\(96\!\cdots\!84\)\( \beta_{4}) q^{70} +(-\)\(22\!\cdots\!58\)\( + \)\(16\!\cdots\!50\)\( \beta_{1} - \)\(23\!\cdots\!50\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!00\)\( \beta_{4}) q^{71} +(-\)\(14\!\cdots\!68\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!24\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!20\)\( \beta_{4}) q^{72} +(-\)\(60\!\cdots\!00\)\( - \)\(28\!\cdots\!70\)\( \beta_{1} + \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(33\!\cdots\!00\)\( \beta_{3} + \)\(14\!\cdots\!90\)\( \beta_{4}) q^{73} +(\)\(18\!\cdots\!26\)\( + \)\(42\!\cdots\!90\)\( \beta_{1} - \)\(17\!\cdots\!60\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3} - \)\(72\!\cdots\!60\)\( \beta_{4}) q^{74} +(\)\(43\!\cdots\!25\)\( - \)\(41\!\cdots\!25\)\( \beta_{1} + \)\(68\!\cdots\!75\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(83\!\cdots\!00\)\( \beta_{4}) q^{75} +(\)\(41\!\cdots\!12\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(89\!\cdots\!68\)\( \beta_{2} - \)\(13\!\cdots\!68\)\( \beta_{3} - \)\(29\!\cdots\!60\)\( \beta_{4}) q^{76} +(\)\(10\!\cdots\!56\)\( - \)\(47\!\cdots\!76\)\( \beta_{1} - \)\(81\!\cdots\!28\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3} + \)\(54\!\cdots\!40\)\( \beta_{4}) q^{77} +(\)\(12\!\cdots\!52\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(36\!\cdots\!36\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!80\)\( \beta_{4}) q^{78} +(\)\(59\!\cdots\!36\)\( + \)\(85\!\cdots\!56\)\( \beta_{1} - \)\(81\!\cdots\!04\)\( \beta_{2} - \)\(63\!\cdots\!44\)\( \beta_{3} - \)\(77\!\cdots\!40\)\( \beta_{4}) q^{79} +(-\)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(42\!\cdots\!76\)\( \beta_{2} - \)\(14\!\cdots\!92\)\( \beta_{3} + \)\(14\!\cdots\!64\)\( \beta_{4}) q^{80} +(-\)\(22\!\cdots\!25\)\( - \)\(39\!\cdots\!26\)\( \beta_{1} + \)\(13\!\cdots\!84\)\( \beta_{2} + \)\(12\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!50\)\( \beta_{4}) q^{81} +(-\)\(35\!\cdots\!18\)\( - \)\(22\!\cdots\!62\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!40\)\( \beta_{4}) q^{82} +(-\)\(94\!\cdots\!57\)\( - \)\(35\!\cdots\!43\)\( \beta_{1} - \)\(14\!\cdots\!27\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} + \)\(10\!\cdots\!00\)\( \beta_{4}) q^{83} +(-\)\(10\!\cdots\!68\)\( - \)\(19\!\cdots\!24\)\( \beta_{1} + \)\(99\!\cdots\!16\)\( \beta_{2} - \)\(20\!\cdots\!84\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4}) q^{84} +(-\)\(78\!\cdots\!50\)\( + \)\(15\!\cdots\!94\)\( \beta_{1} + \)\(65\!\cdots\!36\)\( \beta_{2} - \)\(53\!\cdots\!12\)\( \beta_{3} + \)\(64\!\cdots\!54\)\( \beta_{4}) q^{85} +(\)\(99\!\cdots\!28\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(23\!\cdots\!16\)\( \beta_{2} - \)\(18\!\cdots\!44\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{86} +(\)\(71\!\cdots\!82\)\( + \)\(24\!\cdots\!38\)\( \beta_{1} + \)\(71\!\cdots\!46\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3} - \)\(15\!\cdots\!40\)\( \beta_{4}) q^{87} +(\)\(86\!\cdots\!92\)\( - \)\(88\!\cdots\!12\)\( \beta_{1} - \)\(15\!\cdots\!84\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!40\)\( \beta_{4}) q^{88} +(-\)\(23\!\cdots\!12\)\( + \)\(45\!\cdots\!98\)\( \beta_{1} - \)\(86\!\cdots\!32\)\( \beta_{2} - \)\(12\!\cdots\!32\)\( \beta_{3} - \)\(74\!\cdots\!90\)\( \beta_{4}) q^{89} +(\)\(95\!\cdots\!50\)\( - \)\(15\!\cdots\!82\)\( \beta_{1} + \)\(13\!\cdots\!92\)\( \beta_{2} - \)\(84\!\cdots\!64\)\( \beta_{3} - \)\(25\!\cdots\!12\)\( \beta_{4}) q^{90} +(\)\(46\!\cdots\!08\)\( + \)\(82\!\cdots\!96\)\( \beta_{1} - \)\(38\!\cdots\!64\)\( \beta_{2} + \)\(60\!\cdots\!36\)\( \beta_{3} - \)\(37\!\cdots\!80\)\( \beta_{4}) q^{91} +(-\)\(19\!\cdots\!68\)\( + \)\(38\!\cdots\!88\)\( \beta_{1} + \)\(26\!\cdots\!48\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3} - \)\(25\!\cdots\!40\)\( \beta_{4}) q^{92} +(-\)\(36\!\cdots\!52\)\( + \)\(11\!\cdots\!92\)\( \beta_{1} + \)\(34\!\cdots\!72\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(21\!\cdots\!20\)\( \beta_{4}) q^{93} +(-\)\(98\!\cdots\!40\)\( + \)\(51\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!16\)\( \beta_{2} - \)\(49\!\cdots\!96\)\( \beta_{3} - \)\(11\!\cdots\!40\)\( \beta_{4}) q^{94} +(-\)\(40\!\cdots\!50\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} - \)\(18\!\cdots\!90\)\( \beta_{2} + \)\(52\!\cdots\!80\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4}) q^{95} +(-\)\(44\!\cdots\!24\)\( - \)\(60\!\cdots\!36\)\( \beta_{1} + \)\(25\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!16\)\( \beta_{3} + \)\(50\!\cdots\!20\)\( \beta_{4}) q^{96} +(\)\(57\!\cdots\!24\)\( - \)\(21\!\cdots\!14\)\( \beta_{1} - \)\(76\!\cdots\!16\)\( \beta_{2} + \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(50\!\cdots\!30\)\( \beta_{4}) q^{97} +(\)\(24\!\cdots\!03\)\( - \)\(68\!\cdots\!33\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3} + \)\(79\!\cdots\!60\)\( \beta_{4}) q^{98} +(\)\(32\!\cdots\!53\)\( + \)\(51\!\cdots\!19\)\( \beta_{1} + \)\(62\!\cdots\!79\)\( \beta_{2} - \)\(36\!\cdots\!96\)\( \beta_{3} - \)\(18\!\cdots\!20\)\( \beta_{4}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5q + 5554901256q^{2} + 3443360269119372q^{3} + \)\(35\!\cdots\!40\)\(q^{4} + \)\(33\!\cdots\!50\)\(q^{5} + \)\(14\!\cdots\!60\)\(q^{6} + \)\(33\!\cdots\!56\)\(q^{7} + \)\(32\!\cdots\!80\)\(q^{8} + \)\(27\!\cdots\!85\)\(q^{9} + O(q^{10}) \)	\( 5q + 5554901256q^{2} + 3443360269119372q^{3} + \)\(35\!\cdots\!40\)\(q^{4} + \)\(33\!\cdots\!50\)\(q^{5} + \)\(14\!\cdots\!60\)\(q^{6} + \)\(33\!\cdots\!56\)\(q^{7} + \)\(32\!\cdots\!80\)\(q^{8} + \)\(27\!\cdots\!85\)\(q^{9} - \)\(52\!\cdots\!00\)\(q^{10} + \)\(20\!\cdots\!60\)\(q^{11} - \)\(20\!\cdots\!24\)\(q^{12} + \)\(17\!\cdots\!02\)\(q^{13} + \)\(29\!\cdots\!80\)\(q^{14} - \)\(93\!\cdots\!00\)\(q^{15} + \)\(18\!\cdots\!80\)\(q^{16} + \)\(75\!\cdots\!06\)\(q^{17} - \)\(10\!\cdots\!68\)\(q^{18} + \)\(39\!\cdots\!00\)\(q^{19} - \)\(15\!\cdots\!00\)\(q^{20} - \)\(17\!\cdots\!40\)\(q^{21} + \)\(18\!\cdots\!32\)\(q^{22} - \)\(41\!\cdots\!68\)\(q^{23} - \)\(13\!\cdots\!00\)\(q^{24} + \)\(17\!\cdots\!75\)\(q^{25} + \)\(21\!\cdots\!60\)\(q^{26} + \)\(18\!\cdots\!20\)\(q^{27} + \)\(14\!\cdots\!48\)\(q^{28} + \)\(18\!\cdots\!50\)\(q^{29} + \)\(18\!\cdots\!00\)\(q^{30} + \)\(36\!\cdots\!60\)\(q^{31} + \)\(15\!\cdots\!96\)\(q^{32} + \)\(24\!\cdots\!84\)\(q^{33} + \)\(78\!\cdots\!80\)\(q^{34} + \)\(45\!\cdots\!00\)\(q^{35} - \)\(38\!\cdots\!20\)\(q^{36} - \)\(56\!\cdots\!94\)\(q^{37} - \)\(31\!\cdots\!80\)\(q^{38} - \)\(71\!\cdots\!80\)\(q^{39} - \)\(18\!\cdots\!00\)\(q^{40} + \)\(11\!\cdots\!10\)\(q^{41} + \)\(28\!\cdots\!92\)\(q^{42} + \)\(65\!\cdots\!92\)\(q^{43} + \)\(49\!\cdots\!80\)\(q^{44} + \)\(99\!\cdots\!50\)\(q^{45} + \)\(79\!\cdots\!60\)\(q^{46} - \)\(12\!\cdots\!44\)\(q^{47} - \)\(19\!\cdots\!68\)\(q^{48} - \)\(77\!\cdots\!35\)\(q^{49} - \)\(39\!\cdots\!00\)\(q^{50} - \)\(33\!\cdots\!40\)\(q^{51} + \)\(67\!\cdots\!16\)\(q^{52} + \)\(10\!\cdots\!22\)\(q^{53} + \)\(19\!\cdots\!00\)\(q^{54} + \)\(66\!\cdots\!00\)\(q^{55} + \)\(16\!\cdots\!00\)\(q^{56} - \)\(92\!\cdots\!60\)\(q^{57} - \)\(61\!\cdots\!20\)\(q^{58} - \)\(30\!\cdots\!00\)\(q^{59} - \)\(13\!\cdots\!00\)\(q^{60} - \)\(11\!\cdots\!90\)\(q^{61} - \)\(21\!\cdots\!28\)\(q^{62} - \)\(20\!\cdots\!68\)\(q^{63} + \)\(63\!\cdots\!40\)\(q^{64} + \)\(17\!\cdots\!00\)\(q^{65} + \)\(27\!\cdots\!20\)\(q^{66} - \)\(11\!\cdots\!44\)\(q^{67} + \)\(81\!\cdots\!48\)\(q^{68} + \)\(35\!\cdots\!20\)\(q^{69} - \)\(26\!\cdots\!00\)\(q^{70} - \)\(11\!\cdots\!40\)\(q^{71} - \)\(74\!\cdots\!40\)\(q^{72} - \)\(30\!\cdots\!18\)\(q^{73} + \)\(92\!\cdots\!80\)\(q^{74} + \)\(21\!\cdots\!00\)\(q^{75} + \)\(20\!\cdots\!00\)\(q^{76} + \)\(50\!\cdots\!32\)\(q^{77} + \)\(61\!\cdots\!64\)\(q^{78} + \)\(29\!\cdots\!00\)\(q^{79} - \)\(15\!\cdots\!00\)\(q^{80} - \)\(11\!\cdots\!95\)\(q^{81} - \)\(17\!\cdots\!08\)\(q^{82} - \)\(47\!\cdots\!88\)\(q^{83} - \)\(52\!\cdots\!20\)\(q^{84} - \)\(39\!\cdots\!00\)\(q^{85} + \)\(49\!\cdots\!60\)\(q^{86} + \)\(35\!\cdots\!60\)\(q^{87} + \)\(43\!\cdots\!60\)\(q^{88} - \)\(11\!\cdots\!50\)\(q^{89} + \)\(47\!\cdots\!00\)\(q^{90} + \)\(23\!\cdots\!60\)\(q^{91} - \)\(98\!\cdots\!44\)\(q^{92} - \)\(18\!\cdots\!36\)\(q^{93} - \)\(49\!\cdots\!20\)\(q^{94} - \)\(20\!\cdots\!00\)\(q^{95} - \)\(22\!\cdots\!40\)\(q^{96} + \)\(28\!\cdots\!06\)\(q^{97} + \)\(12\!\cdots\!08\)\(q^{98} + \)\(16\!\cdots\!20\)\(q^{99} + O(q^{100}) \)

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( 24 \nu - 5 \)
\(\beta_{2}\)	\(=\)	\((\)\(-1463 \nu^{4} + 61316688179 \nu^{3} + 1134677465176013610916 \nu^{2} - 175871625402178878144147591024 \nu - 92322819705534579210406721528027067456\)\()/ \)\(37\!\cdots\!28\)\( \)
\(\beta_{3}\)	\(=\)	\((\)\(-1461537 \nu^{4} + 61255371490821 \nu^{3} + 1493224099349187733125372 \nu^{2} - 157864696669790310577560760925712 \nu - 227382046305944113582206155994463766078912\)\()/ \)\(62\!\cdots\!88\)\( \)
\(\beta_{4}\)	\(=\)	\((\)\(4738030953 \nu^{4} - 6783896933686506093 \nu^{3} - 6265292406150844870725632988 \nu^{2} + 4428965898625782506113044897501160080 \nu + 1149735030546567532337013572577831183558493120\)\()/ \)\(15\!\cdots\!20\)\( \)

\(1\)	\(=\)	\(\beta_0\)
\(\nu\)	\(=\)	\((\)\(\beta_{1} + 5\)\()/24\)
\(\nu^{2}\)	\(=\)	\((\)\(\beta_{3} - 5994 \beta_{2} - 1189790397 \beta_{1} + 216434076347341357501\)\()/576\)
\(\nu^{3}\)	\(=\)	\((\)\(-409640 \beta_{4} - 1180151429 \beta_{3} + 3889870865010 \beta_{2} + 43600466505262789881 \beta_{1} - 32188957769729085286587521881\)\()/1728\)
\(\nu^{4}\)	\(=\)	\((\)\(-17168672690120 \beta_{4} + 2277286000240633339 \beta_{3} - 18208831024879688328462 \beta_{2} - 9596312969973284759959813095 \beta_{1} + 393192791921864844386864518484726790407\)\()/1728\)

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