LMFDB - Space of Modular Forms of Level 8 and Weight 14

Defining parameters

Level:	$ N $	=	$ 8 = 2^{3} $
Weight:	$ k $	=	$ 14 $
Nonzero newspaces:			$ 2 $
Newform subspaces:			$ 4 $
Sturm bound:			$56$
Trace bound:			$1$

Dimensions

The following table gives the dimensions of various subspaces of $M_{14}(\Gamma_1(8))$.

	Total	New	Old
Modular forms	29	17	12
Cusp forms	23	15	8
Eisenstein series	6	2	4

Trace form

$ 15q - 2q^{2} + 860q^{3} - 8556q^{4} + 14146q^{5} + 58444q^{6} + 206232q^{7} + 1076872q^{8} - 3319093q^{9} + O(q^{10}) $

\( 15q - 2q^{2} + 860q^{3} - 8556q^{4} + 14146q^{5} + 58444q^{6} + 206232q^{7} + 1076872q^{8} - 3319093q^{9} - 7752648q^{10} + 9989716q^{11} - 8536664q^{12} - 3843462q^{13} + 21101872q^{14} + 42509064q^{15} - 62847216q^{16} - 227240498q^{17} + 262778834q^{18} + 206597196q^{19} + 381237904q^{20} - 1370300832q^{21} + 196775484q^{22} + 1490018760q^{23} + 1739438480q^{24} - 4205723223q^{25} - 685452248q^{26} + 5792047640q^{27} - 978617568q^{28} - 3556537878q^{29} + 539961392q^{30} - 7367666592q^{31} - 8956909792q^{32} + 14055767168q^{33} + 8857451100q^{34} - 15413225904q^{35} + 18075363660q^{36} + 6628831266q^{37} - 23710846420q^{38} + 20630243496q^{39} - 6505554528q^{40} + 47965956118q^{41} - 3443304224q^{42} - 46788121548q^{43} - 28647791928q^{44} + 107010747322q^{45} - 51061300848q^{46} - 201450140016q^{47} - 161587694304q^{48} + 101164221495q^{49} + 246819442102q^{50} + 177530164600q^{51} + 441779820720q^{52} - 125287225454q^{53} - 639572716168q^{54} + 217008668376q^{55} - 699100837568q^{56} - 1007795333472q^{57} + 992195908104q^{58} + 793560430372q^{59} + 1614284565792q^{60} - 698520953142q^{61} - 1019628353344q^{62} - 426693206408q^{63} - 1464077986752q^{64} - 357709159492q^{65} + 2946347707608q^{66} + 1645513736988q^{67} + 2850998874984q^{68} + 1015234563872q^{69} - 5505591528768q^{70} - 4320352976296q^{71} - 8538615449032q^{72} + 3510289845750q^{73} + 9399657781240q^{74} + 844579119460q^{75} + 10892359157736q^{76} + 1067266262304q^{77} - 16235130104944q^{78} - 1836704085264q^{79} - 16787669526720q^{80} + 3452732525415q^{81} + 14910525285612q^{82} + 3890337600524q^{83} + 26784826987072q^{84} + 1613607983652q^{85} - 24718490913284q^{86} - 14659545283608q^{87} - 26046521626416q^{88} - 745298334170q^{89} + 43218339424520q^{90} + 19047242532624q^{91} + 34108342295904q^{92} - 6946080702080q^{93} - 37727942061792q^{94} - 25493775576920q^{95} - 62733552180160q^{96} - 15392053442370q^{97} + 61714328751438q^{98} + 42866396476420q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{14}^{\mathrm{new}}(\Gamma_1(8))$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $ S_k^{\mathrm{new}}(N, \chi) $ we list the newforms together with their dimension.

Label	$\chi$	Newforms	Dimension	$\chi$ degree
8.14.a	$\chi_{8}(1, \cdot)$	8.14.a.a	1	1
8.14.a	$\chi_{8}(1, \cdot)$	8.14.a.b	2	1
8.14.b	$\chi_{8}(5, \cdot)$	8.14.b.a	2	1
8.14.b	$\chi_{8}(5, \cdot)$	8.14.b.b	10	1

Decomposition of $S_{14}^{\mathrm{old}}(\Gamma_1(8))$ into lower level spaces

$ S_{14}^{\mathrm{old}}(\Gamma_1(8)) \cong $ $S_{14}^{\mathrm{new}}(\Gamma_1(2))$$^{\oplus 3}$$\oplus$$S_{14}^{\mathrm{new}}(\Gamma_1(4))$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	()()($ 1 + 112 T + 8192 T^{2} $)($ 1 - 110 T + 8408 T^{2} - 390976 T^{3} - 935936 T^{4} + 3612475392 T^{5} - 7667187712 T^{6} - 26237955211264 T^{7} + 4622346883170304 T^{8} - 495395959010754560 T^{9} + 36893488147419103232 T^{10} $)
$3$	($ 1 + 12 T + 1594323 T^{2} $)($ 1 - 872 T + 179766 T^{2} - 1390249656 T^{3} + 2541865828329 T^{4} $)($ 1 + 2069910 T^{2} + 2541865828329 T^{4} $)($ 1 - 8978642 T^{2} + 41923886773365 T^{4} - $$13\!\cdots\!32$$ T^{6} + $$31\!\cdots\!22$$ T^{8} - $$56\!\cdots\!92$$ T^{10} + $$78\!\cdots\!38$$ T^{12} - $$85\!\cdots\!12$$ T^{14} + $$68\!\cdots\!85$$ T^{16} - $$37\!\cdots\!02$$ T^{18} + $$10\!\cdots\!49$$ T^{20} $)
$5$	($ 1 + 4330 T + 1220703125 T^{2} $)($ 1 - 18476 T + 2066094350 T^{2} - 22553710937500 T^{3} + 1490116119384765625 T^{4} $)($ 1 - 1931729850 T^{2} + 1490116119384765625 T^{4} $)($ 1 - 4565314338 T^{2} + 10147469641592691461 T^{4} - $$14\!\cdots\!00$$ T^{6} + $$15\!\cdots\!50$$ T^{8} - $$17\!\cdots\!00$$ T^{10} + $$23\!\cdots\!50$$ T^{12} - $$32\!\cdots\!00$$ T^{14} + $$33\!\cdots\!25$$ T^{16} - $$22\!\cdots\!50$$ T^{18} + $$73\!\cdots\!25$$ T^{20} $)
$7$	($ 1 + 139992 T + 96889010407 T^{2} $)($ 1 - 110928 T + 39195942926 T^{2} - 10747704146427696 T^{3} + $$93\!\cdots\!49$$ T^{4} $)($ ( 1 + 175832 T + 96889010407 T^{2} )^{2} $)($ ( 1 - 293480 T + 239610643203 T^{2} - 95417056996483936 T^{3} + $$32\!\cdots\!74$$ T^{4} - $$13\!\cdots\!28$$ T^{5} + $$31\!\cdots\!18$$ T^{6} - $$89\!\cdots\!64$$ T^{7} + $$21\!\cdots\!29$$ T^{8} - $$25\!\cdots\!80$$ T^{9} + $$85\!\cdots\!07$$ T^{10} )^{2} $)
$11$	($ 1 + 6484324 T + 34522712143931 T^{2} $)($ 1 - 16474040 T + 135991936296326 T^{2} - $$56\!\cdots\!40$$ T^{3} + $$11\!\cdots\!61$$ T^{4} $)($ 1 - 62100824999962 T^{2} + $$11\!\cdots\!61$$ T^{4} $)($ 1 - 179340783809858 T^{2} + $$16\!\cdots\!41$$ T^{4} - $$10\!\cdots\!00$$ T^{6} + $$50\!\cdots\!78$$ T^{8} - $$19\!\cdots\!64$$ T^{10} + $$60\!\cdots\!58$$ T^{12} - $$14\!\cdots\!00$$ T^{14} + $$27\!\cdots\!21$$ T^{16} - $$36\!\cdots\!78$$ T^{18} + $$24\!\cdots\!01$$ T^{20} $)
$13$	($ 1 + 22588034 T + 302875106592253 T^{2} $)($ 1 - 18744572 T + 344120051417598 T^{2} - $$56\!\cdots\!16$$ T^{3} + $$91\!\cdots\!09$$ T^{4} $)($ 1 + 360392330876150 T^{2} + $$91\!\cdots\!09$$ T^{4} $)($ 1 - 1881700477395890 T^{2} + $$17\!\cdots\!13$$ T^{4} - $$10\!\cdots\!00$$ T^{6} + $$48\!\cdots\!02$$ T^{8} - $$16\!\cdots\!20$$ T^{10} + $$44\!\cdots\!18$$ T^{12} - $$91\!\cdots\!00$$ T^{14} + $$13\!\cdots\!77$$ T^{16} - $$13\!\cdots\!90$$ T^{18} + $$64\!\cdots\!49$$ T^{20} $)
$17$	($ 1 + 23732270 T + 9904578032905937 T^{2} $)($ 1 + 153793628 T + 21058111940247014 T^{2} + $$15\!\cdots\!36$$ T^{3} + $$98\!\cdots\!69$$ T^{4} $)($ ( 1 + 133520302 T + 9904578032905937 T^{2} )^{2} $)($ ( 1 - 108663002 T + 32276400633367229 T^{2} - $$23\!\cdots\!84$$ T^{3} + $$50\!\cdots\!42$$ T^{4} - $$29\!\cdots\!76$$ T^{5} + $$49\!\cdots\!54$$ T^{6} - $$23\!\cdots\!96$$ T^{7} + $$31\!\cdots\!37$$ T^{8} - $$10\!\cdots\!22$$ T^{9} + $$95\!\cdots\!57$$ T^{10} )^{2} $)
$19$	($ 1 - 325344836 T + 42052983462257059 T^{2} $)($ 1 + 118747640 T + 59473815443581782 T^{2} + $$49\!\cdots\!60$$ T^{3} + $$17\!\cdots\!81$$ T^{4} $)($ 1 - 82896428399326282 T^{2} + $$17\!\cdots\!81$$ T^{4} $)($ 1 - 162780863064290354 T^{2} + $$13\!\cdots\!97$$ T^{4} - $$80\!\cdots\!36$$ T^{6} + $$39\!\cdots\!66$$ T^{8} - $$17\!\cdots\!68$$ T^{10} + $$69\!\cdots\!46$$ T^{12} - $$25\!\cdots\!96$$ T^{14} + $$75\!\cdots\!77$$ T^{16} - $$15\!\cdots\!34$$ T^{18} + $$17\!\cdots\!01$$ T^{20} $)
$23$	($ 1 - 921600632 T + 504036361936467383 T^{2} $)($ 1 - 718268912 T + 1097306915486653358 T^{2} - $$36\!\cdots\!96$$ T^{3} + $$25\!\cdots\!89$$ T^{4} $)($ ( 1 + 35585416 T + 504036361936467383 T^{2} )^{2} $)($ ( 1 + 39339976 T + 1242751119340586771 T^{2} + $$42\!\cdots\!92$$ T^{3} + $$76\!\cdots\!02$$ T^{4} + $$38\!\cdots\!08$$ T^{5} + $$38\!\cdots\!66$$ T^{6} + $$10\!\cdots\!88$$ T^{7} + $$15\!\cdots\!77$$ T^{8} + $$25\!\cdots\!96$$ T^{9} + $$32\!\cdots\!43$$ T^{10} )^{2} $)
$29$	($ 1 + 3865879218 T + 10260628712958602189 T^{2} $)($ 1 - 309341340 T + 2504177667132675934 T^{2} - $$31\!\cdots\!60$$ T^{3} + $$10\!\cdots\!21$$ T^{4} $)($ 1 - 18002148940349036842 T^{2} + $$10\!\cdots\!21$$ T^{4} $)($ 1 - 60047671171855484498 T^{2} + $$18\!\cdots\!41$$ T^{4} - $$38\!\cdots\!60$$ T^{6} + $$58\!\cdots\!78$$ T^{8} - $$67\!\cdots\!84$$ T^{10} + $$61\!\cdots\!38$$ T^{12} - $$42\!\cdots\!60$$ T^{14} + $$21\!\cdots\!01$$ T^{16} - $$73\!\cdots\!38$$ T^{18} + $$12\!\cdots\!01$$ T^{20} $)
$31$	($ 1 + 2253401440 T + 24417546297445042591 T^{2} $)($ 1 - 5767504192 T + 50116253247327696702 T^{2} - $$14\!\cdots\!72$$ T^{3} + $$59\!\cdots\!81$$ T^{4} $)($ ( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2} $)($ ( 1 - 324116896 T + 94759416869201467419 T^{2} - $$76\!\cdots\!28$$ T^{3} + $$40\!\cdots\!58$$ T^{4} - $$32\!\cdots\!48$$ T^{5} + $$98\!\cdots\!78$$ T^{6} - $$45\!\cdots\!68$$ T^{7} + $$13\!\cdots\!49$$ T^{8} - $$11\!\cdots\!56$$ T^{9} + $$86\!\cdots\!51$$ T^{10} )^{2} $)
$37$	($ 1 - 18250384566 T + $$24\!\cdots\!97$$ T^{2} $)($ 1 + 11621553300 T + $$18\!\cdots\!38$$ T^{2} + $$28\!\cdots\!00$$ T^{3} + $$59\!\cdots\!09$$ T^{4} $)($ 1 - $$31\!\cdots\!70$$ T^{2} + $$59\!\cdots\!09$$ T^{4} $)($ 1 - $$85\!\cdots\!14$$ T^{2} + $$35\!\cdots\!29$$ T^{4} - $$83\!\cdots\!24$$ T^{6} + $$11\!\cdots\!78$$ T^{8} - $$14\!\cdots\!64$$ T^{10} + $$67\!\cdots\!02$$ T^{12} - $$29\!\cdots\!44$$ T^{14} + $$73\!\cdots\!41$$ T^{16} - $$10\!\cdots\!54$$ T^{18} + $$73\!\cdots\!49$$ T^{20} $)
$41$	($ 1 - 34422845322 T + $$92\!\cdots\!21$$ T^{2} $)($ 1 - 1311168276 T + $$16\!\cdots\!30$$ T^{2} - $$12\!\cdots\!96$$ T^{3} + $$85\!\cdots\!41$$ T^{4} $)($ ( 1 + 23546348918 T + $$92\!\cdots\!21$$ T^{2} )^{2} $)($ ( 1 - 29662320178 T + $$14\!\cdots\!89$$ T^{2} - $$15\!\cdots\!16$$ T^{3} + $$10\!\cdots\!62$$ T^{4} - $$29\!\cdots\!76$$ T^{5} + $$97\!\cdots\!02$$ T^{6} - $$13\!\cdots\!56$$ T^{7} + $$11\!\cdots\!29$$ T^{8} - $$21\!\cdots\!18$$ T^{9} + $$67\!\cdots\!01$$ T^{10} )^{2} $)
$43$	($ 1 + 17192501444 T + $$17\!\cdots\!43$$ T^{2} $)($ 1 + 29595620104 T + $$33\!\cdots\!74$$ T^{2} + $$50\!\cdots\!72$$ T^{3} + $$29\!\cdots\!49$$ T^{4} $)($ 1 - $$32\!\cdots\!30$$ T^{2} + $$29\!\cdots\!49$$ T^{4} $)($ 1 - $$61\!\cdots\!14$$ T^{2} + $$15\!\cdots\!05$$ T^{4} - $$25\!\cdots\!04$$ T^{6} + $$52\!\cdots\!62$$ T^{8} - $$10\!\cdots\!24$$ T^{10} + $$15\!\cdots\!38$$ T^{12} - $$22\!\cdots\!04$$ T^{14} + $$39\!\cdots\!45$$ T^{16} - $$46\!\cdots\!14$$ T^{18} + $$22\!\cdots\!49$$ T^{20} $)
$47$	($ 1 + 67371749904 T + $$54\!\cdots\!27$$ T^{2} $)($ 1 - 12313617888 T + $$10\!\cdots\!90$$ T^{2} - $$67\!\cdots\!76$$ T^{3} + $$29\!\cdots\!29$$ T^{4} $)($ ( 1 + 68107736592 T + $$54\!\cdots\!27$$ T^{2} )^{2} $)($ ( 1 + 5088267408 T + $$10\!\cdots\!43$$ T^{2} + $$16\!\cdots\!28$$ T^{3} + $$79\!\cdots\!70$$ T^{4} + $$90\!\cdots\!04$$ T^{5} + $$43\!\cdots\!90$$ T^{6} + $$48\!\cdots\!12$$ T^{7} + $$17\!\cdots\!69$$ T^{8} + $$45\!\cdots\!28$$ T^{9} + $$48\!\cdots\!07$$ T^{10} )^{2} $)
$53$	($ 1 + 87281218426 T + $$26\!\cdots\!73$$ T^{2} $)($ 1 + 38006007028 T + $$44\!\cdots\!42$$ T^{2} + $$98\!\cdots\!44$$ T^{3} + $$67\!\cdots\!29$$ T^{4} $)($ 1 - $$24\!\cdots\!30$$ T^{2} + $$67\!\cdots\!29$$ T^{4} $)($ 1 - $$15\!\cdots\!98$$ T^{2} + $$12\!\cdots\!49$$ T^{4} - $$63\!\cdots\!52$$ T^{6} + $$24\!\cdots\!06$$ T^{8} - $$72\!\cdots\!20$$ T^{10} + $$16\!\cdots\!74$$ T^{12} - $$29\!\cdots\!32$$ T^{14} + $$37\!\cdots\!61$$ T^{16} - $$32\!\cdots\!38$$ T^{18} + $$14\!\cdots\!49$$ T^{20} $)
$59$	($ 1 - 540214518668 T + $$10\!\cdots\!79$$ T^{2} $)($ 1 - 253345911704 T + $$16\!\cdots\!06$$ T^{2} - $$26\!\cdots\!16$$ T^{3} + $$11\!\cdots\!41$$ T^{4} $)($ 1 - $$19\!\cdots\!22$$ T^{2} + $$11\!\cdots\!41$$ T^{4} $)($ 1 - $$54\!\cdots\!82$$ T^{2} + $$16\!\cdots\!77$$ T^{4} - $$32\!\cdots\!52$$ T^{6} + $$49\!\cdots\!86$$ T^{8} - $$58\!\cdots\!80$$ T^{10} + $$54\!\cdots\!26$$ T^{12} - $$39\!\cdots\!12$$ T^{14} + $$21\!\cdots\!17$$ T^{16} - $$80\!\cdots\!02$$ T^{18} + $$16\!\cdots\!01$$ T^{20} $)
$61$	($ 1 + 51276568850 T + $$16\!\cdots\!81$$ T^{2} $)($ 1 + 647244384292 T + $$42\!\cdots\!62$$ T^{2} + $$10\!\cdots\!52$$ T^{3} + $$26\!\cdots\!61$$ T^{4} $)($ 1 - $$14\!\cdots\!62$$ T^{2} + $$26\!\cdots\!61$$ T^{4} $)($ 1 - $$10\!\cdots\!62$$ T^{2} + $$47\!\cdots\!37$$ T^{4} - $$13\!\cdots\!12$$ T^{6} + $$29\!\cdots\!66$$ T^{8} - $$51\!\cdots\!60$$ T^{10} + $$77\!\cdots\!26$$ T^{12} - $$96\!\cdots\!52$$ T^{14} + $$86\!\cdots\!97$$ T^{16} - $$48\!\cdots\!42$$ T^{18} + $$12\!\cdots\!01$$ T^{20} $)
$67$	($ 1 - 25519930676 T + $$54\!\cdots\!87$$ T^{2} $)($ 1 - 1619993806312 T + $$17\!\cdots\!86$$ T^{2} - $$88\!\cdots\!44$$ T^{3} + $$30\!\cdots\!69$$ T^{4} $)($ 1 - $$95\!\cdots\!30$$ T^{2} + $$30\!\cdots\!69$$ T^{4} $)($ 1 - $$19\!\cdots\!42$$ T^{2} + $$23\!\cdots\!49$$ T^{4} - $$18\!\cdots\!88$$ T^{6} + $$12\!\cdots\!86$$ T^{8} - $$71\!\cdots\!60$$ T^{10} + $$37\!\cdots\!34$$ T^{12} - $$17\!\cdots\!68$$ T^{14} + $$64\!\cdots\!41$$ T^{16} - $$16\!\cdots\!82$$ T^{18} + $$24\!\cdots\!49$$ T^{20} $)
$71$	($ 1 + 1387500699032 T + $$11\!\cdots\!11$$ T^{2} $)($ 1 + 1040270142512 T + $$22\!\cdots\!82$$ T^{2} + $$12\!\cdots\!32$$ T^{3} + $$13\!\cdots\!21$$ T^{4} $)($ ( 1 + 1309471657368 T + $$11\!\cdots\!11$$ T^{2} )^{2} $)($ ( 1 - 363180589992 T + $$34\!\cdots\!95$$ T^{2} - $$14\!\cdots\!16$$ T^{3} + $$61\!\cdots\!94$$ T^{4} - $$22\!\cdots\!64$$ T^{5} + $$71\!\cdots\!34$$ T^{6} - $$20\!\cdots\!36$$ T^{7} + $$53\!\cdots\!45$$ T^{8} - $$66\!\cdots\!72$$ T^{9} + $$21\!\cdots\!51$$ T^{10} )^{2} $)
$73$	($ 1 + 819049441238 T + $$16\!\cdots\!33$$ T^{2} $)($ 1 - 4005283908692 T + $$73\!\cdots\!18$$ T^{2} - $$66\!\cdots\!36$$ T^{3} + $$27\!\cdots\!89$$ T^{4} $)($ ( 1 - 478647871914 T + $$16\!\cdots\!33$$ T^{2} )^{2} $)($ ( 1 + 316620182766 T + $$64\!\cdots\!53$$ T^{2} + $$24\!\cdots\!84$$ T^{3} + $$18\!\cdots\!54$$ T^{4} + $$60\!\cdots\!52$$ T^{5} + $$31\!\cdots\!82$$ T^{6} + $$67\!\cdots\!76$$ T^{7} + $$29\!\cdots\!61$$ T^{8} + $$24\!\cdots\!86$$ T^{9} + $$13\!\cdots\!93$$ T^{10} )^{2} $)
$79$	($ 1 + 4030935615344 T + $$46\!\cdots\!39$$ T^{2} $)($ 1 + 2521777572064 T + $$10\!\cdots\!58$$ T^{2} + $$11\!\cdots\!96$$ T^{3} + $$21\!\cdots\!21$$ T^{4} $)($ ( 1 + 364547231600 T + $$46\!\cdots\!39$$ T^{2} )^{2} $)($ ( 1 - 2722551782672 T + $$12\!\cdots\!19$$ T^{2} - $$22\!\cdots\!64$$ T^{3} + $$55\!\cdots\!14$$ T^{4} - $$94\!\cdots\!68$$ T^{5} + $$25\!\cdots\!46$$ T^{6} - $$49\!\cdots\!44$$ T^{7} + $$12\!\cdots\!61$$ T^{8} - $$12\!\cdots\!52$$ T^{9} + $$22\!\cdots\!99$$ T^{10} )^{2} $)
$83$	($ 1 - 4180823831428 T + $$88\!\cdots\!63$$ T^{2} $)($ 1 + 290486230904 T + $$14\!\cdots\!54$$ T^{2} + $$25\!\cdots\!52$$ T^{3} + $$78\!\cdots\!69$$ T^{4} $)($ 1 - $$16\!\cdots\!70$$ T^{2} + $$78\!\cdots\!69$$ T^{4} $)($ 1 - $$43\!\cdots\!74$$ T^{2} + $$87\!\cdots\!65$$ T^{4} - $$12\!\cdots\!64$$ T^{6} + $$14\!\cdots\!62$$ T^{8} - $$14\!\cdots\!44$$ T^{10} + $$11\!\cdots\!78$$ T^{12} - $$76\!\cdots\!04$$ T^{14} + $$42\!\cdots\!85$$ T^{16} - $$16\!\cdots\!54$$ T^{18} + $$30\!\cdots\!49$$ T^{20} $)
$89$	($ 1 - 2677027798266 T + $$21\!\cdots\!69$$ T^{2} $)($ 1 + 8723755657740 T + $$61\!\cdots\!42$$ T^{2} + $$19\!\cdots\!60$$ T^{3} + $$48\!\cdots\!61$$ T^{4} $)($ ( 1 + 102457641350 T + $$21\!\cdots\!69$$ T^{2} )^{2} $)($ ( 1 - 2753172404002 T + $$62\!\cdots\!29$$ T^{2} - $$14\!\cdots\!84$$ T^{3} + $$21\!\cdots\!54$$ T^{4} - $$47\!\cdots\!48$$ T^{5} + $$47\!\cdots\!26$$ T^{6} - $$70\!\cdots\!24$$ T^{7} + $$66\!\cdots\!61$$ T^{8} - $$64\!\cdots\!42$$ T^{9} + $$51\!\cdots\!49$$ T^{10} )^{2} $)
$97$	($ 1 + 14039464316446 T + $$67\!\cdots\!77$$ T^{2} $)($ 1 - 9601712299972 T + $$81\!\cdots\!50$$ T^{2} - $$64\!\cdots\!44$$ T^{3} + $$45\!\cdots\!29$$ T^{4} $)($ ( 1 + 6157717373342 T + $$67\!\cdots\!77$$ T^{2} )^{2} $)($ ( 1 - 680566660394 T + $$17\!\cdots\!53$$ T^{2} + $$11\!\cdots\!08$$ T^{3} + $$15\!\cdots\!70$$ T^{4} + $$20\!\cdots\!28$$ T^{5} + $$10\!\cdots\!90$$ T^{6} + $$53\!\cdots\!32$$ T^{7} + $$54\!\cdots\!49$$ T^{8} - $$13\!\cdots\!54$$ T^{9} + $$13\!\cdots\!57$$ T^{10} )^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 8 = 2^{3} \)
Weight:	\( k \)	=	\( 14 \)
Nonzero newspaces:			\( 2 \)
Newform subspaces:			\( 4 \)
Sturm bound:			\(56\)
Trace bound:			\(1\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{14}^{\mathrm{new}}(\Gamma_1(8))\)

Decomposition of \(S_{14}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

Hecke Characteristic Polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8.14

Level	8
Weight	14
Dimension	15
Nonzero newspaces	2
Newform subspaces	4
Sturm bound	56
Trace bound	1

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
8.14.a	\(\chi_{8}(1, \cdot)\)	8.14.a.a	1	1
8.14.a	\(\chi_{8}(1, \cdot)\)	8.14.a.b	2	1
8.14.b	\(\chi_{8}(5, \cdot)\)	8.14.b.a	2	1
8.14.b	\(\chi_{8}(5, \cdot)\)	8.14.b.b	10	1

$p$	$F_p(T)$
$2$	(\( \))(\( \))(\( 1 + 112 T + 8192 T^{2} \))(\( 1 - 110 T + 8408 T^{2} - 390976 T^{3} - 935936 T^{4} + 3612475392 T^{5} - 7667187712 T^{6} - 26237955211264 T^{7} + 4622346883170304 T^{8} - 495395959010754560 T^{9} + 36893488147419103232 T^{10} \))
$3$	(\( 1 + 12 T + 1594323 T^{2} \))(\( 1 - 872 T + 179766 T^{2} - 1390249656 T^{3} + 2541865828329 T^{4} \))(\( 1 + 2069910 T^{2} + 2541865828329 T^{4} \))(\( 1 - 8978642 T^{2} + 41923886773365 T^{4} - \)\(13\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!22\)\( T^{8} - \)\(56\!\cdots\!92\)\( T^{10} + \)\(78\!\cdots\!38\)\( T^{12} - \)\(85\!\cdots\!12\)\( T^{14} + \)\(68\!\cdots\!85\)\( T^{16} - \)\(37\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))
$5$	(\( 1 + 4330 T + 1220703125 T^{2} \))(\( 1 - 18476 T + 2066094350 T^{2} - 22553710937500 T^{3} + 1490116119384765625 T^{4} \))(\( 1 - 1931729850 T^{2} + 1490116119384765625 T^{4} \))(\( 1 - 4565314338 T^{2} + 10147469641592691461 T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!50\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!25\)\( T^{16} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(73\!\cdots\!25\)\( T^{20} \))
$7$	(\( 1 + 139992 T + 96889010407 T^{2} \))(\( 1 - 110928 T + 39195942926 T^{2} - 10747704146427696 T^{3} + \)\(93\!\cdots\!49\)\( T^{4} \))(\( ( 1 + 175832 T + 96889010407 T^{2} )^{2} \))(\( ( 1 - 293480 T + 239610643203 T^{2} - 95417056996483936 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!28\)\( T^{5} + \)\(31\!\cdots\!18\)\( T^{6} - \)\(89\!\cdots\!64\)\( T^{7} + \)\(21\!\cdots\!29\)\( T^{8} - \)\(25\!\cdots\!80\)\( T^{9} + \)\(85\!\cdots\!07\)\( T^{10} )^{2} \))
$11$	(\( 1 + 6484324 T + 34522712143931 T^{2} \))(\( 1 - 16474040 T + 135991936296326 T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - 62100824999962 T^{2} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - 179340783809858 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!78\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{10} + \)\(60\!\cdots\!58\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!21\)\( T^{16} - \)\(36\!\cdots\!78\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))
$13$	(\( 1 + 22588034 T + 302875106592253 T^{2} \))(\( 1 - 18744572 T + 344120051417598 T^{2} - \)\(56\!\cdots\!16\)\( T^{3} + \)\(91\!\cdots\!09\)\( T^{4} \))(\( 1 + 360392330876150 T^{2} + \)\(91\!\cdots\!09\)\( T^{4} \))(\( 1 - 1881700477395890 T^{2} + \)\(17\!\cdots\!13\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(48\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!18\)\( T^{12} - \)\(91\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!77\)\( T^{16} - \)\(13\!\cdots\!90\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} \))
$17$	(\( 1 + 23732270 T + 9904578032905937 T^{2} \))(\( 1 + 153793628 T + 21058111940247014 T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(98\!\cdots\!69\)\( T^{4} \))(\( ( 1 + 133520302 T + 9904578032905937 T^{2} )^{2} \))(\( ( 1 - 108663002 T + 32276400633367229 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!42\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!54\)\( T^{6} - \)\(23\!\cdots\!96\)\( T^{7} + \)\(31\!\cdots\!37\)\( T^{8} - \)\(10\!\cdots\!22\)\( T^{9} + \)\(95\!\cdots\!57\)\( T^{10} )^{2} \))
$19$	(\( 1 - 325344836 T + 42052983462257059 T^{2} \))(\( 1 + 118747640 T + 59473815443581782 T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 82896428399326282 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 162780863064290354 T^{2} + \)\(13\!\cdots\!97\)\( T^{4} - \)\(80\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!66\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(69\!\cdots\!46\)\( T^{12} - \)\(25\!\cdots\!96\)\( T^{14} + \)\(75\!\cdots\!77\)\( T^{16} - \)\(15\!\cdots\!34\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \))
$23$	(\( 1 - 921600632 T + 504036361936467383 T^{2} \))(\( 1 - 718268912 T + 1097306915486653358 T^{2} - \)\(36\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!89\)\( T^{4} \))(\( ( 1 + 35585416 T + 504036361936467383 T^{2} )^{2} \))(\( ( 1 + 39339976 T + 1242751119340586771 T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!02\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(38\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(15\!\cdots\!77\)\( T^{8} + \)\(25\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!43\)\( T^{10} )^{2} \))
$29$	(\( 1 + 3865879218 T + 10260628712958602189 T^{2} \))(\( 1 - 309341340 T + 2504177667132675934 T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 18002148940349036842 T^{2} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 60047671171855484498 T^{2} + \)\(18\!\cdots\!41\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(58\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!84\)\( T^{10} + \)\(61\!\cdots\!38\)\( T^{12} - \)\(42\!\cdots\!60\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} - \)\(73\!\cdots\!38\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$31$	(\( 1 + 2253401440 T + 24417546297445042591 T^{2} \))(\( 1 - 5767504192 T + 50116253247327696702 T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(59\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2} \))(\( ( 1 - 324116896 T + 94759416869201467419 T^{2} - \)\(76\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(98\!\cdots\!78\)\( T^{6} - \)\(45\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(86\!\cdots\!51\)\( T^{10} )^{2} \))
$37$	(\( 1 - 18250384566 T + \)\(24\!\cdots\!97\)\( T^{2} \))(\( 1 + 11621553300 T + \)\(18\!\cdots\!38\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(31\!\cdots\!70\)\( T^{2} + \)\(59\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(85\!\cdots\!14\)\( T^{2} + \)\(35\!\cdots\!29\)\( T^{4} - \)\(83\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(67\!\cdots\!02\)\( T^{12} - \)\(29\!\cdots\!44\)\( T^{14} + \)\(73\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!54\)\( T^{18} + \)\(73\!\cdots\!49\)\( T^{20} \))
$41$	(\( 1 - 34422845322 T + \)\(92\!\cdots\!21\)\( T^{2} \))(\( 1 - 1311168276 T + \)\(16\!\cdots\!30\)\( T^{2} - \)\(12\!\cdots\!96\)\( T^{3} + \)\(85\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 23546348918 T + \)\(92\!\cdots\!21\)\( T^{2} )^{2} \))(\( ( 1 - 29662320178 T + \)\(14\!\cdots\!89\)\( T^{2} - \)\(15\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(97\!\cdots\!02\)\( T^{6} - \)\(13\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!29\)\( T^{8} - \)\(21\!\cdots\!18\)\( T^{9} + \)\(67\!\cdots\!01\)\( T^{10} )^{2} \))
$43$	(\( 1 + 17192501444 T + \)\(17\!\cdots\!43\)\( T^{2} \))(\( 1 + 29595620104 T + \)\(33\!\cdots\!74\)\( T^{2} + \)\(50\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(32\!\cdots\!30\)\( T^{2} + \)\(29\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(61\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!05\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{6} + \)\(52\!\cdots\!62\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!04\)\( T^{14} + \)\(39\!\cdots\!45\)\( T^{16} - \)\(46\!\cdots\!14\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))
$47$	(\( 1 + 67371749904 T + \)\(54\!\cdots\!27\)\( T^{2} \))(\( 1 - 12313617888 T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(67\!\cdots\!76\)\( T^{3} + \)\(29\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 68107736592 T + \)\(54\!\cdots\!27\)\( T^{2} )^{2} \))(\( ( 1 + 5088267408 T + \)\(10\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(79\!\cdots\!70\)\( T^{4} + \)\(90\!\cdots\!04\)\( T^{5} + \)\(43\!\cdots\!90\)\( T^{6} + \)\(48\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!69\)\( T^{8} + \)\(45\!\cdots\!28\)\( T^{9} + \)\(48\!\cdots\!07\)\( T^{10} )^{2} \))
$53$	(\( 1 + 87281218426 T + \)\(26\!\cdots\!73\)\( T^{2} \))(\( 1 + 38006007028 T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(98\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(15\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!49\)\( T^{4} - \)\(63\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!20\)\( T^{10} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{14} + \)\(37\!\cdots\!61\)\( T^{16} - \)\(32\!\cdots\!38\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} \))
$59$	(\( 1 - 540214518668 T + \)\(10\!\cdots\!79\)\( T^{2} \))(\( 1 - 253345911704 T + \)\(16\!\cdots\!06\)\( T^{2} - \)\(26\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!22\)\( T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(54\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!77\)\( T^{4} - \)\(32\!\cdots\!52\)\( T^{6} + \)\(49\!\cdots\!86\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} + \)\(54\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{14} + \)\(21\!\cdots\!17\)\( T^{16} - \)\(80\!\cdots\!02\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \))
$61$	(\( 1 + 51276568850 T + \)\(16\!\cdots\!81\)\( T^{2} \))(\( 1 + 647244384292 T + \)\(42\!\cdots\!62\)\( T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(14\!\cdots\!62\)\( T^{2} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(10\!\cdots\!62\)\( T^{2} + \)\(47\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{6} + \)\(29\!\cdots\!66\)\( T^{8} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(77\!\cdots\!26\)\( T^{12} - \)\(96\!\cdots\!52\)\( T^{14} + \)\(86\!\cdots\!97\)\( T^{16} - \)\(48\!\cdots\!42\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$67$	(\( 1 - 25519930676 T + \)\(54\!\cdots\!87\)\( T^{2} \))(\( 1 - 1619993806312 T + \)\(17\!\cdots\!86\)\( T^{2} - \)\(88\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(95\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!42\)\( T^{2} + \)\(23\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(71\!\cdots\!60\)\( T^{10} + \)\(37\!\cdots\!34\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!41\)\( T^{16} - \)\(16\!\cdots\!82\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} \))
$71$	(\( 1 + 1387500699032 T + \)\(11\!\cdots\!11\)\( T^{2} \))(\( 1 + 1040270142512 T + \)\(22\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 1309471657368 T + \)\(11\!\cdots\!11\)\( T^{2} )^{2} \))(\( ( 1 - 363180589992 T + \)\(34\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(61\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!34\)\( T^{6} - \)\(20\!\cdots\!36\)\( T^{7} + \)\(53\!\cdots\!45\)\( T^{8} - \)\(66\!\cdots\!72\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} )^{2} \))
$73$	(\( 1 + 819049441238 T + \)\(16\!\cdots\!33\)\( T^{2} \))(\( 1 - 4005283908692 T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))(\( ( 1 - 478647871914 T + \)\(16\!\cdots\!33\)\( T^{2} )^{2} \))(\( ( 1 + 316620182766 T + \)\(64\!\cdots\!53\)\( T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(60\!\cdots\!52\)\( T^{5} + \)\(31\!\cdots\!82\)\( T^{6} + \)\(67\!\cdots\!76\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} + \)\(24\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!93\)\( T^{10} )^{2} \))
$79$	(\( 1 + 4030935615344 T + \)\(46\!\cdots\!39\)\( T^{2} \))(\( 1 + 2521777572064 T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 364547231600 T + \)\(46\!\cdots\!39\)\( T^{2} )^{2} \))(\( ( 1 - 2722551782672 T + \)\(12\!\cdots\!19\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(55\!\cdots\!14\)\( T^{4} - \)\(94\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!46\)\( T^{6} - \)\(49\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} )^{2} \))
$83$	(\( 1 - 4180823831428 T + \)\(88\!\cdots\!63\)\( T^{2} \))(\( 1 + 290486230904 T + \)\(14\!\cdots\!54\)\( T^{2} + \)\(25\!\cdots\!52\)\( T^{3} + \)\(78\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!70\)\( T^{2} + \)\(78\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(43\!\cdots\!74\)\( T^{2} + \)\(87\!\cdots\!65\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(76\!\cdots\!04\)\( T^{14} + \)\(42\!\cdots\!85\)\( T^{16} - \)\(16\!\cdots\!54\)\( T^{18} + \)\(30\!\cdots\!49\)\( T^{20} \))
$89$	(\( 1 - 2677027798266 T + \)\(21\!\cdots\!69\)\( T^{2} \))(\( 1 + 8723755657740 T + \)\(61\!\cdots\!42\)\( T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(48\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 102457641350 T + \)\(21\!\cdots\!69\)\( T^{2} )^{2} \))(\( ( 1 - 2753172404002 T + \)\(62\!\cdots\!29\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!48\)\( T^{5} + \)\(47\!\cdots\!26\)\( T^{6} - \)\(70\!\cdots\!24\)\( T^{7} + \)\(66\!\cdots\!61\)\( T^{8} - \)\(64\!\cdots\!42\)\( T^{9} + \)\(51\!\cdots\!49\)\( T^{10} )^{2} \))
$97$	(\( 1 + 14039464316446 T + \)\(67\!\cdots\!77\)\( T^{2} \))(\( 1 - 9601712299972 T + \)\(81\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(45\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 6157717373342 T + \)\(67\!\cdots\!77\)\( T^{2} )^{2} \))(\( ( 1 - 680566660394 T + \)\(17\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!90\)\( T^{6} + \)\(53\!\cdots\!32\)\( T^{7} + \)\(54\!\cdots\!49\)\( T^{8} - \)\(13\!\cdots\!54\)\( T^{9} + \)\(13\!\cdots\!57\)\( T^{10} )^{2} \))
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