# Properties

 Label 8.18.b Level 8 Weight 18 Character orbit b Rep. character $$\chi_{8}(5,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 1 Sturm bound 18 Trace bound 0

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## Defining parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$18$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$18$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{18}(8, [\chi])$$.

Total New Old
Modular forms 18 18 0
Cusp forms 16 16 0
Eisenstein series 2 2 0

## Trace form

 $$16q + 270q^{2} - 27436q^{4} + 5839948q^{6} + 11529600q^{7} + 24334920q^{8} - 602654096q^{9} + O(q^{10})$$ $$16q + 270q^{2} - 27436q^{4} + 5839948q^{6} + 11529600q^{7} + 24334920q^{8} - 602654096q^{9} + 131002712q^{10} - 2795125400q^{12} + 16363788528q^{14} - 9993282176q^{15} + 26500434192q^{16} - 7489125600q^{17} - 113450563870q^{18} - 209445719856q^{20} + 223126527100q^{22} + 746845345920q^{23} - 1099415493232q^{24} - 1809682431664q^{25} + 2467726531080q^{26} + 3220542267040q^{28} - 1188624268048q^{30} - 318979758592q^{31} + 1455647316000q^{32} + 5633526177600q^{33} - 4461251980292q^{34} - 33088278002484q^{36} + 24076283913900q^{38} - 18457706051456q^{39} + 60626292962592q^{40} + 7482251536032q^{41} - 51630378688160q^{42} + 193654716236040q^{44} - 195097141003568q^{46} - 376698804821760q^{47} - 329350060416480q^{48} + 127691292101520q^{49} + 474997408872102q^{50} - 272251877663120q^{52} + 735354219382520q^{54} + 2209036687713152q^{55} - 162767516076480q^{56} - 190521298294720q^{57} - 623262610679960q^{58} - 1973616194963808q^{60} + 695695648144320q^{62} - 8131096607338880q^{63} + 1111931745501248q^{64} + 2385987975356160q^{65} + 3598826202828312q^{66} + 5981109959771880q^{68} - 10044559836180288q^{70} + 9025926285576576q^{71} - 19918679666289160q^{72} + 11332002046118560q^{73} + 11098735408189464q^{74} + 5959440926938280q^{76} + 4184252259031760q^{78} - 45299671392008448q^{79} + 1337342539452480q^{80} + 20101901999290832q^{81} + 15639739637081420q^{82} + 19796542864700224q^{84} - 14252032276026564q^{86} + 25965768920837760q^{87} - 66964872768837680q^{88} - 69879174608766048q^{89} + 136151511125051240q^{90} + 57336249810701280q^{92} - 192318922166254176q^{94} + 93790444358203776q^{95} - 342799224184788928q^{96} + 95593398602180640q^{97} + 339641261743253790q^{98} + O(q^{100})$$

## Decomposition of $$S_{18}^{\mathrm{new}}(8, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
8.18.b.a $$16$$ $$14.658$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$270$$ $$0$$ $$0$$ $$11529600$$ $$q+(17-\beta _{1})q^{2}+(3\beta _{1}-\beta _{2})q^{3}+(-1712+\cdots)q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 270 T + 50168 T^{2} - 15096000 T^{3} - 3619419136 T^{4} + 1008340992000 T^{5} - 342095875276800 T^{6} + 981059695269642240 T^{7} -$$$$42\!\cdots\!80$$$$T^{8} +$$$$12\!\cdots\!80$$$$T^{9} -$$$$58\!\cdots\!00$$$$T^{10} +$$$$22\!\cdots\!00$$$$T^{11} -$$$$10\!\cdots\!16$$$$T^{12} -$$$$58\!\cdots\!00$$$$T^{13} +$$$$25\!\cdots\!72$$$$T^{14} -$$$$17\!\cdots\!60$$$$T^{15} +$$$$87\!\cdots\!36$$$$T^{16}$$
$3$ $$1 - 731794256 T^{2} + 282192653104988472 T^{4} -$$$$75\!\cdots\!72$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{8} -$$$$24\!\cdots\!24$$$$T^{10} +$$$$32\!\cdots\!00$$$$T^{12} -$$$$39\!\cdots\!56$$$$T^{14} +$$$$49\!\cdots\!54$$$$T^{16} -$$$$66\!\cdots\!64$$$$T^{18} +$$$$91\!\cdots\!00$$$$T^{20} -$$$$11\!\cdots\!16$$$$T^{22} +$$$$11\!\cdots\!00$$$$T^{24} -$$$$97\!\cdots\!28$$$$T^{26} +$$$$60\!\cdots\!32$$$$T^{28} -$$$$26\!\cdots\!84$$$$T^{30} +$$$$59\!\cdots\!41$$$$T^{32}$$
$5$ $$1 - 5198674409168 T^{2} +$$$$13\!\cdots\!56$$$$T^{4} -$$$$24\!\cdots\!00$$$$T^{6} +$$$$33\!\cdots\!00$$$$T^{8} -$$$$37\!\cdots\!00$$$$T^{10} +$$$$36\!\cdots\!00$$$$T^{12} -$$$$31\!\cdots\!00$$$$T^{14} +$$$$25\!\cdots\!50$$$$T^{16} -$$$$18\!\cdots\!00$$$$T^{18} +$$$$12\!\cdots\!00$$$$T^{20} -$$$$73\!\cdots\!00$$$$T^{22} +$$$$38\!\cdots\!00$$$$T^{24} -$$$$16\!\cdots\!00$$$$T^{26} +$$$$53\!\cdots\!00$$$$T^{28} -$$$$11\!\cdots\!00$$$$T^{30} +$$$$13\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 - 5764800 T + 915215692443448 T^{2} -$$$$65\!\cdots\!20$$$$T^{3} +$$$$50\!\cdots\!24$$$$T^{4} -$$$$34\!\cdots\!80$$$$T^{5} +$$$$18\!\cdots\!60$$$$T^{6} -$$$$12\!\cdots\!60$$$$T^{7} +$$$$50\!\cdots\!90$$$$T^{8} -$$$$27\!\cdots\!20$$$$T^{9} +$$$$10\!\cdots\!40$$$$T^{10} -$$$$44\!\cdots\!40$$$$T^{11} +$$$$14\!\cdots\!24$$$$T^{12} -$$$$44\!\cdots\!40$$$$T^{13} +$$$$14\!\cdots\!52$$$$T^{14} -$$$$21\!\cdots\!00$$$$T^{15} +$$$$85\!\cdots\!01$$$$T^{16} )^{2}$$
$11$ $$1 - 3877608573076448976 T^{2} +$$$$79\!\cdots\!84$$$$T^{4} -$$$$11\!\cdots\!04$$$$T^{6} +$$$$12\!\cdots\!88$$$$T^{8} -$$$$11\!\cdots\!72$$$$T^{10} +$$$$81\!\cdots\!96$$$$T^{12} -$$$$51\!\cdots\!28$$$$T^{14} +$$$$28\!\cdots\!22$$$$T^{16} -$$$$13\!\cdots\!48$$$$T^{18} +$$$$53\!\cdots\!76$$$$T^{20} -$$$$18\!\cdots\!12$$$$T^{22} +$$$$52\!\cdots\!68$$$$T^{24} -$$$$12\!\cdots\!04$$$$T^{26} +$$$$22\!\cdots\!44$$$$T^{28} -$$$$27\!\cdots\!56$$$$T^{30} +$$$$18\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 63371249746529137488 T^{2} +$$$$20\!\cdots\!16$$$$T^{4} -$$$$48\!\cdots\!88$$$$T^{6} +$$$$85\!\cdots\!68$$$$T^{8} -$$$$12\!\cdots\!16$$$$T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$15\!\cdots\!62$$$$T^{16} -$$$$12\!\cdots\!84$$$$T^{18} +$$$$85\!\cdots\!44$$$$T^{20} -$$$$52\!\cdots\!04$$$$T^{22} +$$$$26\!\cdots\!88$$$$T^{24} -$$$$11\!\cdots\!12$$$$T^{26} +$$$$36\!\cdots\!76$$$$T^{28} -$$$$83\!\cdots\!52$$$$T^{30} +$$$$98\!\cdots\!81$$$$T^{32}$$
$17$ $$( 1 + 3744562800 T +$$$$34\!\cdots\!44$$$$T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!60$$$$T^{4} -$$$$36\!\cdots\!00$$$$T^{5} +$$$$81\!\cdots\!48$$$$T^{6} -$$$$51\!\cdots\!00$$$$T^{7} +$$$$77\!\cdots\!98$$$$T^{8} -$$$$42\!\cdots\!00$$$$T^{9} +$$$$55\!\cdots\!92$$$$T^{10} -$$$$20\!\cdots\!00$$$$T^{11} +$$$$29\!\cdots\!60$$$$T^{12} -$$$$39\!\cdots\!00$$$$T^{13} +$$$$11\!\cdots\!16$$$$T^{14} +$$$$99\!\cdots\!00$$$$T^{15} +$$$$21\!\cdots\!81$$$$T^{16} )^{2}$$
$19$ $$1 -$$$$50\!\cdots\!16$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$21\!\cdots\!48$$$$T^{6} +$$$$27\!\cdots\!00$$$$T^{8} -$$$$27\!\cdots\!68$$$$T^{10} +$$$$22\!\cdots\!64$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{14} +$$$$94\!\cdots\!82$$$$T^{16} -$$$$47\!\cdots\!24$$$$T^{18} +$$$$20\!\cdots\!24$$$$T^{20} -$$$$74\!\cdots\!48$$$$T^{22} +$$$$22\!\cdots\!00$$$$T^{24} -$$$$52\!\cdots\!48$$$$T^{26} +$$$$94\!\cdots\!60$$$$T^{28} -$$$$11\!\cdots\!56$$$$T^{30} +$$$$66\!\cdots\!61$$$$T^{32}$$
$23$ $$( 1 - 373422672960 T +$$$$64\!\cdots\!76$$$$T^{2} -$$$$17\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!52$$$$T^{4} -$$$$41\!\cdots\!20$$$$T^{5} +$$$$40\!\cdots\!92$$$$T^{6} -$$$$69\!\cdots\!60$$$$T^{7} +$$$$63\!\cdots\!74$$$$T^{8} -$$$$98\!\cdots\!80$$$$T^{9} +$$$$79\!\cdots\!28$$$$T^{10} -$$$$11\!\cdots\!40$$$$T^{11} +$$$$77\!\cdots\!12$$$$T^{12} -$$$$96\!\cdots\!40$$$$T^{13} +$$$$50\!\cdots\!04$$$$T^{14} -$$$$41\!\cdots\!20$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 -$$$$62\!\cdots\!20$$$$T^{2} +$$$$20\!\cdots\!80$$$$T^{4} -$$$$44\!\cdots\!20$$$$T^{6} +$$$$75\!\cdots\!16$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{10} +$$$$11\!\cdots\!20$$$$T^{12} -$$$$10\!\cdots\!80$$$$T^{14} +$$$$81\!\cdots\!06$$$$T^{16} -$$$$54\!\cdots\!80$$$$T^{18} +$$$$30\!\cdots\!20$$$$T^{20} -$$$$14\!\cdots\!80$$$$T^{22} +$$$$57\!\cdots\!36$$$$T^{24} -$$$$18\!\cdots\!20$$$$T^{26} +$$$$43\!\cdots\!80$$$$T^{28} -$$$$70\!\cdots\!20$$$$T^{30} +$$$$59\!\cdots\!41$$$$T^{32}$$
$31$ $$( 1 + 159489879296 T +$$$$10\!\cdots\!04$$$$T^{2} +$$$$16\!\cdots\!28$$$$T^{3} +$$$$51\!\cdots\!12$$$$T^{4} +$$$$11\!\cdots\!72$$$$T^{5} +$$$$18\!\cdots\!80$$$$T^{6} +$$$$39\!\cdots\!28$$$$T^{7} +$$$$49\!\cdots\!98$$$$T^{8} +$$$$88\!\cdots\!08$$$$T^{9} +$$$$93\!\cdots\!80$$$$T^{10} +$$$$13\!\cdots\!32$$$$T^{11} +$$$$13\!\cdots\!92$$$$T^{12} +$$$$94\!\cdots\!28$$$$T^{13} +$$$$13\!\cdots\!44$$$$T^{14} +$$$$47\!\cdots\!16$$$$T^{15} +$$$$66\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$1 -$$$$39\!\cdots\!44$$$$T^{2} +$$$$80\!\cdots\!76$$$$T^{4} -$$$$10\!\cdots\!72$$$$T^{6} +$$$$11\!\cdots\!64$$$$T^{8} -$$$$90\!\cdots\!04$$$$T^{10} +$$$$60\!\cdots\!24$$$$T^{12} -$$$$34\!\cdots\!60$$$$T^{14} +$$$$17\!\cdots\!90$$$$T^{16} -$$$$72\!\cdots\!40$$$$T^{18} +$$$$26\!\cdots\!04$$$$T^{20} -$$$$81\!\cdots\!76$$$$T^{22} +$$$$20\!\cdots\!24$$$$T^{24} -$$$$42\!\cdots\!28$$$$T^{26} +$$$$65\!\cdots\!36$$$$T^{28} -$$$$67\!\cdots\!76$$$$T^{30} +$$$$35\!\cdots\!81$$$$T^{32}$$
$41$ $$( 1 - 3741125768016 T +$$$$70\!\cdots\!80$$$$T^{2} -$$$$14\!\cdots\!84$$$$T^{3} +$$$$29\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$99\!\cdots\!52$$$$T^{6} +$$$$61\!\cdots\!84$$$$T^{7} +$$$$26\!\cdots\!42$$$$T^{8} +$$$$15\!\cdots\!04$$$$T^{9} +$$$$67\!\cdots\!72$$$$T^{10} +$$$$25\!\cdots\!92$$$$T^{11} +$$$$13\!\cdots\!00$$$$T^{12} -$$$$17\!\cdots\!84$$$$T^{13} +$$$$22\!\cdots\!80$$$$T^{14} -$$$$31\!\cdots\!76$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$1 -$$$$41\!\cdots\!32$$$$T^{2} +$$$$84\!\cdots\!12$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$11\!\cdots\!40$$$$T^{8} -$$$$97\!\cdots\!68$$$$T^{10} +$$$$74\!\cdots\!32$$$$T^{12} -$$$$50\!\cdots\!72$$$$T^{14} +$$$$31\!\cdots\!30$$$$T^{16} -$$$$17\!\cdots\!28$$$$T^{18} +$$$$88\!\cdots\!32$$$$T^{20} -$$$$40\!\cdots\!32$$$$T^{22} +$$$$16\!\cdots\!40$$$$T^{24} -$$$$54\!\cdots\!72$$$$T^{26} +$$$$14\!\cdots\!12$$$$T^{28} -$$$$24\!\cdots\!68$$$$T^{30} +$$$$20\!\cdots\!01$$$$T^{32}$$
$47$ $$( 1 + 188349402410880 T +$$$$19\!\cdots\!72$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!52$$$$T^{4} +$$$$23\!\cdots\!00$$$$T^{5} +$$$$87\!\cdots\!84$$$$T^{6} +$$$$99\!\cdots\!60$$$$T^{7} +$$$$28\!\cdots\!54$$$$T^{8} +$$$$26\!\cdots\!20$$$$T^{9} +$$$$62\!\cdots\!96$$$$T^{10} +$$$$43\!\cdots\!00$$$$T^{11} +$$$$85\!\cdots\!72$$$$T^{12} +$$$$42\!\cdots\!00$$$$T^{13} +$$$$69\!\cdots\!48$$$$T^{14} +$$$$17\!\cdots\!40$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$1 -$$$$12\!\cdots\!88$$$$T^{2} +$$$$92\!\cdots\!64$$$$T^{4} -$$$$47\!\cdots\!52$$$$T^{6} +$$$$18\!\cdots\!96$$$$T^{8} -$$$$62\!\cdots\!40$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{14} +$$$$94\!\cdots\!02$$$$T^{16} -$$$$18\!\cdots\!16$$$$T^{18} +$$$$31\!\cdots\!88$$$$T^{20} -$$$$46\!\cdots\!60$$$$T^{22} +$$$$59\!\cdots\!16$$$$T^{24} -$$$$63\!\cdots\!48$$$$T^{26} +$$$$52\!\cdots\!84$$$$T^{28} -$$$$30\!\cdots\!32$$$$T^{30} +$$$$10\!\cdots\!41$$$$T^{32}$$
$59$ $$1 -$$$$11\!\cdots\!84$$$$T^{2} +$$$$69\!\cdots\!76$$$$T^{4} -$$$$25\!\cdots\!84$$$$T^{6} +$$$$69\!\cdots\!16$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{10} +$$$$24\!\cdots\!32$$$$T^{12} -$$$$35\!\cdots\!68$$$$T^{14} +$$$$46\!\cdots\!62$$$$T^{16} -$$$$56\!\cdots\!48$$$$T^{18} +$$$$63\!\cdots\!72$$$$T^{20} -$$$$60\!\cdots\!00$$$$T^{22} +$$$$47\!\cdots\!56$$$$T^{24} -$$$$28\!\cdots\!84$$$$T^{26} +$$$$12\!\cdots\!36$$$$T^{28} -$$$$34\!\cdots\!64$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 -$$$$16\!\cdots\!44$$$$T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$64\!\cdots\!84$$$$T^{6} +$$$$24\!\cdots\!36$$$$T^{8} -$$$$82\!\cdots\!80$$$$T^{10} +$$$$24\!\cdots\!72$$$$T^{12} -$$$$64\!\cdots\!28$$$$T^{14} +$$$$15\!\cdots\!02$$$$T^{16} -$$$$32\!\cdots\!48$$$$T^{18} +$$$$61\!\cdots\!32$$$$T^{20} -$$$$10\!\cdots\!80$$$$T^{22} +$$$$15\!\cdots\!96$$$$T^{24} -$$$$20\!\cdots\!84$$$$T^{26} +$$$$20\!\cdots\!96$$$$T^{28} -$$$$13\!\cdots\!64$$$$T^{30} +$$$$40\!\cdots\!21$$$$T^{32}$$
$67$ $$1 -$$$$82\!\cdots\!12$$$$T^{2} +$$$$35\!\cdots\!44$$$$T^{4} -$$$$10\!\cdots\!88$$$$T^{6} +$$$$24\!\cdots\!96$$$$T^{8} -$$$$47\!\cdots\!60$$$$T^{10} +$$$$75\!\cdots\!48$$$$T^{12} -$$$$10\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!82$$$$T^{16} -$$$$12\!\cdots\!04$$$$T^{18} +$$$$11\!\cdots\!68$$$$T^{20} -$$$$85\!\cdots\!40$$$$T^{22} +$$$$55\!\cdots\!76$$$$T^{24} -$$$$29\!\cdots\!12$$$$T^{26} +$$$$11\!\cdots\!24$$$$T^{28} -$$$$33\!\cdots\!08$$$$T^{30} +$$$$49\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 4512963142788288 T +$$$$90\!\cdots\!40$$$$T^{2} -$$$$19\!\cdots\!68$$$$T^{3} +$$$$31\!\cdots\!80$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$78\!\cdots\!64$$$$T^{6} +$$$$17\!\cdots\!64$$$$T^{7} +$$$$22\!\cdots\!90$$$$T^{8} +$$$$51\!\cdots\!24$$$$T^{9} +$$$$68\!\cdots\!84$$$$T^{10} +$$$$36\!\cdots\!52$$$$T^{11} +$$$$24\!\cdots\!80$$$$T^{12} -$$$$43\!\cdots\!68$$$$T^{13} +$$$$60\!\cdots\!40$$$$T^{14} -$$$$89\!\cdots\!28$$$$T^{15} +$$$$59\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 5666001023059280 T +$$$$17\!\cdots\!28$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!64$$$$T^{4} -$$$$66\!\cdots\!60$$$$T^{5} +$$$$61\!\cdots\!08$$$$T^{6} -$$$$24\!\cdots\!40$$$$T^{7} +$$$$27\!\cdots\!70$$$$T^{8} -$$$$11\!\cdots\!20$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} -$$$$71\!\cdots\!20$$$$T^{11} +$$$$65\!\cdots\!84$$$$T^{12} -$$$$24\!\cdots\!20$$$$T^{13} +$$$$19\!\cdots\!12$$$$T^{14} -$$$$30\!\cdots\!60$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 + 22649835696004224 T +$$$$78\!\cdots\!40$$$$T^{2} +$$$$10\!\cdots\!12$$$$T^{3} +$$$$22\!\cdots\!48$$$$T^{4} +$$$$15\!\cdots\!08$$$$T^{5} +$$$$30\!\cdots\!08$$$$T^{6} -$$$$78\!\cdots\!52$$$$T^{7} +$$$$34\!\cdots\!66$$$$T^{8} -$$$$14\!\cdots\!68$$$$T^{9} +$$$$10\!\cdots\!48$$$$T^{10} +$$$$94\!\cdots\!32$$$$T^{11} +$$$$24\!\cdots\!28$$$$T^{12} +$$$$20\!\cdots\!88$$$$T^{13} +$$$$28\!\cdots\!40$$$$T^{14} +$$$$14\!\cdots\!56$$$$T^{15} +$$$$11\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$1 -$$$$26\!\cdots\!92$$$$T^{2} +$$$$34\!\cdots\!32$$$$T^{4} -$$$$29\!\cdots\!68$$$$T^{6} +$$$$21\!\cdots\!00$$$$T^{8} -$$$$13\!\cdots\!28$$$$T^{10} +$$$$71\!\cdots\!12$$$$T^{12} -$$$$34\!\cdots\!12$$$$T^{14} +$$$$15\!\cdots\!30$$$$T^{16} -$$$$61\!\cdots\!48$$$$T^{18} +$$$$22\!\cdots\!92$$$$T^{20} -$$$$73\!\cdots\!92$$$$T^{22} +$$$$21\!\cdots\!00$$$$T^{24} -$$$$52\!\cdots\!32$$$$T^{26} +$$$$10\!\cdots\!72$$$$T^{28} -$$$$14\!\cdots\!28$$$$T^{30} +$$$$97\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 + 34939587304383024 T +$$$$39\!\cdots\!00$$$$T^{2} +$$$$75\!\cdots\!32$$$$T^{3} +$$$$62\!\cdots\!48$$$$T^{4} +$$$$60\!\cdots\!48$$$$T^{5} +$$$$64\!\cdots\!08$$$$T^{6} +$$$$76\!\cdots\!28$$$$T^{7} +$$$$64\!\cdots\!66$$$$T^{8} +$$$$10\!\cdots\!12$$$$T^{9} +$$$$12\!\cdots\!28$$$$T^{10} +$$$$15\!\cdots\!72$$$$T^{11} +$$$$22\!\cdots\!88$$$$T^{12} +$$$$37\!\cdots\!68$$$$T^{13} +$$$$27\!\cdots\!00$$$$T^{14} +$$$$33\!\cdots\!16$$$$T^{15} +$$$$13\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 47796699301090320 T +$$$$34\!\cdots\!40$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$54\!\cdots\!44$$$$T^{4} -$$$$32\!\cdots\!40$$$$T^{5} +$$$$55\!\cdots\!60$$$$T^{6} -$$$$31\!\cdots\!00$$$$T^{7} +$$$$39\!\cdots\!50$$$$T^{8} -$$$$18\!\cdots\!00$$$$T^{9} +$$$$19\!\cdots\!40$$$$T^{10} -$$$$68\!\cdots\!20$$$$T^{11} +$$$$68\!\cdots\!84$$$$T^{12} -$$$$14\!\cdots\!40$$$$T^{13} +$$$$15\!\cdots\!60$$$$T^{14} -$$$$12\!\cdots\!60$$$$T^{15} +$$$$15\!\cdots\!21$$$$T^{16} )^{2}$$
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