# Properties

 Label 160.6.d Level 160 Weight 6 Character orbit d Rep. character $$\chi_{160}(81,\cdot)$$ Character field $$\Q$$ Dimension 20 Newform subspaces 1 Sturm bound 144 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$144$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(160, [\chi])$$.

Total New Old
Modular forms 128 20 108
Cusp forms 112 20 92
Eisenstein series 16 0 16

## Trace form

 $$20q + 196q^{7} - 1620q^{9} + O(q^{10})$$ $$20q + 196q^{7} - 1620q^{9} - 900q^{15} + 4676q^{23} - 12500q^{25} - 7160q^{31} + 5672q^{33} + 44904q^{39} + 11608q^{41} - 44180q^{47} + 18756q^{49} + 24200q^{55} + 5032q^{57} - 240620q^{63} + 200312q^{71} - 105136q^{73} - 282080q^{79} + 65172q^{81} + 332592q^{87} - 3160q^{89} - 144400q^{95} + 147376q^{97} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(160, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
160.6.d.a $$20$$ $$25.661$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$196$$ $$q-\beta _{1}q^{3}+\beta _{2}q^{5}+(10+\beta _{4})q^{7}+(-3^{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(160, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(160, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 3}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 1620 T^{2} + 1394322 T^{4} - 834927892 T^{6} + 392823148221 T^{8} - 155295909553872 T^{10} + 53814329151823576 T^{12} - 16772396183204761872 T^{14} +$$$$47\!\cdots\!30$$$$T^{16} -$$$$12\!\cdots\!00$$$$T^{18} +$$$$31\!\cdots\!00$$$$T^{20} -$$$$75\!\cdots\!00$$$$T^{22} +$$$$16\!\cdots\!30$$$$T^{24} -$$$$34\!\cdots\!28$$$$T^{26} +$$$$65\!\cdots\!76$$$$T^{28} -$$$$11\!\cdots\!28$$$$T^{30} +$$$$16\!\cdots\!21$$$$T^{32} -$$$$20\!\cdots\!08$$$$T^{34} +$$$$20\!\cdots\!22$$$$T^{36} -$$$$14\!\cdots\!80$$$$T^{38} +$$$$51\!\cdots\!01$$$$T^{40}$$
$5$ $$( 1 + 625 T^{2} )^{10}$$
$7$ $$( 1 - 98 T + 84148 T^{2} - 8017214 T^{3} + 3556570901 T^{4} - 345044190776 T^{5} + 103920201897616 T^{6} - 10159390994080936 T^{7} + 2363994840482709802 T^{8} -$$$$22\!\cdots\!76$$$$T^{9} +$$$$43\!\cdots\!72$$$$T^{10} -$$$$37\!\cdots\!32$$$$T^{11} +$$$$66\!\cdots\!98$$$$T^{12} -$$$$48\!\cdots\!48$$$$T^{13} +$$$$82\!\cdots\!16$$$$T^{14} -$$$$46\!\cdots\!32$$$$T^{15} +$$$$80\!\cdots\!49$$$$T^{16} -$$$$30\!\cdots\!02$$$$T^{17} +$$$$53\!\cdots\!48$$$$T^{18} -$$$$10\!\cdots\!86$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20} )^{2}$$
$11$ $$1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} +$$$$22\!\cdots\!17$$$$T^{8} -$$$$70\!\cdots\!04$$$$T^{10} +$$$$18\!\cdots\!36$$$$T^{12} -$$$$40\!\cdots\!64$$$$T^{14} +$$$$79\!\cdots\!82$$$$T^{16} -$$$$14\!\cdots\!04$$$$T^{18} +$$$$23\!\cdots\!76$$$$T^{20} -$$$$36\!\cdots\!04$$$$T^{22} +$$$$53\!\cdots\!82$$$$T^{24} -$$$$70\!\cdots\!64$$$$T^{26} +$$$$82\!\cdots\!36$$$$T^{28} -$$$$82\!\cdots\!04$$$$T^{30} +$$$$69\!\cdots\!17$$$$T^{32} -$$$$46\!\cdots\!24$$$$T^{34} +$$$$23\!\cdots\!26$$$$T^{36} -$$$$80\!\cdots\!04$$$$T^{38} +$$$$13\!\cdots\!01$$$$T^{40}$$
$13$ $$1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} +$$$$67\!\cdots\!57$$$$T^{8} -$$$$50\!\cdots\!12$$$$T^{10} +$$$$32\!\cdots\!64$$$$T^{12} -$$$$17\!\cdots\!28$$$$T^{14} +$$$$87\!\cdots\!42$$$$T^{16} -$$$$38\!\cdots\!32$$$$T^{18} +$$$$14\!\cdots\!64$$$$T^{20} -$$$$52\!\cdots\!68$$$$T^{22} +$$$$16\!\cdots\!42$$$$T^{24} -$$$$46\!\cdots\!72$$$$T^{26} +$$$$11\!\cdots\!64$$$$T^{28} -$$$$25\!\cdots\!88$$$$T^{30} +$$$$46\!\cdots\!57$$$$T^{32} -$$$$70\!\cdots\!52$$$$T^{34} +$$$$81\!\cdots\!54$$$$T^{36} -$$$$63\!\cdots\!68$$$$T^{38} +$$$$24\!\cdots\!01$$$$T^{40}$$
$17$ $$( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} +$$$$14\!\cdots\!00$$$$T^{7} +$$$$56\!\cdots\!10$$$$T^{8} +$$$$21\!\cdots\!00$$$$T^{9} +$$$$84\!\cdots\!72$$$$T^{10} +$$$$31\!\cdots\!00$$$$T^{11} +$$$$11\!\cdots\!90$$$$T^{12} +$$$$41\!\cdots\!00$$$$T^{13} +$$$$13\!\cdots\!52$$$$T^{14} +$$$$42\!\cdots\!00$$$$T^{15} +$$$$12\!\cdots\!05$$$$T^{16} +$$$$21\!\cdots\!00$$$$T^{17} +$$$$89\!\cdots\!62$$$$T^{18} +$$$$33\!\cdots\!49$$$$T^{20} )^{2}$$
$19$ $$1 - 23638692 T^{2} + 296336418074734 T^{4} -$$$$25\!\cdots\!32$$$$T^{6} +$$$$17\!\cdots\!97$$$$T^{8} -$$$$93\!\cdots\!80$$$$T^{10} +$$$$42\!\cdots\!52$$$$T^{12} -$$$$16\!\cdots\!48$$$$T^{14} +$$$$56\!\cdots\!66$$$$T^{16} -$$$$16\!\cdots\!12$$$$T^{18} +$$$$44\!\cdots\!28$$$$T^{20} -$$$$10\!\cdots\!12$$$$T^{22} +$$$$21\!\cdots\!66$$$$T^{24} -$$$$38\!\cdots\!48$$$$T^{26} +$$$$60\!\cdots\!52$$$$T^{28} -$$$$80\!\cdots\!80$$$$T^{30} +$$$$91\!\cdots\!97$$$$T^{32} -$$$$83\!\cdots\!32$$$$T^{34} +$$$$59\!\cdots\!34$$$$T^{36} -$$$$28\!\cdots\!92$$$$T^{38} +$$$$75\!\cdots\!01$$$$T^{40}$$
$23$ $$( 1 - 2338 T + 35997660 T^{2} - 45007042654 T^{3} + 520972471777845 T^{4} - 50426407997609208 T^{5} +$$$$39\!\cdots\!60$$$$T^{6} +$$$$66\!\cdots\!96$$$$T^{7} +$$$$17\!\cdots\!10$$$$T^{8} +$$$$91\!\cdots\!52$$$$T^{9} +$$$$76\!\cdots\!60$$$$T^{10} +$$$$58\!\cdots\!36$$$$T^{11} +$$$$74\!\cdots\!90$$$$T^{12} +$$$$17\!\cdots\!72$$$$T^{13} +$$$$68\!\cdots\!60$$$$T^{14} -$$$$55\!\cdots\!44$$$$T^{15} +$$$$37\!\cdots\!05$$$$T^{16} -$$$$20\!\cdots\!78$$$$T^{17} +$$$$10\!\cdots\!60$$$$T^{18} -$$$$44\!\cdots\!34$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20} )^{2}$$
$29$ $$1 - 215092900 T^{2} + 23243889276296494 T^{4} -$$$$16\!\cdots\!00$$$$T^{6} +$$$$90\!\cdots\!13$$$$T^{8} -$$$$38\!\cdots\!00$$$$T^{10} +$$$$13\!\cdots\!92$$$$T^{12} -$$$$42\!\cdots\!00$$$$T^{14} +$$$$11\!\cdots\!86$$$$T^{16} -$$$$27\!\cdots\!00$$$$T^{18} +$$$$58\!\cdots\!28$$$$T^{20} -$$$$11\!\cdots\!00$$$$T^{22} +$$$$20\!\cdots\!86$$$$T^{24} -$$$$31\!\cdots\!00$$$$T^{26} +$$$$43\!\cdots\!92$$$$T^{28} -$$$$51\!\cdots\!00$$$$T^{30} +$$$$50\!\cdots\!13$$$$T^{32} -$$$$39\!\cdots\!00$$$$T^{34} +$$$$22\!\cdots\!94$$$$T^{36} -$$$$88\!\cdots\!00$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$( 1 + 3580 T + 132421614 T^{2} + 484936307876 T^{3} + 8983138546835629 T^{4} + 33755686429634218608 T^{5} +$$$$43\!\cdots\!88$$$$T^{6} +$$$$16\!\cdots\!96$$$$T^{7} +$$$$17\!\cdots\!38$$$$T^{8} +$$$$60\!\cdots\!32$$$$T^{9} +$$$$54\!\cdots\!44$$$$T^{10} +$$$$17\!\cdots\!32$$$$T^{11} +$$$$13\!\cdots\!38$$$$T^{12} +$$$$38\!\cdots\!96$$$$T^{13} +$$$$29\!\cdots\!88$$$$T^{14} +$$$$64\!\cdots\!08$$$$T^{15} +$$$$49\!\cdots\!29$$$$T^{16} +$$$$76\!\cdots\!76$$$$T^{17} +$$$$59\!\cdots\!14$$$$T^{18} +$$$$46\!\cdots\!80$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$
$37$ $$1 - 565372668 T^{2} + 171401952644913934 T^{4} -$$$$36\!\cdots\!00$$$$T^{6} +$$$$59\!\cdots\!09$$$$T^{8} -$$$$80\!\cdots\!44$$$$T^{10} +$$$$93\!\cdots\!16$$$$T^{12} -$$$$93\!\cdots\!40$$$$T^{14} +$$$$83\!\cdots\!86$$$$T^{16} -$$$$67\!\cdots\!48$$$$T^{18} +$$$$48\!\cdots\!08$$$$T^{20} -$$$$32\!\cdots\!52$$$$T^{22} +$$$$19\!\cdots\!86$$$$T^{24} -$$$$10\!\cdots\!60$$$$T^{26} +$$$$49\!\cdots\!16$$$$T^{28} -$$$$20\!\cdots\!56$$$$T^{30} +$$$$73\!\cdots\!09$$$$T^{32} -$$$$21\!\cdots\!00$$$$T^{34} +$$$$48\!\cdots\!34$$$$T^{36} -$$$$77\!\cdots\!32$$$$T^{38} +$$$$66\!\cdots\!01$$$$T^{40}$$
$41$ $$( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} -$$$$13\!\cdots\!48$$$$T^{5} +$$$$39\!\cdots\!40$$$$T^{6} -$$$$28\!\cdots\!68$$$$T^{7} +$$$$63\!\cdots\!42$$$$T^{8} -$$$$43\!\cdots\!24$$$$T^{9} +$$$$82\!\cdots\!24$$$$T^{10} -$$$$49\!\cdots\!24$$$$T^{11} +$$$$85\!\cdots\!42$$$$T^{12} -$$$$44\!\cdots\!68$$$$T^{13} +$$$$70\!\cdots\!40$$$$T^{14} -$$$$28\!\cdots\!48$$$$T^{15} +$$$$43\!\cdots\!93$$$$T^{16} -$$$$11\!\cdots\!28$$$$T^{17} +$$$$18\!\cdots\!34$$$$T^{18} -$$$$21\!\cdots\!04$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20} )^{2}$$
$43$ $$1 - 1208126740 T^{2} + 787740581508660018 T^{4} -$$$$36\!\cdots\!96$$$$T^{6} +$$$$12\!\cdots\!73$$$$T^{8} -$$$$37\!\cdots\!80$$$$T^{10} +$$$$95\!\cdots\!40$$$$T^{12} -$$$$20\!\cdots\!56$$$$T^{14} +$$$$40\!\cdots\!70$$$$T^{16} -$$$$70\!\cdots\!00$$$$T^{18} +$$$$10\!\cdots\!96$$$$T^{20} -$$$$15\!\cdots\!00$$$$T^{22} +$$$$18\!\cdots\!70$$$$T^{24} -$$$$21\!\cdots\!44$$$$T^{26} +$$$$20\!\cdots\!40$$$$T^{28} -$$$$17\!\cdots\!20$$$$T^{30} +$$$$13\!\cdots\!73$$$$T^{32} -$$$$79\!\cdots\!04$$$$T^{34} +$$$$37\!\cdots\!18$$$$T^{36} -$$$$12\!\cdots\!60$$$$T^{38} +$$$$22\!\cdots\!01$$$$T^{40}$$
$47$ $$( 1 + 22090 T + 1548510076 T^{2} + 25779450599270 T^{3} + 1065218725316011845 T^{4} +$$$$14\!\cdots\!20$$$$T^{5} +$$$$45\!\cdots\!96$$$$T^{6} +$$$$51\!\cdots\!60$$$$T^{7} +$$$$14\!\cdots\!10$$$$T^{8} +$$$$14\!\cdots\!00$$$$T^{9} +$$$$36\!\cdots\!56$$$$T^{10} +$$$$32\!\cdots\!00$$$$T^{11} +$$$$76\!\cdots\!90$$$$T^{12} +$$$$62\!\cdots\!80$$$$T^{13} +$$$$12\!\cdots\!96$$$$T^{14} +$$$$90\!\cdots\!40$$$$T^{15} +$$$$15\!\cdots\!05$$$$T^{16} +$$$$86\!\cdots\!10$$$$T^{17} +$$$$11\!\cdots\!76$$$$T^{18} +$$$$38\!\cdots\!30$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20} )^{2}$$
$53$ $$1 - 4003669356 T^{2} + 7954725909905649454 T^{4} -$$$$10\!\cdots\!60$$$$T^{6} +$$$$10\!\cdots\!77$$$$T^{8} -$$$$91\!\cdots\!76$$$$T^{10} +$$$$65\!\cdots\!16$$$$T^{12} -$$$$40\!\cdots\!80$$$$T^{14} +$$$$22\!\cdots\!54$$$$T^{16} -$$$$11\!\cdots\!56$$$$T^{18} +$$$$48\!\cdots\!96$$$$T^{20} -$$$$19\!\cdots\!44$$$$T^{22} +$$$$68\!\cdots\!54$$$$T^{24} -$$$$21\!\cdots\!20$$$$T^{26} +$$$$61\!\cdots\!16$$$$T^{28} -$$$$14\!\cdots\!24$$$$T^{30} +$$$$31\!\cdots\!77$$$$T^{32} -$$$$53\!\cdots\!40$$$$T^{34} +$$$$69\!\cdots\!54$$$$T^{36} -$$$$61\!\cdots\!44$$$$T^{38} +$$$$26\!\cdots\!01$$$$T^{40}$$
$59$ $$1 - 9996949828 T^{2} + 49650800837889885966 T^{4} -$$$$16\!\cdots\!20$$$$T^{6} +$$$$39\!\cdots\!17$$$$T^{8} -$$$$73\!\cdots\!28$$$$T^{10} +$$$$11\!\cdots\!44$$$$T^{12} -$$$$14\!\cdots\!80$$$$T^{14} +$$$$15\!\cdots\!54$$$$T^{16} -$$$$13\!\cdots\!88$$$$T^{18} +$$$$10\!\cdots\!24$$$$T^{20} -$$$$70\!\cdots\!88$$$$T^{22} +$$$$39\!\cdots\!54$$$$T^{24} -$$$$19\!\cdots\!80$$$$T^{26} +$$$$77\!\cdots\!44$$$$T^{28} -$$$$25\!\cdots\!28$$$$T^{30} +$$$$69\!\cdots\!17$$$$T^{32} -$$$$14\!\cdots\!20$$$$T^{34} +$$$$23\!\cdots\!66$$$$T^{36} -$$$$23\!\cdots\!28$$$$T^{38} +$$$$12\!\cdots\!01$$$$T^{40}$$
$61$ $$1 - 7152852348 T^{2} + 26523669582205677166 T^{4} -$$$$67\!\cdots\!00$$$$T^{6} +$$$$13\!\cdots\!17$$$$T^{8} -$$$$22\!\cdots\!48$$$$T^{10} +$$$$31\!\cdots\!84$$$$T^{12} -$$$$38\!\cdots\!60$$$$T^{14} +$$$$42\!\cdots\!14$$$$T^{16} -$$$$41\!\cdots\!08$$$$T^{18} +$$$$36\!\cdots\!64$$$$T^{20} -$$$$29\!\cdots\!08$$$$T^{22} +$$$$21\!\cdots\!14$$$$T^{24} -$$$$13\!\cdots\!60$$$$T^{26} +$$$$81\!\cdots\!84$$$$T^{28} -$$$$41\!\cdots\!48$$$$T^{30} +$$$$17\!\cdots\!17$$$$T^{32} -$$$$63\!\cdots\!00$$$$T^{34} +$$$$17\!\cdots\!66$$$$T^{36} -$$$$34\!\cdots\!48$$$$T^{38} +$$$$34\!\cdots\!01$$$$T^{40}$$
$67$ $$1 - 11828518964 T^{2} + 68669252417166303634 T^{4} -$$$$26\!\cdots\!60$$$$T^{6} +$$$$76\!\cdots\!57$$$$T^{8} -$$$$18\!\cdots\!04$$$$T^{10} +$$$$38\!\cdots\!16$$$$T^{12} -$$$$70\!\cdots\!40$$$$T^{14} +$$$$11\!\cdots\!54$$$$T^{16} -$$$$18\!\cdots\!64$$$$T^{18} +$$$$25\!\cdots\!76$$$$T^{20} -$$$$33\!\cdots\!36$$$$T^{22} +$$$$39\!\cdots\!54$$$$T^{24} -$$$$42\!\cdots\!60$$$$T^{26} +$$$$42\!\cdots\!16$$$$T^{28} -$$$$36\!\cdots\!96$$$$T^{30} +$$$$28\!\cdots\!57$$$$T^{32} -$$$$17\!\cdots\!40$$$$T^{34} +$$$$83\!\cdots\!34$$$$T^{36} -$$$$26\!\cdots\!36$$$$T^{38} +$$$$40\!\cdots\!01$$$$T^{40}$$
$71$ $$( 1 - 100156 T + 13448448446 T^{2} - 957445823975748 T^{3} + 76873855601451932317 T^{4} -$$$$43\!\cdots\!20$$$$T^{5} +$$$$26\!\cdots\!28$$$$T^{6} -$$$$12\!\cdots\!92$$$$T^{7} +$$$$67\!\cdots\!74$$$$T^{8} -$$$$28\!\cdots\!84$$$$T^{9} +$$$$13\!\cdots\!68$$$$T^{10} -$$$$51\!\cdots\!84$$$$T^{11} +$$$$21\!\cdots\!74$$$$T^{12} -$$$$74\!\cdots\!92$$$$T^{13} +$$$$28\!\cdots\!28$$$$T^{14} -$$$$82\!\cdots\!20$$$$T^{15} +$$$$26\!\cdots\!17$$$$T^{16} -$$$$59\!\cdots\!48$$$$T^{17} +$$$$15\!\cdots\!46$$$$T^{18} -$$$$20\!\cdots\!56$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$
$73$ $$( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} +$$$$11\!\cdots\!56$$$$T^{5} +$$$$82\!\cdots\!04$$$$T^{6} +$$$$19\!\cdots\!04$$$$T^{7} +$$$$11\!\cdots\!58$$$$T^{8} +$$$$24\!\cdots\!96$$$$T^{9} +$$$$16\!\cdots\!04$$$$T^{10} +$$$$51\!\cdots\!28$$$$T^{11} +$$$$49\!\cdots\!42$$$$T^{12} +$$$$17\!\cdots\!28$$$$T^{13} +$$$$15\!\cdots\!04$$$$T^{14} +$$$$44\!\cdots\!08$$$$T^{15} +$$$$29\!\cdots\!41$$$$T^{16} +$$$$61\!\cdots\!52$$$$T^{17} +$$$$31\!\cdots\!94$$$$T^{18} +$$$$37\!\cdots\!24$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20} )^{2}$$
$79$ $$( 1 + 141040 T + 30508439526 T^{2} + 3027236690742416 T^{3} +$$$$38\!\cdots\!93$$$$T^{4} +$$$$29\!\cdots\!88$$$$T^{5} +$$$$27\!\cdots\!48$$$$T^{6} +$$$$17\!\cdots\!16$$$$T^{7} +$$$$13\!\cdots\!30$$$$T^{8} +$$$$73\!\cdots\!40$$$$T^{9} +$$$$47\!\cdots\!04$$$$T^{10} +$$$$22\!\cdots\!60$$$$T^{11} +$$$$12\!\cdots\!30$$$$T^{12} +$$$$51\!\cdots\!84$$$$T^{13} +$$$$24\!\cdots\!48$$$$T^{14} +$$$$81\!\cdots\!12$$$$T^{15} +$$$$32\!\cdots\!93$$$$T^{16} +$$$$79\!\cdots\!84$$$$T^{17} +$$$$24\!\cdots\!26$$$$T^{18} +$$$$34\!\cdots\!60$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20} )^{2}$$
$83$ $$1 - 34768653380 T^{2} +$$$$55\!\cdots\!58$$$$T^{4} -$$$$53\!\cdots\!56$$$$T^{6} +$$$$34\!\cdots\!33$$$$T^{8} -$$$$15\!\cdots\!20$$$$T^{10} +$$$$44\!\cdots\!40$$$$T^{12} -$$$$24\!\cdots\!36$$$$T^{14} -$$$$64\!\cdots\!10$$$$T^{16} +$$$$49\!\cdots\!20$$$$T^{18} -$$$$23\!\cdots\!44$$$$T^{20} +$$$$76\!\cdots\!80$$$$T^{22} -$$$$15\!\cdots\!10$$$$T^{24} -$$$$90\!\cdots\!64$$$$T^{26} +$$$$25\!\cdots\!40$$$$T^{28} -$$$$14\!\cdots\!80$$$$T^{30} +$$$$48\!\cdots\!33$$$$T^{32} -$$$$11\!\cdots\!44$$$$T^{34} +$$$$18\!\cdots\!58$$$$T^{36} -$$$$18\!\cdots\!20$$$$T^{38} +$$$$80\!\cdots\!01$$$$T^{40}$$
$89$ $$( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} +$$$$38\!\cdots\!93$$$$T^{4} +$$$$90\!\cdots\!72$$$$T^{5} +$$$$38\!\cdots\!08$$$$T^{6} +$$$$95\!\cdots\!24$$$$T^{7} +$$$$29\!\cdots\!70$$$$T^{8} +$$$$73\!\cdots\!60$$$$T^{9} +$$$$18\!\cdots\!64$$$$T^{10} +$$$$40\!\cdots\!40$$$$T^{11} +$$$$91\!\cdots\!70$$$$T^{12} +$$$$16\!\cdots\!76$$$$T^{13} +$$$$37\!\cdots\!08$$$$T^{14} +$$$$49\!\cdots\!28$$$$T^{15} +$$$$11\!\cdots\!93$$$$T^{16} +$$$$77\!\cdots\!16$$$$T^{17} +$$$$25\!\cdots\!46$$$$T^{18} +$$$$83\!\cdots\!20$$$$T^{19} +$$$$29\!\cdots\!01$$$$T^{20} )^{2}$$
$97$ $$( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} +$$$$98\!\cdots\!25$$$$T^{4} -$$$$59\!\cdots\!28$$$$T^{5} +$$$$15\!\cdots\!52$$$$T^{6} -$$$$87\!\cdots\!16$$$$T^{7} +$$$$18\!\cdots\!70$$$$T^{8} -$$$$96\!\cdots\!88$$$$T^{9} +$$$$17\!\cdots\!72$$$$T^{10} -$$$$82\!\cdots\!16$$$$T^{11} +$$$$13\!\cdots\!30$$$$T^{12} -$$$$55\!\cdots\!88$$$$T^{13} +$$$$84\!\cdots\!52$$$$T^{14} -$$$$27\!\cdots\!96$$$$T^{15} +$$$$39\!\cdots\!25$$$$T^{16} -$$$$97\!\cdots\!08$$$$T^{17} +$$$$12\!\cdots\!62$$$$T^{18} -$$$$18\!\cdots\!16$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20} )^{2}$$