# Properties

 Label 108.6.b Level 108 Weight 6 Character orbit b Rep. character $$\chi_{108}(107,\cdot)$$ Character field $$\Q$$ Dimension 40 Newform subspaces 3 Sturm bound 108 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$108$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(108, [\chi])$$.

Total New Old
Modular forms 96 40 56
Cusp forms 84 40 44
Eisenstein series 12 0 12

## Trace form

 $$40q + 10q^{4} + O(q^{10})$$ $$40q + 10q^{4} - 334q^{10} + 116q^{13} - 2678q^{16} + 4746q^{22} - 26364q^{25} - 5142q^{28} - 4828q^{34} - 16132q^{37} + 24326q^{40} - 768q^{46} - 127184q^{49} + 40376q^{52} + 4316q^{58} - 50260q^{61} + 120982q^{64} + 267522q^{70} - 75568q^{73} + 88020q^{76} + 214328q^{82} + 112744q^{85} - 174402q^{88} + 56820q^{94} - 275752q^{97} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(108, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
108.6.b.a $$4$$ $$17.321$$ $$\Q(\sqrt{3}, \sqrt{-29})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}-\beta _{2})q^{2}+(-26-2\beta _{3})q^{4}+\cdots$$
108.6.b.b $$16$$ $$17.321$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(6-\beta _{4})q^{4}+(-3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$
108.6.b.c $$20$$ $$17.321$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}+(1+\beta _{2})q^{4}+\beta _{15}q^{5}-\beta _{5}q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(108, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(108, [\chi]) \cong$$ $$S_{6}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 52 T^{2} + 1024 T^{4}$$)($$1 - 47 T^{2} + 1060 T^{4} - 30080 T^{6} + 762880 T^{8} - 30801920 T^{10} + 1111490560 T^{12} - 50465865728 T^{14} + 1099511627776 T^{16}$$)($$1 - 10 T^{2} + 1092 T^{4} - 9504 T^{6} - 1286400 T^{8} - 761856 T^{10} - 1317273600 T^{12} - 9965666304 T^{14} + 1172526071808 T^{16} - 10995116277760 T^{18} + 1125899906842624 T^{20}$$)
$3$ 1
$5$ ($$( 1 + 2131 T^{2} + 9765625 T^{4} )^{2}$$)($$( 1 - 14972 T^{2} + 119342578 T^{4} - 617133972224 T^{6} + 2271820128564475 T^{8} - 6026698947500000000 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$13\!\cdots\!00$$$$T^{14} +$$$$90\!\cdots\!25$$$$T^{16} )^{2}$$)($$( 1 - 11818 T^{2} + 71630853 T^{4} - 313823973048 T^{6} + 1185932779192818 T^{8} - 3986228637264265212 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$29\!\cdots\!00$$$$T^{14} +$$$$66\!\cdots\!25$$$$T^{16} -$$$$10\!\cdots\!50$$$$T^{18} +$$$$88\!\cdots\!25$$$$T^{20} )^{2}$$)
$7$ ($$( 1 - 8471 T^{2} + 282475249 T^{4} )^{2}$$)($$( 1 - 54884 T^{2} + 1251149938 T^{4} - 15826547698400 T^{6} + 189674874534565723 T^{8} -$$$$44\!\cdots\!00$$$$T^{10} +$$$$99\!\cdots\!38$$$$T^{12} -$$$$12\!\cdots\!16$$$$T^{14} +$$$$63\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 72919 T^{2} + 2924383791 T^{4} - 81340986295962 T^{6} + 1774418094261308301 T^{8} -$$$$32\!\cdots\!57$$$$T^{10} +$$$$50\!\cdots\!49$$$$T^{12} -$$$$64\!\cdots\!62$$$$T^{14} +$$$$65\!\cdots\!59$$$$T^{16} -$$$$46\!\cdots\!19$$$$T^{18} +$$$$17\!\cdots\!49$$$$T^{20} )^{2}$$)
$11$ ($$( 1 - 14573 T^{2} + 25937424601 T^{4} )^{2}$$)($$( 1 + 741148 T^{2} + 299964078250 T^{4} + 79656648391789648 T^{6} +$$$$15\!\cdots\!99$$$$T^{8} +$$$$20\!\cdots\!48$$$$T^{10} +$$$$20\!\cdots\!50$$$$T^{12} +$$$$12\!\cdots\!48$$$$T^{14} +$$$$45\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 822230 T^{2} + 360495679845 T^{4} + 108431998645925448 T^{6} +$$$$24\!\cdots\!70$$$$T^{8} +$$$$44\!\cdots\!92$$$$T^{10} +$$$$64\!\cdots\!70$$$$T^{12} +$$$$72\!\cdots\!48$$$$T^{14} +$$$$62\!\cdots\!45$$$$T^{16} +$$$$37\!\cdots\!30$$$$T^{18} +$$$$11\!\cdots\!01$$$$T^{20} )^{2}$$)
$13$ ($$( 1 + 166 T + 371293 T^{2} )^{4}$$)($$( 1 - 224 T + 554236 T^{2} + 77263648 T^{3} + 200668790230 T^{4} + 28687451656864 T^{5} + 76406139088422364 T^{6} - 11465640035156329568 T^{7} +$$$$19\!\cdots\!01$$$$T^{8} )^{4}$$)($$( 1 + 29 T + 791115 T^{2} + 39777006 T^{3} + 380980388709 T^{4} + 35518566649227 T^{5} + 141455351464930737 T^{6} + 5483598057428624094 T^{7} +$$$$40\!\cdots\!55$$$$T^{8} +$$$$55\!\cdots\!29$$$$T^{9} +$$$$70\!\cdots\!93$$$$T^{10} )^{4}$$)
$17$ ($$( 1 - 2151950 T^{2} + 2015993900449 T^{4} )^{2}$$)($$( 1 - 4946936 T^{2} + 15515554849948 T^{4} - 34385249721915417800 T^{6} +$$$$55\!\cdots\!90$$$$T^{8} -$$$$69\!\cdots\!00$$$$T^{10} +$$$$63\!\cdots\!48$$$$T^{12} -$$$$40\!\cdots\!64$$$$T^{14} +$$$$16\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 6202690 T^{2} + 18689060069517 T^{4} - 34705007722821565080 T^{6} +$$$$46\!\cdots\!02$$$$T^{8} -$$$$60\!\cdots\!16$$$$T^{10} +$$$$94\!\cdots\!98$$$$T^{12} -$$$$14\!\cdots\!80$$$$T^{14} +$$$$15\!\cdots\!33$$$$T^{16} -$$$$10\!\cdots\!90$$$$T^{18} +$$$$33\!\cdots\!49$$$$T^{20} )^{2}$$)
$19$ ($$( 1 - 4501190 T^{2} + 6131066257801 T^{4} )^{2}$$)($$( 1 - 9713816 T^{2} + 55349997016060 T^{4} -$$$$21\!\cdots\!72$$$$T^{6} +$$$$61\!\cdots\!02$$$$T^{8} -$$$$13\!\cdots\!72$$$$T^{10} +$$$$20\!\cdots\!60$$$$T^{12} -$$$$22\!\cdots\!16$$$$T^{14} +$$$$14\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 13782439 T^{2} + 98755621335207 T^{4} -$$$$47\!\cdots\!14$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8} -$$$$48\!\cdots\!01$$$$T^{10} +$$$$10\!\cdots\!01$$$$T^{12} -$$$$17\!\cdots\!14$$$$T^{14} +$$$$22\!\cdots\!07$$$$T^{16} -$$$$19\!\cdots\!39$$$$T^{18} +$$$$86\!\cdots\!01$$$$T^{20} )^{2}$$)
$23$ ($$( 1 - 2019266 T^{2} + 41426511213649 T^{4} )^{2}$$)($$( 1 + 29994040 T^{2} + 487786705761244 T^{4} +$$$$51\!\cdots\!44$$$$T^{6} +$$$$39\!\cdots\!14$$$$T^{8} +$$$$21\!\cdots\!56$$$$T^{10} +$$$$83\!\cdots\!44$$$$T^{12} +$$$$21\!\cdots\!60$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 42721646 T^{2} + 918206452664445 T^{4} +$$$$12\!\cdots\!28$$$$T^{6} +$$$$12\!\cdots\!34$$$$T^{8} +$$$$95\!\cdots\!68$$$$T^{10} +$$$$53\!\cdots\!66$$$$T^{12} +$$$$22\!\cdots\!28$$$$T^{14} +$$$$65\!\cdots\!05$$$$T^{16} +$$$$12\!\cdots\!46$$$$T^{18} +$$$$12\!\cdots\!49$$$$T^{20} )^{2}$$)
$29$ ($$( 1 - 29365574 T^{2} + 420707233300201 T^{4} )^{2}$$)($$( 1 - 26575640 T^{2} + 859790443267708 T^{4} -$$$$25\!\cdots\!88$$$$T^{6} +$$$$44\!\cdots\!26$$$$T^{8} -$$$$10\!\cdots\!88$$$$T^{10} +$$$$15\!\cdots\!08$$$$T^{12} -$$$$19\!\cdots\!40$$$$T^{14} +$$$$31\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 142578466 T^{2} + 9766751584325109 T^{4} -$$$$42\!\cdots\!00$$$$T^{6} +$$$$13\!\cdots\!22$$$$T^{8} -$$$$31\!\cdots\!16$$$$T^{10} +$$$$56\!\cdots\!22$$$$T^{12} -$$$$75\!\cdots\!00$$$$T^{14} +$$$$72\!\cdots\!09$$$$T^{16} -$$$$44\!\cdots\!66$$$$T^{18} +$$$$13\!\cdots\!01$$$$T^{20} )^{2}$$)
$31$ ($$( 1 - 17147735 T^{2} + 819628286980801 T^{4} )^{2}$$)($$( 1 - 160216820 T^{2} + 12412163878108546 T^{4} -$$$$60\!\cdots\!08$$$$T^{6} +$$$$20\!\cdots\!87$$$$T^{8} -$$$$49\!\cdots\!08$$$$T^{10} +$$$$83\!\cdots\!46$$$$T^{12} -$$$$88\!\cdots\!20$$$$T^{14} +$$$$45\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 53059450 T^{2} + 1830387109253997 T^{4} -$$$$54\!\cdots\!08$$$$T^{6} +$$$$17\!\cdots\!98$$$$T^{8} -$$$$63\!\cdots\!76$$$$T^{10} +$$$$14\!\cdots\!98$$$$T^{12} -$$$$36\!\cdots\!08$$$$T^{14} +$$$$10\!\cdots\!97$$$$T^{16} -$$$$23\!\cdots\!50$$$$T^{18} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$)
$37$ ($$( 1 - 15332 T + 69343957 T^{2} )^{4}$$)($$( 1 + 17752 T + 254119924 T^{2} + 2137611581224 T^{3} + 19827261455867542 T^{4} +$$$$14\!\cdots\!68$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$59\!\cdots\!36$$$$T^{7} +$$$$23\!\cdots\!01$$$$T^{8} )^{4}$$)($$( 1 + 1613 T + 123026979 T^{2} + 343711884654 T^{3} + 5803261980890325 T^{4} + 31670451235762477275 T^{5} +$$$$40\!\cdots\!25$$$$T^{6} +$$$$16\!\cdots\!46$$$$T^{7} +$$$$41\!\cdots\!47$$$$T^{8} +$$$$37\!\cdots\!13$$$$T^{9} +$$$$16\!\cdots\!57$$$$T^{10} )^{4}$$)
$41$ ($$( 1 - 129650498 T^{2} + 13422659310152401 T^{4} )^{2}$$)($$( 1 - 667999304 T^{2} + 215857510496501980 T^{4} -$$$$43\!\cdots\!84$$$$T^{6} +$$$$60\!\cdots\!14$$$$T^{8} -$$$$58\!\cdots\!84$$$$T^{10} +$$$$38\!\cdots\!80$$$$T^{12} -$$$$16\!\cdots\!04$$$$T^{14} +$$$$32\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 204662074 T^{2} + 6421654936070589 T^{4} +$$$$14\!\cdots\!28$$$$T^{6} -$$$$52\!\cdots\!06$$$$T^{8} -$$$$96\!\cdots\!76$$$$T^{10} -$$$$70\!\cdots\!06$$$$T^{12} +$$$$26\!\cdots\!28$$$$T^{14} +$$$$15\!\cdots\!89$$$$T^{16} -$$$$66\!\cdots\!74$$$$T^{18} +$$$$43\!\cdots\!01$$$$T^{20} )^{2}$$)
$43$ ($$( 1 - 284996378 T^{2} + 21611482313284249 T^{4} )^{2}$$)($$( 1 - 271892936 T^{2} + 81822516500494204 T^{4} -$$$$14\!\cdots\!72$$$$T^{6} +$$$$26\!\cdots\!02$$$$T^{8} -$$$$30\!\cdots\!28$$$$T^{10} +$$$$38\!\cdots\!04$$$$T^{12} -$$$$27\!\cdots\!64$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 1038109906 T^{2} + 527969585590823397 T^{4} -$$$$17\!\cdots\!40$$$$T^{6} +$$$$39\!\cdots\!02$$$$T^{8} -$$$$67\!\cdots\!12$$$$T^{10} +$$$$85\!\cdots\!98$$$$T^{12} -$$$$80\!\cdots\!40$$$$T^{14} +$$$$53\!\cdots\!53$$$$T^{16} -$$$$22\!\cdots\!06$$$$T^{18} +$$$$47\!\cdots\!49$$$$T^{20} )^{2}$$)
$47$ ($$( 1 + 408701842 T^{2} + 52599132235830049 T^{4} )^{2}$$)($$( 1 + 1268760328 T^{2} + 775645167851419804 T^{4} +$$$$30\!\cdots\!92$$$$T^{6} +$$$$81\!\cdots\!58$$$$T^{8} +$$$$15\!\cdots\!08$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{12} +$$$$18\!\cdots\!72$$$$T^{14} +$$$$76\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 920931614 T^{2} + 499070896097327373 T^{4} +$$$$18\!\cdots\!24$$$$T^{6} +$$$$53\!\cdots\!58$$$$T^{8} +$$$$12\!\cdots\!24$$$$T^{10} +$$$$27\!\cdots\!42$$$$T^{12} +$$$$50\!\cdots\!24$$$$T^{14} +$$$$72\!\cdots\!77$$$$T^{16} +$$$$70\!\cdots\!14$$$$T^{18} +$$$$40\!\cdots\!49$$$$T^{20} )^{2}$$)
$53$ ($$( 1 - 537758237 T^{2} + 174887470365513049 T^{4} )^{2}$$)($$( 1 - 1665548972 T^{2} + 1494189343182800962 T^{4} -$$$$95\!\cdots\!20$$$$T^{6} +$$$$46\!\cdots\!75$$$$T^{8} -$$$$16\!\cdots\!80$$$$T^{10} +$$$$45\!\cdots\!62$$$$T^{12} -$$$$89\!\cdots\!28$$$$T^{14} +$$$$93\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 1301597650 T^{2} + 1197727639041193893 T^{4} -$$$$82\!\cdots\!04$$$$T^{6} +$$$$45\!\cdots\!18$$$$T^{8} -$$$$21\!\cdots\!32$$$$T^{10} +$$$$79\!\cdots\!82$$$$T^{12} -$$$$25\!\cdots\!04$$$$T^{14} +$$$$64\!\cdots\!57$$$$T^{16} -$$$$12\!\cdots\!50$$$$T^{18} +$$$$16\!\cdots\!49$$$$T^{20} )^{2}$$)
$59$ ($$( 1 + 621590410 T^{2} + 511116753300641401 T^{4} )^{2}$$)($$( 1 + 4822767208 T^{2} + 10641078380871180220 T^{4} +$$$$14\!\cdots\!92$$$$T^{6} +$$$$12\!\cdots\!22$$$$T^{8} +$$$$71\!\cdots\!92$$$$T^{10} +$$$$27\!\cdots\!20$$$$T^{12} +$$$$64\!\cdots\!08$$$$T^{14} +$$$$68\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 1806503990 T^{2} + 2218805640623321733 T^{4} +$$$$21\!\cdots\!92$$$$T^{6} +$$$$17\!\cdots\!70$$$$T^{8} +$$$$12\!\cdots\!12$$$$T^{10} +$$$$87\!\cdots\!70$$$$T^{12} +$$$$55\!\cdots\!92$$$$T^{14} +$$$$29\!\cdots\!33$$$$T^{16} +$$$$12\!\cdots\!90$$$$T^{18} +$$$$34\!\cdots\!01$$$$T^{20} )^{2}$$)
$61$ ($$( 1 + 53188 T + 844596301 T^{2} )^{4}$$)($$( 1 - 19472 T + 1261489684 T^{2} - 43949943879536 T^{3} + 833752337244254902 T^{4} -$$$$37\!\cdots\!36$$$$T^{5} +$$$$89\!\cdots\!84$$$$T^{6} -$$$$11\!\cdots\!72$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8} )^{4}$$)($$( 1 - 21151 T + 2606330283 T^{2} - 19051394797770 T^{3} + 2604033901923348357 T^{4} -$$$$93\!\cdots\!13$$$$T^{5} +$$$$21\!\cdots\!57$$$$T^{6} -$$$$13\!\cdots\!70$$$$T^{7} +$$$$15\!\cdots\!83$$$$T^{8} -$$$$10\!\cdots\!51$$$$T^{9} +$$$$42\!\cdots\!01$$$$T^{10} )^{4}$$)
$67$ ($$( 1 - 1014398666 T^{2} + 1822837804551761449 T^{4} )^{2}$$)($$( 1 - 1447449416 T^{2} + 2915553445416739516 T^{4} -$$$$52\!\cdots\!68$$$$T^{6} +$$$$74\!\cdots\!58$$$$T^{8} -$$$$95\!\cdots\!32$$$$T^{10} +$$$$96\!\cdots\!16$$$$T^{12} -$$$$87\!\cdots\!84$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 10598201455 T^{2} + 53515912648993384359 T^{4} -$$$$16\!\cdots\!10$$$$T^{6} +$$$$36\!\cdots\!89$$$$T^{8} -$$$$58\!\cdots\!01$$$$T^{10} +$$$$67\!\cdots\!61$$$$T^{12} -$$$$56\!\cdots\!10$$$$T^{14} +$$$$32\!\cdots\!91$$$$T^{16} -$$$$11\!\cdots\!55$$$$T^{18} +$$$$20\!\cdots\!49$$$$T^{20} )^{2}$$)
$71$ ($$( 1 + 2889010114 T^{2} + 3255243551009881201 T^{4} )^{2}$$)($$( 1 + 9830208232 T^{2} + 46886702724166517020 T^{4} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$30\!\cdots\!22$$$$T^{8} +$$$$46\!\cdots\!88$$$$T^{10} +$$$$49\!\cdots\!20$$$$T^{12} +$$$$33\!\cdots\!32$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 6724308230 T^{2} + 20388189580420161501 T^{4} +$$$$45\!\cdots\!64$$$$T^{6} +$$$$10\!\cdots\!26$$$$T^{8} +$$$$22\!\cdots\!68$$$$T^{10} +$$$$34\!\cdots\!26$$$$T^{12} +$$$$48\!\cdots\!64$$$$T^{14} +$$$$70\!\cdots\!01$$$$T^{16} +$$$$75\!\cdots\!30$$$$T^{18} +$$$$36\!\cdots\!01$$$$T^{20} )^{2}$$)
$73$ ($$( 1 + 30739 T + 2073071593 T^{2} )^{4}$$)($$( 1 + 9508 T + 3097043794 T^{2} - 21796113866240 T^{3} + 7953703459188333211 T^{4} -$$$$45\!\cdots\!20$$$$T^{5} +$$$$13\!\cdots\!06$$$$T^{6} +$$$$84\!\cdots\!56$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{4}$$)($$( 1 - 21355 T + 2385405975 T^{2} - 91039324043106 T^{3} + 7775807385428270205 T^{4} -$$$$26\!\cdots\!29$$$$T^{5} +$$$$16\!\cdots\!65$$$$T^{6} -$$$$39\!\cdots\!94$$$$T^{7} +$$$$21\!\cdots\!75$$$$T^{8} -$$$$39\!\cdots\!55$$$$T^{9} +$$$$38\!\cdots\!93$$$$T^{10} )^{4}$$)
$79$ ($$( 1 - 1226373986 T^{2} + 9468276082626847201 T^{4} )^{2}$$)($$( 1 + 99394360 T^{2} + 19778301042529431580 T^{4} -$$$$22\!\cdots\!96$$$$T^{6} +$$$$20\!\cdots\!66$$$$T^{8} -$$$$21\!\cdots\!96$$$$T^{10} +$$$$17\!\cdots\!80$$$$T^{12} +$$$$84\!\cdots\!60$$$$T^{14} +$$$$80\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 13863408967 T^{2} + 96370465581362357535 T^{4} -$$$$46\!\cdots\!74$$$$T^{6} +$$$$17\!\cdots\!65$$$$T^{8} -$$$$59\!\cdots\!89$$$$T^{10} +$$$$16\!\cdots\!65$$$$T^{12} -$$$$41\!\cdots\!74$$$$T^{14} +$$$$81\!\cdots\!35$$$$T^{16} -$$$$11\!\cdots\!67$$$$T^{18} +$$$$76\!\cdots\!01$$$$T^{20} )^{2}$$)
$83$ ($$( 1 + 7748073619 T^{2} + 15516041187205853449 T^{4} )^{2}$$)($$( 1 + 20559730876 T^{2} +$$$$21\!\cdots\!14$$$$T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$70\!\cdots\!39$$$$T^{8} +$$$$23\!\cdots\!32$$$$T^{10} +$$$$52\!\cdots\!14$$$$T^{12} +$$$$76\!\cdots\!24$$$$T^{14} +$$$$57\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 + 15102812222 T^{2} +$$$$16\!\cdots\!49$$$$T^{4} +$$$$11\!\cdots\!68$$$$T^{6} +$$$$64\!\cdots\!90$$$$T^{8} +$$$$28\!\cdots\!80$$$$T^{10} +$$$$10\!\cdots\!10$$$$T^{12} +$$$$27\!\cdots\!68$$$$T^{14} +$$$$60\!\cdots\!01$$$$T^{16} +$$$$87\!\cdots\!22$$$$T^{18} +$$$$89\!\cdots\!49$$$$T^{20} )^{2}$$)
$89$ ($$( 1 - 6450847934 T^{2} + 31181719929966183601 T^{4} )^{2}$$)($$( 1 - 30119375768 T^{2} +$$$$43\!\cdots\!12$$$$T^{4} -$$$$41\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!18$$$$T^{8} -$$$$12\!\cdots\!60$$$$T^{10} +$$$$42\!\cdots\!12$$$$T^{12} -$$$$91\!\cdots\!68$$$$T^{14} +$$$$94\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 46714027858 T^{2} +$$$$10\!\cdots\!53$$$$T^{4} -$$$$13\!\cdots\!72$$$$T^{6} +$$$$12\!\cdots\!82$$$$T^{8} -$$$$83\!\cdots\!68$$$$T^{10} +$$$$39\!\cdots\!82$$$$T^{12} -$$$$13\!\cdots\!72$$$$T^{14} +$$$$31\!\cdots\!53$$$$T^{16} -$$$$44\!\cdots\!58$$$$T^{18} +$$$$29\!\cdots\!01$$$$T^{20} )^{2}$$)
$97$ ($$( 1 + 13717 T + 8587340257 T^{2} )^{4}$$)($$( 1 + 244 T + 6580710202 T^{2} + 1109461517844208 T^{3} - 18256445891874654653 T^{4} +$$$$95\!\cdots\!56$$$$T^{5} +$$$$48\!\cdots\!98$$$$T^{6} +$$$$15\!\cdots\!92$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8} )^{4}$$)($$( 1 + 54977 T + 25395595455 T^{2} + 1194992001523350 T^{3} +$$$$29\!\cdots\!17$$$$T^{4} +$$$$12\!\cdots\!15$$$$T^{5} +$$$$24\!\cdots\!69$$$$T^{6} +$$$$88\!\cdots\!50$$$$T^{7} +$$$$16\!\cdots\!15$$$$T^{8} +$$$$29\!\cdots\!77$$$$T^{9} +$$$$46\!\cdots\!57$$$$T^{10} )^{4}$$)