Properties

Label 108.6
Level 108
Weight 6
Dimension 731
Nonzero newspaces 6
Newform subspaces 11
Sturm bound 3888
Trace bound 1

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Defining parameters

Level: \( N \) = \( 108\( 108 = 2^{2} \cdot 3^{3} \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 11 \)
Sturm bound: \(3888\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(108))\).

Total New Old
Modular forms 1695 763 932
Cusp forms 1545 731 814
Eisenstein series 150 32 118

Trace form

\( 731q - 3q^{2} + 3q^{4} - 72q^{5} - 6q^{6} - 2q^{7} - 9q^{8} + 318q^{9} + O(q^{10}) \) \( 731q - 3q^{2} + 3q^{4} - 72q^{5} - 6q^{6} - 2q^{7} - 9q^{8} + 318q^{9} - 405q^{10} - 1434q^{11} + 1173q^{12} + 568q^{13} + 3027q^{14} + 531q^{15} - 2685q^{16} - 5766q^{17} - 5697q^{18} - 209q^{19} + 2475q^{20} + 12882q^{21} + 4803q^{22} + 7707q^{23} + 8988q^{24} - 6736q^{25} - 17415q^{27} - 7206q^{28} + 33375q^{29} - 18135q^{30} + 2896q^{31} + 14457q^{32} + 12342q^{33} + 1431q^{34} - 45375q^{35} + 35664q^{36} - 40325q^{37} + 29745q^{38} - 10545q^{39} + 13419q^{40} + 174162q^{41} - 25506q^{42} + 57448q^{43} - 158049q^{44} - 35781q^{45} - 27651q^{46} - 31932q^{47} + 14469q^{48} - 93090q^{49} + 66756q^{50} + 42831q^{51} - 3465q^{52} - 120990q^{53} + 231414q^{54} - 75942q^{55} + 172449q^{56} + 72048q^{57} + 10743q^{58} + 95925q^{59} - 326058q^{60} + 92380q^{61} - 461448q^{62} - 98115q^{63} + 14637q^{64} - 95976q^{65} - 24771q^{66} + 16903q^{67} + 357870q^{68} + 345801q^{69} + 227715q^{70} - 161556q^{71} + 110388q^{72} - 292772q^{73} + 40803q^{74} - 75273q^{75} + 111135q^{76} + 183600q^{77} + 178248q^{78} + 62164q^{79} - 7926q^{81} + 63030q^{82} + 152226q^{83} - 307308q^{84} - 125652q^{85} - 558483q^{86} - 201483q^{87} - 198645q^{88} - 222162q^{89} - 840048q^{90} - 421192q^{91} - 600627q^{92} - 526815q^{93} - 156753q^{94} + 251274q^{95} + 992022q^{96} + 190771q^{97} + 1307970q^{98} + 13635q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(108))\)

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
108.6.a \(\chi_{108}(1, \cdot)\) 108.6.a.a 1 1
108.6.a.b 2
108.6.a.c 2
108.6.a.d 2
108.6.b \(\chi_{108}(107, \cdot)\) 108.6.b.a 4 1
108.6.b.b 16
108.6.b.c 20
108.6.e \(\chi_{108}(37, \cdot)\) 108.6.e.a 10 2
108.6.h \(\chi_{108}(35, \cdot)\) 108.6.h.a 56 2
108.6.i \(\chi_{108}(13, \cdot)\) 108.6.i.a 90 6
108.6.l \(\chi_{108}(11, \cdot)\) 108.6.l.a 528 6

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(108))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(108)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 2}\)

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
$5$ (\( 1 + 3125 T^{2} \))(\( 1 + 2929 T^{2} + 9765625 T^{4} \))(\( 1 + 2929 T^{2} + 9765625 T^{4} \))(\( 1 - 1526 T^{2} + 9765625 T^{4} \))(\( ( 1 + 2131 T^{2} + 9765625 T^{4} )^{2} \))(\( ( 1 - 14972 T^{2} + 119342578 T^{4} - 617133972224 T^{6} + 2271820128564475 T^{8} - 6026698947500000000 T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(90\!\cdots\!25\)\( T^{16} )^{2} \))(\( ( 1 - 11818 T^{2} + 71630853 T^{4} - 313823973048 T^{6} + 1185932779192818 T^{8} - 3986228637264265212 T^{10} + \)\(11\!\cdots\!50\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(66\!\cdots\!25\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(88\!\cdots\!25\)\( T^{20} )^{2} \))(\( 1 - 21 T - 5203 T^{2} + 519930 T^{3} + 14035794 T^{4} - 2854822770 T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 11924309583255600 T^{9} + 2533145723872694124 T^{10} - 37263467447673750000 T^{11} - \)\(58\!\cdots\!25\)\( T^{12} + \)\(29\!\cdots\!25\)\( T^{13} + \)\(73\!\cdots\!75\)\( T^{14} - \)\(85\!\cdots\!50\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} + \)\(15\!\cdots\!50\)\( T^{17} - \)\(47\!\cdots\!75\)\( T^{18} - \)\(59\!\cdots\!25\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 + 25 T + 16807 T^{2} \))(\( 1 + 32 T + 3981 T^{2} + 537824 T^{3} + 282475249 T^{4} \))(\( 1 + 32 T + 3981 T^{2} + 537824 T^{3} + 282475249 T^{4} \))(\( ( 1 - 29 T + 16807 T^{2} )^{2} \))(\( ( 1 - 8471 T^{2} + 282475249 T^{4} )^{2} \))(\( ( 1 - 54884 T^{2} + 1251149938 T^{4} - 15826547698400 T^{6} + 189674874534565723 T^{8} - \)\(44\!\cdots\!00\)\( T^{10} + \)\(99\!\cdots\!38\)\( T^{12} - \)\(12\!\cdots\!16\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 72919 T^{2} + 2924383791 T^{4} - 81340986295962 T^{6} + 1774418094261308301 T^{8} - \)\(32\!\cdots\!57\)\( T^{10} + \)\(50\!\cdots\!49\)\( T^{12} - \)\(64\!\cdots\!62\)\( T^{14} + \)\(65\!\cdots\!59\)\( T^{16} - \)\(46\!\cdots\!19\)\( T^{18} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} - \)\(69\!\cdots\!60\)\( T^{10} + \)\(26\!\cdots\!24\)\( T^{11} + \)\(67\!\cdots\!43\)\( T^{12} - \)\(26\!\cdots\!39\)\( T^{13} - \)\(22\!\cdots\!03\)\( T^{14} - \)\(10\!\cdots\!72\)\( T^{15} + \)\(99\!\cdots\!64\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{17} - \)\(25\!\cdots\!69\)\( T^{18} - \)\(31\!\cdots\!03\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \))
$11$ (\( 1 + 161051 T^{2} \))(\( 1 + 486 T + 168607 T^{2} + 78270786 T^{3} + 25937424601 T^{4} \))(\( 1 - 486 T + 168607 T^{2} - 78270786 T^{3} + 25937424601 T^{4} \))(\( 1 + 314326 T^{2} + 25937424601 T^{4} \))(\( ( 1 - 14573 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 + 741148 T^{2} + 299964078250 T^{4} + 79656648391789648 T^{6} + \)\(15\!\cdots\!99\)\( T^{8} + \)\(20\!\cdots\!48\)\( T^{10} + \)\(20\!\cdots\!50\)\( T^{12} + \)\(12\!\cdots\!48\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 822230 T^{2} + 360495679845 T^{4} + 108431998645925448 T^{6} + \)\(24\!\cdots\!70\)\( T^{8} + \)\(44\!\cdots\!92\)\( T^{10} + \)\(64\!\cdots\!70\)\( T^{12} + \)\(72\!\cdots\!48\)\( T^{14} + \)\(62\!\cdots\!45\)\( T^{16} + \)\(37\!\cdots\!30\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} - \)\(10\!\cdots\!51\)\( T^{8} - \)\(50\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} - \)\(81\!\cdots\!82\)\( T^{11} - \)\(27\!\cdots\!51\)\( T^{12} + \)\(35\!\cdots\!07\)\( T^{13} - \)\(46\!\cdots\!06\)\( T^{14} - \)\(36\!\cdots\!51\)\( T^{15} + \)\(18\!\cdots\!82\)\( T^{16} + \)\(20\!\cdots\!19\)\( T^{17} - \)\(17\!\cdots\!32\)\( T^{18} + \)\(12\!\cdots\!27\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 + 427 T + 371293 T^{2} \))(\( 1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4} \))(\( 1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4} \))(\( ( 1 - 329 T + 371293 T^{2} )^{2} \))(\( ( 1 + 166 T + 371293 T^{2} )^{4} \))(\( ( 1 - 224 T + 554236 T^{2} + 77263648 T^{3} + 200668790230 T^{4} + 28687451656864 T^{5} + 76406139088422364 T^{6} - 11465640035156329568 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 29 T + 791115 T^{2} + 39777006 T^{3} + 380980388709 T^{4} + 35518566649227 T^{5} + 141455351464930737 T^{6} + 5483598057428624094 T^{7} + \)\(40\!\cdots\!55\)\( T^{8} + \)\(55\!\cdots\!29\)\( T^{9} + \)\(70\!\cdots\!93\)\( T^{10} )^{4} \))(\( 1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} + \)\(98\!\cdots\!51\)\( T^{8} + \)\(77\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!24\)\( T^{10} + \)\(28\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!29\)\( T^{13} - \)\(36\!\cdots\!73\)\( T^{14} - \)\(59\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(14\!\cdots\!74\)\( T^{17} - \)\(36\!\cdots\!31\)\( T^{18} + \)\(24\!\cdots\!33\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \))
$17$ (\( 1 + 1419857 T^{2} \))(\( 1 + 1944 T + 2456098 T^{2} + 2760202008 T^{3} + 2015993900449 T^{4} \))(\( 1 - 1944 T + 2456098 T^{2} - 2760202008 T^{3} + 2015993900449 T^{4} \))(\( 1 - 2020286 T^{2} + 2015993900449 T^{4} \))(\( ( 1 - 2151950 T^{2} + 2015993900449 T^{4} )^{2} \))(\( ( 1 - 4946936 T^{2} + 15515554849948 T^{4} - 34385249721915417800 T^{6} + \)\(55\!\cdots\!90\)\( T^{8} - \)\(69\!\cdots\!00\)\( T^{10} + \)\(63\!\cdots\!48\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 6202690 T^{2} + 18689060069517 T^{4} - 34705007722821565080 T^{6} + \)\(46\!\cdots\!02\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(94\!\cdots\!98\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!33\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \))(\( ( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} + \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} )^{2} \))
$19$ (\( 1 + 1711 T + 2476099 T^{2} \))(\( 1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4} \))(\( 1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4} \))(\( ( 1 - 1799 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 4501190 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 9713816 T^{2} + 55349997016060 T^{4} - \)\(21\!\cdots\!72\)\( T^{6} + \)\(61\!\cdots\!02\)\( T^{8} - \)\(13\!\cdots\!72\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(22\!\cdots\!16\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 13782439 T^{2} + 98755621335207 T^{4} - \)\(47\!\cdots\!14\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} - \)\(48\!\cdots\!01\)\( T^{10} + \)\(10\!\cdots\!01\)\( T^{12} - \)\(17\!\cdots\!14\)\( T^{14} + \)\(22\!\cdots\!07\)\( T^{16} - \)\(19\!\cdots\!39\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} )^{2} \))
$23$ (\( 1 + 6436343 T^{2} \))(\( 1 + 6804 T + 24393154 T^{2} + 43792877772 T^{3} + 41426511213649 T^{4} \))(\( 1 - 6804 T + 24393154 T^{2} - 43792877772 T^{3} + 41426511213649 T^{4} \))(\( 1 - 198770 T^{2} + 41426511213649 T^{4} \))(\( ( 1 - 2019266 T^{2} + 41426511213649 T^{4} )^{2} \))(\( ( 1 + 29994040 T^{2} + 487786705761244 T^{4} + \)\(51\!\cdots\!44\)\( T^{6} + \)\(39\!\cdots\!14\)\( T^{8} + \)\(21\!\cdots\!56\)\( T^{10} + \)\(83\!\cdots\!44\)\( T^{12} + \)\(21\!\cdots\!60\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 42721646 T^{2} + 918206452664445 T^{4} + \)\(12\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{8} + \)\(95\!\cdots\!68\)\( T^{10} + \)\(53\!\cdots\!66\)\( T^{12} + \)\(22\!\cdots\!28\)\( T^{14} + \)\(65\!\cdots\!05\)\( T^{16} + \)\(12\!\cdots\!46\)\( T^{18} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} + \)\(48\!\cdots\!51\)\( T^{7} - \)\(71\!\cdots\!41\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(81\!\cdots\!80\)\( T^{10} - \)\(74\!\cdots\!36\)\( T^{11} - \)\(29\!\cdots\!09\)\( T^{12} + \)\(12\!\cdots\!57\)\( T^{13} - \)\(83\!\cdots\!63\)\( T^{14} - \)\(62\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!24\)\( T^{16} + \)\(17\!\cdots\!40\)\( T^{17} - \)\(47\!\cdots\!41\)\( T^{18} + \)\(75\!\cdots\!57\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \))
$29$ (\( 1 + 20511149 T^{2} \))(\( 1 + 11664 T + 70238998 T^{2} + 239242041936 T^{3} + 420707233300201 T^{4} \))(\( 1 - 11664 T + 70238998 T^{2} - 239242041936 T^{3} + 420707233300201 T^{4} \))(\( 1 + 39031642 T^{2} + 420707233300201 T^{4} \))(\( ( 1 - 29365574 T^{2} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 26575640 T^{2} + 859790443267708 T^{4} - \)\(25\!\cdots\!88\)\( T^{6} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{12} - \)\(19\!\cdots\!40\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 142578466 T^{2} + 9766751584325109 T^{4} - \)\(42\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!16\)\( T^{10} + \)\(56\!\cdots\!22\)\( T^{12} - \)\(75\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!09\)\( T^{16} - \)\(44\!\cdots\!66\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} + \)\(64\!\cdots\!39\)\( T^{6} - \)\(82\!\cdots\!55\)\( T^{7} - \)\(90\!\cdots\!17\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!16\)\( T^{10} - \)\(29\!\cdots\!40\)\( T^{11} - \)\(38\!\cdots\!17\)\( T^{12} - \)\(71\!\cdots\!95\)\( T^{13} + \)\(11\!\cdots\!39\)\( T^{14} - \)\(24\!\cdots\!42\)\( T^{15} + \)\(21\!\cdots\!54\)\( T^{16} - \)\(48\!\cdots\!66\)\( T^{17} + \)\(11\!\cdots\!57\)\( T^{18} - \)\(38\!\cdots\!17\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \))
$31$ (\( 1 + 10324 T + 28629151 T^{2} \))(\( 1 - 3328 T + 38238117 T^{2} - 95277814528 T^{3} + 819628286980801 T^{4} \))(\( 1 - 3328 T + 38238117 T^{2} - 95277814528 T^{3} + 819628286980801 T^{4} \))(\( ( 1 - 5228 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 17147735 T^{2} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 160216820 T^{2} + 12412163878108546 T^{4} - \)\(60\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!87\)\( T^{8} - \)\(49\!\cdots\!08\)\( T^{10} + \)\(83\!\cdots\!46\)\( T^{12} - \)\(88\!\cdots\!20\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 53059450 T^{2} + 1830387109253997 T^{4} - \)\(54\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!98\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{10} + \)\(14\!\cdots\!98\)\( T^{12} - \)\(36\!\cdots\!08\)\( T^{14} + \)\(10\!\cdots\!97\)\( T^{16} - \)\(23\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} - \)\(32\!\cdots\!47\)\( T^{6} - \)\(42\!\cdots\!39\)\( T^{7} - \)\(13\!\cdots\!49\)\( T^{8} + \)\(36\!\cdots\!48\)\( T^{9} + \)\(63\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{11} - \)\(11\!\cdots\!49\)\( T^{12} - \)\(99\!\cdots\!89\)\( T^{13} - \)\(21\!\cdots\!47\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!00\)\( T^{16} - \)\(29\!\cdots\!72\)\( T^{17} - \)\(24\!\cdots\!77\)\( T^{18} - \)\(35\!\cdots\!09\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \))
$37$ (\( 1 + 6661 T + 69343957 T^{2} \))(\( 1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4} \))(\( ( 1 - 8783 T + 69343957 T^{2} )^{2} \))(\( ( 1 - 15332 T + 69343957 T^{2} )^{4} \))(\( ( 1 + 17752 T + 254119924 T^{2} + 2137611581224 T^{3} + 19827261455867542 T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(59\!\cdots\!36\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 1613 T + 123026979 T^{2} + 343711884654 T^{3} + 5803261980890325 T^{4} + 31670451235762477275 T^{5} + \)\(40\!\cdots\!25\)\( T^{6} + \)\(16\!\cdots\!46\)\( T^{7} + \)\(41\!\cdots\!47\)\( T^{8} + \)\(37\!\cdots\!13\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{4} \))(\( ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{2} \))
$41$ (\( 1 + 115856201 T^{2} \))(\( 1 + 13608 T + 266834974 T^{2} + 1576571183208 T^{3} + 13422659310152401 T^{4} \))(\( 1 - 13608 T + 266834974 T^{2} - 1576571183208 T^{3} + 13422659310152401 T^{4} \))(\( 1 - 9156974 T^{2} + 13422659310152401 T^{4} \))(\( ( 1 - 129650498 T^{2} + 13422659310152401 T^{4} )^{2} \))(\( ( 1 - 667999304 T^{2} + 215857510496501980 T^{4} - \)\(43\!\cdots\!84\)\( T^{6} + \)\(60\!\cdots\!14\)\( T^{8} - \)\(58\!\cdots\!84\)\( T^{10} + \)\(38\!\cdots\!80\)\( T^{12} - \)\(16\!\cdots\!04\)\( T^{14} + \)\(32\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 204662074 T^{2} + 6421654936070589 T^{4} + \)\(14\!\cdots\!28\)\( T^{6} - \)\(52\!\cdots\!06\)\( T^{8} - \)\(96\!\cdots\!76\)\( T^{10} - \)\(70\!\cdots\!06\)\( T^{12} + \)\(26\!\cdots\!28\)\( T^{14} + \)\(15\!\cdots\!89\)\( T^{16} - \)\(66\!\cdots\!74\)\( T^{18} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} - \)\(52\!\cdots\!97\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!43\)\( T^{7} - \)\(31\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!42\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!42\)\( T^{11} - \)\(41\!\cdots\!03\)\( T^{12} + \)\(62\!\cdots\!43\)\( T^{13} + \)\(30\!\cdots\!40\)\( T^{14} - \)\(10\!\cdots\!97\)\( T^{15} - \)\(12\!\cdots\!22\)\( T^{16} + \)\(12\!\cdots\!75\)\( T^{17} - \)\(38\!\cdots\!42\)\( T^{18} - \)\(69\!\cdots\!35\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \))
$43$ (\( 1 + 3352 T + 147008443 T^{2} \))(\( 1 - 4960 T + 213010962 T^{2} - 729161877280 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 4960 T + 213010962 T^{2} - 729161877280 T^{3} + 21611482313284249 T^{4} \))(\( ( 1 - 19976 T + 147008443 T^{2} )^{2} \))(\( ( 1 - 284996378 T^{2} + 21611482313284249 T^{4} )^{2} \))(\( ( 1 - 271892936 T^{2} + 81822516500494204 T^{4} - \)\(14\!\cdots\!72\)\( T^{6} + \)\(26\!\cdots\!02\)\( T^{8} - \)\(30\!\cdots\!28\)\( T^{10} + \)\(38\!\cdots\!04\)\( T^{12} - \)\(27\!\cdots\!64\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 1038109906 T^{2} + 527969585590823397 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{8} - \)\(67\!\cdots\!12\)\( T^{10} + \)\(85\!\cdots\!98\)\( T^{12} - \)\(80\!\cdots\!40\)\( T^{14} + \)\(53\!\cdots\!53\)\( T^{16} - \)\(22\!\cdots\!06\)\( T^{18} + \)\(47\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} - \)\(75\!\cdots\!27\)\( T^{5} + \)\(55\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!57\)\( T^{7} - \)\(29\!\cdots\!39\)\( T^{8} + \)\(61\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!84\)\( T^{10} + \)\(90\!\cdots\!66\)\( T^{11} - \)\(64\!\cdots\!11\)\( T^{12} - \)\(38\!\cdots\!99\)\( T^{13} + \)\(26\!\cdots\!62\)\( T^{14} - \)\(52\!\cdots\!61\)\( T^{15} + \)\(12\!\cdots\!66\)\( T^{16} + \)\(59\!\cdots\!89\)\( T^{17} - \)\(59\!\cdots\!36\)\( T^{18} - \)\(47\!\cdots\!67\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \))
$47$ (\( 1 + 229345007 T^{2} \))(\( 1 + 18468 T + 312496354 T^{2} + 4235543589276 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 18468 T + 312496354 T^{2} - 4235543589276 T^{3} + 52599132235830049 T^{4} \))(\( 1 + 341046910 T^{2} + 52599132235830049 T^{4} \))(\( ( 1 + 408701842 T^{2} + 52599132235830049 T^{4} )^{2} \))(\( ( 1 + 1268760328 T^{2} + 775645167851419804 T^{4} + \)\(30\!\cdots\!92\)\( T^{6} + \)\(81\!\cdots\!58\)\( T^{8} + \)\(15\!\cdots\!08\)\( T^{10} + \)\(21\!\cdots\!04\)\( T^{12} + \)\(18\!\cdots\!72\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 920931614 T^{2} + 499070896097327373 T^{4} + \)\(18\!\cdots\!24\)\( T^{6} + \)\(53\!\cdots\!58\)\( T^{8} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(27\!\cdots\!42\)\( T^{12} + \)\(50\!\cdots\!24\)\( T^{14} + \)\(72\!\cdots\!77\)\( T^{16} + \)\(70\!\cdots\!14\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} - \)\(41\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!27\)\( T^{6} + \)\(55\!\cdots\!85\)\( T^{7} + \)\(12\!\cdots\!11\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{9} - \)\(37\!\cdots\!60\)\( T^{10} - \)\(96\!\cdots\!92\)\( T^{11} + \)\(67\!\cdots\!39\)\( T^{12} + \)\(67\!\cdots\!55\)\( T^{13} - \)\(71\!\cdots\!27\)\( T^{14} - \)\(26\!\cdots\!84\)\( T^{15} + \)\(14\!\cdots\!40\)\( T^{16} + \)\(47\!\cdots\!92\)\( T^{17} - \)\(30\!\cdots\!33\)\( T^{18} - \)\(44\!\cdots\!85\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \))
$53$ (\( 1 + 418195493 T^{2} \))(\( 1 - 11664 T + 484234009 T^{2} - 4877832230352 T^{3} + 174887470365513049 T^{4} \))(\( 1 + 11664 T + 484234009 T^{2} + 4877832230352 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 31068470 T^{2} + 174887470365513049 T^{4} \))(\( ( 1 - 537758237 T^{2} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 - 1665548972 T^{2} + 1494189343182800962 T^{4} - \)\(95\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!75\)\( T^{8} - \)\(16\!\cdots\!80\)\( T^{10} + \)\(45\!\cdots\!62\)\( T^{12} - \)\(89\!\cdots\!28\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 1301597650 T^{2} + 1197727639041193893 T^{4} - \)\(82\!\cdots\!04\)\( T^{6} + \)\(45\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{10} + \)\(79\!\cdots\!82\)\( T^{12} - \)\(25\!\cdots\!04\)\( T^{14} + \)\(64\!\cdots\!57\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \))(\( ( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} + \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} )^{2} \))
$59$ (\( 1 + 714924299 T^{2} \))(\( 1 + 1944 T + 961283686 T^{2} + 1389812837256 T^{3} + 511116753300641401 T^{4} \))(\( 1 - 1944 T + 961283686 T^{2} - 1389812837256 T^{3} + 511116753300641401 T^{4} \))(\( 1 + 1396994998 T^{2} + 511116753300641401 T^{4} \))(\( ( 1 + 621590410 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 4822767208 T^{2} + 10641078380871180220 T^{4} + \)\(14\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!22\)\( T^{8} + \)\(71\!\cdots\!92\)\( T^{10} + \)\(27\!\cdots\!20\)\( T^{12} + \)\(64\!\cdots\!08\)\( T^{14} + \)\(68\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 1806503990 T^{2} + 2218805640623321733 T^{4} + \)\(21\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(12\!\cdots\!12\)\( T^{10} + \)\(87\!\cdots\!70\)\( T^{12} + \)\(55\!\cdots\!92\)\( T^{14} + \)\(29\!\cdots\!33\)\( T^{16} + \)\(12\!\cdots\!90\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} + \)\(10\!\cdots\!09\)\( T^{5} + \)\(69\!\cdots\!78\)\( T^{6} - \)\(39\!\cdots\!93\)\( T^{7} + \)\(84\!\cdots\!01\)\( T^{8} + \)\(17\!\cdots\!74\)\( T^{9} - \)\(12\!\cdots\!20\)\( T^{10} + \)\(12\!\cdots\!26\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} - \)\(14\!\cdots\!07\)\( T^{13} + \)\(18\!\cdots\!78\)\( T^{14} + \)\(19\!\cdots\!91\)\( T^{15} - \)\(12\!\cdots\!78\)\( T^{16} - \)\(13\!\cdots\!51\)\( T^{17} + \)\(19\!\cdots\!40\)\( T^{18} - \)\(44\!\cdots\!63\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 - 56927 T + 844596301 T^{2} \))(\( 1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4} \))(\( 1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 + 1069 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 53188 T + 844596301 T^{2} )^{4} \))(\( ( 1 - 19472 T + 1261489684 T^{2} - 43949943879536 T^{3} + 833752337244254902 T^{4} - \)\(37\!\cdots\!36\)\( T^{5} + \)\(89\!\cdots\!84\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 21151 T + 2606330283 T^{2} - 19051394797770 T^{3} + 2604033901923348357 T^{4} - \)\(93\!\cdots\!13\)\( T^{5} + \)\(21\!\cdots\!57\)\( T^{6} - \)\(13\!\cdots\!70\)\( T^{7} + \)\(15\!\cdots\!83\)\( T^{8} - \)\(10\!\cdots\!51\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \))(\( 1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(10\!\cdots\!33\)\( T^{6} - \)\(70\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!67\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{11} + \)\(80\!\cdots\!67\)\( T^{12} - \)\(42\!\cdots\!89\)\( T^{13} - \)\(51\!\cdots\!33\)\( T^{14} + \)\(23\!\cdots\!90\)\( T^{15} + \)\(25\!\cdots\!98\)\( T^{16} - \)\(13\!\cdots\!46\)\( T^{17} - \)\(91\!\cdots\!83\)\( T^{18} - \)\(30\!\cdots\!03\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \))
$67$ (\( 1 + 37939 T + 1350125107 T^{2} \))(\( 1 - 90064 T + 4055628738 T^{2} - 121597667636848 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 90064 T + 4055628738 T^{2} - 121597667636848 T^{3} + 1822837804551761449 T^{4} \))(\( ( 1 + 62077 T + 1350125107 T^{2} )^{2} \))(\( ( 1 - 1014398666 T^{2} + 1822837804551761449 T^{4} )^{2} \))(\( ( 1 - 1447449416 T^{2} + 2915553445416739516 T^{4} - \)\(52\!\cdots\!68\)\( T^{6} + \)\(74\!\cdots\!58\)\( T^{8} - \)\(95\!\cdots\!32\)\( T^{10} + \)\(96\!\cdots\!16\)\( T^{12} - \)\(87\!\cdots\!84\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 10598201455 T^{2} + 53515912648993384359 T^{4} - \)\(16\!\cdots\!10\)\( T^{6} + \)\(36\!\cdots\!89\)\( T^{8} - \)\(58\!\cdots\!01\)\( T^{10} + \)\(67\!\cdots\!61\)\( T^{12} - \)\(56\!\cdots\!10\)\( T^{14} + \)\(32\!\cdots\!91\)\( T^{16} - \)\(11\!\cdots\!55\)\( T^{18} + \)\(20\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} + \)\(30\!\cdots\!39\)\( T^{5} - \)\(79\!\cdots\!54\)\( T^{6} - \)\(69\!\cdots\!51\)\( T^{7} - \)\(33\!\cdots\!67\)\( T^{8} + \)\(37\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!76\)\( T^{10} + \)\(51\!\cdots\!22\)\( T^{11} - \)\(60\!\cdots\!83\)\( T^{12} - \)\(16\!\cdots\!93\)\( T^{13} - \)\(26\!\cdots\!54\)\( T^{14} + \)\(13\!\cdots\!73\)\( T^{15} + \)\(60\!\cdots\!70\)\( T^{16} - \)\(63\!\cdots\!29\)\( T^{17} - \)\(42\!\cdots\!44\)\( T^{18} - \)\(20\!\cdots\!49\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \))
$71$ (\( 1 + 1804229351 T^{2} \))(\( 1 - 44712 T + 2046094414 T^{2} - 80670702741912 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 44712 T + 2046094414 T^{2} + 80670702741912 T^{3} + 3255243551009881201 T^{4} \))(\( 1 + 1457026126 T^{2} + 3255243551009881201 T^{4} \))(\( ( 1 + 2889010114 T^{2} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 9830208232 T^{2} + 46886702724166517020 T^{4} + \)\(14\!\cdots\!88\)\( T^{6} + \)\(30\!\cdots\!22\)\( T^{8} + \)\(46\!\cdots\!88\)\( T^{10} + \)\(49\!\cdots\!20\)\( T^{12} + \)\(33\!\cdots\!32\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 6724308230 T^{2} + 20388189580420161501 T^{4} + \)\(45\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(22\!\cdots\!68\)\( T^{10} + \)\(34\!\cdots\!26\)\( T^{12} + \)\(48\!\cdots\!64\)\( T^{14} + \)\(70\!\cdots\!01\)\( T^{16} + \)\(75\!\cdots\!30\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} + \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} + \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} + \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} )^{2} \))
$73$ (\( 1 - 79577 T + 2073071593 T^{2} \))(\( 1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4} \))(\( ( 1 + 48079 T + 2073071593 T^{2} )^{2} \))(\( ( 1 + 30739 T + 2073071593 T^{2} )^{4} \))(\( ( 1 + 9508 T + 3097043794 T^{2} - 21796113866240 T^{3} + 7953703459188333211 T^{4} - \)\(45\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!06\)\( T^{6} + \)\(84\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 - 21355 T + 2385405975 T^{2} - 91039324043106 T^{3} + 7775807385428270205 T^{4} - \)\(26\!\cdots\!29\)\( T^{5} + \)\(16\!\cdots\!65\)\( T^{6} - \)\(39\!\cdots\!94\)\( T^{7} + \)\(21\!\cdots\!75\)\( T^{8} - \)\(39\!\cdots\!55\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{4} \))(\( ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{2} \))
$79$ (\( 1 - 90857 T + 3077056399 T^{2} \))(\( 1 - 28768 T + 4017714654 T^{2} - 88520758486432 T^{3} + 9468276082626847201 T^{4} \))(\( 1 - 28768 T + 4017714654 T^{2} - 88520758486432 T^{3} + 9468276082626847201 T^{4} \))(\( ( 1 - 49979 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 1226373986 T^{2} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 99394360 T^{2} + 19778301042529431580 T^{4} - \)\(22\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!66\)\( T^{8} - \)\(21\!\cdots\!96\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(84\!\cdots\!60\)\( T^{14} + \)\(80\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 13863408967 T^{2} + 96370465581362357535 T^{4} - \)\(46\!\cdots\!74\)\( T^{6} + \)\(17\!\cdots\!65\)\( T^{8} - \)\(59\!\cdots\!89\)\( T^{10} + \)\(16\!\cdots\!65\)\( T^{12} - \)\(41\!\cdots\!74\)\( T^{14} + \)\(81\!\cdots\!35\)\( T^{16} - \)\(11\!\cdots\!67\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} + \)\(21\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!85\)\( T^{6} - \)\(84\!\cdots\!45\)\( T^{7} - \)\(35\!\cdots\!45\)\( T^{8} + \)\(81\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{11} - \)\(33\!\cdots\!45\)\( T^{12} - \)\(24\!\cdots\!55\)\( T^{13} + \)\(24\!\cdots\!85\)\( T^{14} + \)\(60\!\cdots\!48\)\( T^{15} + \)\(22\!\cdots\!36\)\( T^{16} - \)\(93\!\cdots\!32\)\( T^{17} - \)\(43\!\cdots\!29\)\( T^{18} - \)\(74\!\cdots\!07\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \))
$83$ (\( 1 + 3939040643 T^{2} \))(\( 1 - 15066 T - 2258995409 T^{2} - 59345586327438 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 15066 T - 2258995409 T^{2} + 59345586327438 T^{3} + 15516041187205853449 T^{4} \))(\( 1 + 4552161670 T^{2} + 15516041187205853449 T^{4} \))(\( ( 1 + 7748073619 T^{2} + 15516041187205853449 T^{4} )^{2} \))(\( ( 1 + 20559730876 T^{2} + \)\(21\!\cdots\!14\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(70\!\cdots\!39\)\( T^{8} + \)\(23\!\cdots\!32\)\( T^{10} + \)\(52\!\cdots\!14\)\( T^{12} + \)\(76\!\cdots\!24\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 + 15102812222 T^{2} + \)\(16\!\cdots\!49\)\( T^{4} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(64\!\cdots\!90\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!10\)\( T^{12} + \)\(27\!\cdots\!68\)\( T^{14} + \)\(60\!\cdots\!01\)\( T^{16} + \)\(87\!\cdots\!22\)\( T^{18} + \)\(89\!\cdots\!49\)\( T^{20} )^{2} \))(\( 1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} - \)\(36\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!01\)\( T^{6} + \)\(11\!\cdots\!41\)\( T^{7} - \)\(18\!\cdots\!73\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} - \)\(88\!\cdots\!72\)\( T^{10} + \)\(55\!\cdots\!12\)\( T^{11} - \)\(28\!\cdots\!77\)\( T^{12} + \)\(67\!\cdots\!87\)\( T^{13} + \)\(28\!\cdots\!01\)\( T^{14} - \)\(34\!\cdots\!24\)\( T^{15} + \)\(23\!\cdots\!32\)\( T^{16} - \)\(17\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!83\)\( T^{18} - \)\(52\!\cdots\!93\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \))
$89$ (\( 1 + 5584059449 T^{2} \))(\( 1 - 178848 T + 18739138030 T^{2} - 998697864334752 T^{3} + 31181719929966183601 T^{4} \))(\( 1 + 178848 T + 18739138030 T^{2} + 998697864334752 T^{3} + 31181719929966183601 T^{4} \))(\( 1 + 3438704914 T^{2} + 31181719929966183601 T^{4} \))(\( ( 1 - 6450847934 T^{2} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 - 30119375768 T^{2} + \)\(43\!\cdots\!12\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{6} + \)\(27\!\cdots\!18\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{10} + \)\(42\!\cdots\!12\)\( T^{12} - \)\(91\!\cdots\!68\)\( T^{14} + \)\(94\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 46714027858 T^{2} + \)\(10\!\cdots\!53\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{8} - \)\(83\!\cdots\!68\)\( T^{10} + \)\(39\!\cdots\!82\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(31\!\cdots\!53\)\( T^{16} - \)\(44\!\cdots\!58\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} + \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} )^{2} \))
$97$ (\( 1 - 177725 T + 8587340257 T^{2} \))(\( 1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4} \))(\( ( 1 - 12917 T + 8587340257 T^{2} )^{2} \))(\( ( 1 + 13717 T + 8587340257 T^{2} )^{4} \))(\( ( 1 + 244 T + 6580710202 T^{2} + 1109461517844208 T^{3} - 18256445891874654653 T^{4} + \)\(95\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!98\)\( T^{6} + \)\(15\!\cdots\!92\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )^{4} \))(\( ( 1 + 54977 T + 25395595455 T^{2} + 1194992001523350 T^{3} + \)\(29\!\cdots\!17\)\( T^{4} + \)\(12\!\cdots\!15\)\( T^{5} + \)\(24\!\cdots\!69\)\( T^{6} + \)\(88\!\cdots\!50\)\( T^{7} + \)\(16\!\cdots\!15\)\( T^{8} + \)\(29\!\cdots\!77\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} )^{4} \))(\( 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - \)\(30\!\cdots\!95\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{7} - \)\(20\!\cdots\!15\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(56\!\cdots\!42\)\( T^{11} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!97\)\( T^{13} + \)\(56\!\cdots\!20\)\( T^{14} - \)\(14\!\cdots\!15\)\( T^{15} + \)\(39\!\cdots\!90\)\( T^{16} + \)\(77\!\cdots\!49\)\( T^{17} - \)\(52\!\cdots\!38\)\( T^{18} - \)\(10\!\cdots\!37\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \))
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