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

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