Properties

Label 40.6.d
Level 40
Weight 6
Character orbit d
Rep. character \(\chi_{40}(21,\cdot)\)
Character field \(\Q\)
Dimension 20
Newform subspaces 1
Sturm bound 36
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(36\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(40, [\chi])\).

Total New Old
Modular forms 32 20 12
Cusp forms 28 20 8
Eisenstein series 4 0 4

Trace form

\( 20q + 2q^{2} - 32q^{4} + 204q^{6} - 196q^{7} + 248q^{8} - 1620q^{9} + O(q^{10}) \) \( 20q + 2q^{2} - 32q^{4} + 204q^{6} - 196q^{7} + 248q^{8} - 1620q^{9} - 50q^{10} - 1876q^{12} + 2708q^{14} + 900q^{15} + 3080q^{16} - 5294q^{18} - 1900q^{20} + 13836q^{22} - 4676q^{23} + 1032q^{24} - 12500q^{25} - 8084q^{26} + 2108q^{28} + 5800q^{30} + 7160q^{31} + 6792q^{32} + 5672q^{33} + 21132q^{34} + 18344q^{36} - 19580q^{38} - 44904q^{39} + 6200q^{40} + 11608q^{41} - 17116q^{42} + 72296q^{44} - 28516q^{46} + 44180q^{47} - 88856q^{48} + 18756q^{49} - 1250q^{50} - 39680q^{52} - 100584q^{54} - 24200q^{55} - 53624q^{56} + 5032q^{57} + 59496q^{58} - 31300q^{60} + 59824q^{62} + 240620q^{63} - 11264q^{64} - 56688q^{66} + 11576q^{68} + 29800q^{70} - 200312q^{71} + 235912q^{72} - 105136q^{73} + 78876q^{74} - 153872q^{76} + 95864q^{78} + 282080q^{79} + 16000q^{80} + 65172q^{81} - 223032q^{82} - 297128q^{84} + 27452q^{86} - 332592q^{87} + 86896q^{88} - 3160q^{89} + 51750q^{90} + 107916q^{92} + 148820q^{94} + 144400q^{95} + 395168q^{96} + 147376q^{97} + 216942q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(40, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
40.6.d.a \(20\) \(6.415\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(2\) \(0\) \(0\) \(-196\) \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{3})q^{4}+\beta _{5}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(40, [\chi])\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 18 T^{2} - 116 T^{3} - 444 T^{4} - 1584 T^{5} - 2144 T^{6} + 186496 T^{7} - 497408 T^{8} + 567296 T^{9} - 25550848 T^{10} + 18153472 T^{11} - 509345792 T^{12} + 6111100928 T^{13} - 2248146944 T^{14} - 53150220288 T^{15} - 476741369856 T^{16} - 3985729650688 T^{17} + 19791209299968 T^{18} - 70368744177664 T^{19} + 1125899906842624 T^{20} \)
$3$ \( 1 - 1620 T^{2} + 1394322 T^{4} - 834927892 T^{6} + 392823148221 T^{8} - 155295909553872 T^{10} + 53814329151823576 T^{12} - 16772396183204761872 T^{14} + \)\(47\!\cdots\!30\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(31\!\cdots\!00\)\( T^{20} - \)\(75\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!30\)\( T^{24} - \)\(34\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!76\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{30} + \)\(16\!\cdots\!21\)\( T^{32} - \)\(20\!\cdots\!08\)\( T^{34} + \)\(20\!\cdots\!22\)\( T^{36} - \)\(14\!\cdots\!80\)\( T^{38} + \)\(51\!\cdots\!01\)\( T^{40} \)
$5$ \( ( 1 + 625 T^{2} )^{10} \)
$7$ \( ( 1 + 98 T + 84148 T^{2} + 8017214 T^{3} + 3556570901 T^{4} + 345044190776 T^{5} + 103920201897616 T^{6} + 10159390994080936 T^{7} + 2363994840482709802 T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} + \)\(37\!\cdots\!32\)\( T^{11} + \)\(66\!\cdots\!98\)\( T^{12} + \)\(48\!\cdots\!48\)\( T^{13} + \)\(82\!\cdots\!16\)\( T^{14} + \)\(46\!\cdots\!32\)\( T^{15} + \)\(80\!\cdots\!49\)\( T^{16} + \)\(30\!\cdots\!02\)\( T^{17} + \)\(53\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!86\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \)
$11$ \( 1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(53\!\cdots\!82\)\( T^{24} - \)\(70\!\cdots\!64\)\( T^{26} + \)\(82\!\cdots\!36\)\( T^{28} - \)\(82\!\cdots\!04\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(46\!\cdots\!24\)\( T^{34} + \)\(23\!\cdots\!26\)\( T^{36} - \)\(80\!\cdots\!04\)\( T^{38} + \)\(13\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} + \)\(67\!\cdots\!57\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{16} - \)\(38\!\cdots\!32\)\( T^{18} + \)\(14\!\cdots\!64\)\( T^{20} - \)\(52\!\cdots\!68\)\( T^{22} + \)\(16\!\cdots\!42\)\( T^{24} - \)\(46\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!64\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{30} + \)\(46\!\cdots\!57\)\( T^{32} - \)\(70\!\cdots\!52\)\( T^{34} + \)\(81\!\cdots\!54\)\( T^{36} - \)\(63\!\cdots\!68\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \)
$17$ \( ( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!90\)\( T^{12} + \)\(41\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} + \)\(42\!\cdots\!00\)\( T^{15} + \)\(12\!\cdots\!05\)\( T^{16} + \)\(21\!\cdots\!00\)\( T^{17} + \)\(89\!\cdots\!62\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \)
$19$ \( 1 - 23638692 T^{2} + 296336418074734 T^{4} - \)\(25\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(93\!\cdots\!80\)\( T^{10} + \)\(42\!\cdots\!52\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{14} + \)\(56\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!12\)\( T^{22} + \)\(21\!\cdots\!66\)\( T^{24} - \)\(38\!\cdots\!48\)\( T^{26} + \)\(60\!\cdots\!52\)\( T^{28} - \)\(80\!\cdots\!80\)\( T^{30} + \)\(91\!\cdots\!97\)\( T^{32} - \)\(83\!\cdots\!32\)\( T^{34} + \)\(59\!\cdots\!34\)\( T^{36} - \)\(28\!\cdots\!92\)\( T^{38} + \)\(75\!\cdots\!01\)\( T^{40} \)
$23$ \( ( 1 + 2338 T + 35997660 T^{2} + 45007042654 T^{3} + 520972471777845 T^{4} + 50426407997609208 T^{5} + \)\(39\!\cdots\!60\)\( T^{6} - \)\(66\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(91\!\cdots\!52\)\( T^{9} + \)\(76\!\cdots\!60\)\( T^{10} - \)\(58\!\cdots\!36\)\( T^{11} + \)\(74\!\cdots\!90\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(68\!\cdots\!60\)\( T^{14} + \)\(55\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!05\)\( T^{16} + \)\(20\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!60\)\( T^{18} + \)\(44\!\cdots\!34\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \)
$29$ \( 1 - 215092900 T^{2} + 23243889276296494 T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!13\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!28\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(20\!\cdots\!86\)\( T^{24} - \)\(31\!\cdots\!00\)\( T^{26} + \)\(43\!\cdots\!92\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(50\!\cdots\!13\)\( T^{32} - \)\(39\!\cdots\!00\)\( T^{34} + \)\(22\!\cdots\!94\)\( T^{36} - \)\(88\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( ( 1 - 3580 T + 132421614 T^{2} - 484936307876 T^{3} + 8983138546835629 T^{4} - 33755686429634218608 T^{5} + \)\(43\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} - \)\(60\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} - \)\(17\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!38\)\( T^{12} - \)\(38\!\cdots\!96\)\( T^{13} + \)\(29\!\cdots\!88\)\( T^{14} - \)\(64\!\cdots\!08\)\( T^{15} + \)\(49\!\cdots\!29\)\( T^{16} - \)\(76\!\cdots\!76\)\( T^{17} + \)\(59\!\cdots\!14\)\( T^{18} - \)\(46\!\cdots\!80\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$37$ \( 1 - 565372668 T^{2} + 171401952644913934 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(59\!\cdots\!09\)\( T^{8} - \)\(80\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(83\!\cdots\!86\)\( T^{16} - \)\(67\!\cdots\!48\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(32\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!86\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(73\!\cdots\!09\)\( T^{32} - \)\(21\!\cdots\!00\)\( T^{34} + \)\(48\!\cdots\!34\)\( T^{36} - \)\(77\!\cdots\!32\)\( T^{38} + \)\(66\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!68\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!24\)\( T^{9} + \)\(82\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!24\)\( T^{11} + \)\(85\!\cdots\!42\)\( T^{12} - \)\(44\!\cdots\!68\)\( T^{13} + \)\(70\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!48\)\( T^{15} + \)\(43\!\cdots\!93\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!34\)\( T^{18} - \)\(21\!\cdots\!04\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \)
$43$ \( 1 - 1208126740 T^{2} + 787740581508660018 T^{4} - \)\(36\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!73\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!70\)\( T^{16} - \)\(70\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!70\)\( T^{24} - \)\(21\!\cdots\!44\)\( T^{26} + \)\(20\!\cdots\!40\)\( T^{28} - \)\(17\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!73\)\( T^{32} - \)\(79\!\cdots\!04\)\( T^{34} + \)\(37\!\cdots\!18\)\( T^{36} - \)\(12\!\cdots\!60\)\( T^{38} + \)\(22\!\cdots\!01\)\( T^{40} \)
$47$ \( ( 1 - 22090 T + 1548510076 T^{2} - 25779450599270 T^{3} + 1065218725316011845 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} - \)\(32\!\cdots\!00\)\( T^{11} + \)\(76\!\cdots\!90\)\( T^{12} - \)\(62\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!96\)\( T^{14} - \)\(90\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(86\!\cdots\!10\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} - \)\(38\!\cdots\!30\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \)
$53$ \( 1 - 4003669356 T^{2} + 7954725909905649454 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(91\!\cdots\!76\)\( T^{10} + \)\(65\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(48\!\cdots\!96\)\( T^{20} - \)\(19\!\cdots\!44\)\( T^{22} + \)\(68\!\cdots\!54\)\( T^{24} - \)\(21\!\cdots\!20\)\( T^{26} + \)\(61\!\cdots\!16\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(31\!\cdots\!77\)\( T^{32} - \)\(53\!\cdots\!40\)\( T^{34} + \)\(69\!\cdots\!54\)\( T^{36} - \)\(61\!\cdots\!44\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 - 9996949828 T^{2} + 49650800837889885966 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!17\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!88\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(19\!\cdots\!80\)\( T^{26} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(14\!\cdots\!20\)\( T^{34} + \)\(23\!\cdots\!66\)\( T^{36} - \)\(23\!\cdots\!28\)\( T^{38} + \)\(12\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 7152852348 T^{2} + 26523669582205677166 T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!17\)\( T^{8} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!14\)\( T^{16} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{20} - \)\(29\!\cdots\!08\)\( T^{22} + \)\(21\!\cdots\!14\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(81\!\cdots\!84\)\( T^{28} - \)\(41\!\cdots\!48\)\( T^{30} + \)\(17\!\cdots\!17\)\( T^{32} - \)\(63\!\cdots\!00\)\( T^{34} + \)\(17\!\cdots\!66\)\( T^{36} - \)\(34\!\cdots\!48\)\( T^{38} + \)\(34\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 11828518964 T^{2} + 68669252417166303634 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!04\)\( T^{10} + \)\(38\!\cdots\!16\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!64\)\( T^{18} + \)\(25\!\cdots\!76\)\( T^{20} - \)\(33\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(42\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!16\)\( T^{28} - \)\(36\!\cdots\!96\)\( T^{30} + \)\(28\!\cdots\!57\)\( T^{32} - \)\(17\!\cdots\!40\)\( T^{34} + \)\(83\!\cdots\!34\)\( T^{36} - \)\(26\!\cdots\!36\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \)
$71$ \( ( 1 + 100156 T + 13448448446 T^{2} + 957445823975748 T^{3} + 76873855601451932317 T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!68\)\( T^{10} + \)\(51\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!74\)\( T^{12} + \)\(74\!\cdots\!92\)\( T^{13} + \)\(28\!\cdots\!28\)\( T^{14} + \)\(82\!\cdots\!20\)\( T^{15} + \)\(26\!\cdots\!17\)\( T^{16} + \)\(59\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} + \)\(20\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \)
$73$ \( ( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} + \)\(51\!\cdots\!28\)\( T^{11} + \)\(49\!\cdots\!42\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(15\!\cdots\!04\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{17} + \)\(31\!\cdots\!94\)\( T^{18} + \)\(37\!\cdots\!24\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \)
$79$ \( ( 1 - 141040 T + 30508439526 T^{2} - 3027236690742416 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} - \)\(29\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(73\!\cdots\!40\)\( T^{9} + \)\(47\!\cdots\!04\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(51\!\cdots\!84\)\( T^{13} + \)\(24\!\cdots\!48\)\( T^{14} - \)\(81\!\cdots\!12\)\( T^{15} + \)\(32\!\cdots\!93\)\( T^{16} - \)\(79\!\cdots\!84\)\( T^{17} + \)\(24\!\cdots\!26\)\( T^{18} - \)\(34\!\cdots\!60\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \)
$83$ \( 1 - 34768653380 T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!40\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{14} - \)\(64\!\cdots\!10\)\( T^{16} + \)\(49\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!44\)\( T^{20} + \)\(76\!\cdots\!80\)\( T^{22} - \)\(15\!\cdots\!10\)\( T^{24} - \)\(90\!\cdots\!64\)\( T^{26} + \)\(25\!\cdots\!40\)\( T^{28} - \)\(14\!\cdots\!80\)\( T^{30} + \)\(48\!\cdots\!33\)\( T^{32} - \)\(11\!\cdots\!44\)\( T^{34} + \)\(18\!\cdots\!58\)\( T^{36} - \)\(18\!\cdots\!20\)\( T^{38} + \)\(80\!\cdots\!01\)\( T^{40} \)
$89$ \( ( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(73\!\cdots\!60\)\( T^{9} + \)\(18\!\cdots\!64\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(91\!\cdots\!70\)\( T^{12} + \)\(16\!\cdots\!76\)\( T^{13} + \)\(37\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!28\)\( T^{15} + \)\(11\!\cdots\!93\)\( T^{16} + \)\(77\!\cdots\!16\)\( T^{17} + \)\(25\!\cdots\!46\)\( T^{18} + \)\(83\!\cdots\!20\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \)
$97$ \( ( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(82\!\cdots\!16\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} - \)\(55\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!52\)\( T^{14} - \)\(27\!\cdots\!96\)\( T^{15} + \)\(39\!\cdots\!25\)\( T^{16} - \)\(97\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!62\)\( T^{18} - \)\(18\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \)
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