Properties

Label 8.20
Level 8
Weight 20
Dimension 23
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 80
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 20 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(80\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_1(8))\).

Total New Old
Modular forms 41 25 16
Cusp forms 35 23 12
Eisenstein series 6 2 4

Trace form

\( 23q - 458q^{2} - 4180q^{3} - 412108q^{4} + 3366838q^{5} - 47240948q^{6} + 63655016q^{7} - 173313752q^{8} - 4808787337q^{9} + O(q^{10}) \) \( 23q - 458q^{2} - 4180q^{3} - 412108q^{4} + 3366838q^{5} - 47240948q^{6} + 63655016q^{7} - 173313752q^{8} - 4808787337q^{9} + 758800104q^{10} - 7461180572q^{11} - 9351642296q^{12} - 24989324626q^{13} - 122453668784q^{14} + 313775442840q^{15} - 696212072432q^{16} + 745408812286q^{17} + 1813497117770q^{18} + 91789194652q^{19} + 6517087595632q^{20} + 1413706100256q^{21} - 11074654117412q^{22} + 12366423799224q^{23} - 36473014189168q^{24} - 4283851921075q^{25} + 26782030269304q^{26} + 11408404421240q^{27} - 97002327802784q^{28} + 47417600266014q^{29} + 327847200544208q^{30} + 56684472288928q^{31} + 122449430282912q^{32} + 261008379402584q^{33} - 478448330325748q^{34} - 954185847061488q^{35} - 1053851053424436q^{36} + 1600526867615382q^{37} - 486194587539796q^{38} - 4629801688702440q^{39} - 1697818250610976q^{40} + 6559782861515446q^{41} + 4964410757677984q^{42} - 12296885808859324q^{43} + 7031781193616424q^{44} + 26569910848304302q^{45} - 14917443628457488q^{46} - 21582002902394640q^{47} - 3197964747991392q^{48} + 79583802656715631q^{49} - 34734151321861826q^{50} - 46515035377714280q^{51} - 26877456911772208q^{52} + 46649219708073478q^{53} + 32102189624395064q^{54} - 139301757145687448q^{55} + 52858270161183424q^{56} + 99401193783928920q^{57} - 20470056808312424q^{58} - 55774821261089228q^{59} - 25825294995349152q^{60} - 124702231193815042q^{61} + 93766686763697984q^{62} + 623085861964036072q^{63} - 253721318369185216q^{64} - 745666944375540604q^{65} + 479551889901302712q^{66} + 226785809908434796q^{67} + 201388986348307752q^{68} - 441174309724134304q^{69} - 340698473416023872q^{70} - 371529087787501144q^{71} - 115475454880420648q^{72} - 1032388677649304106q^{73} - 93558253821761240q^{74} + 3003253016266453684q^{75} - 1090379619519933176q^{76} - 2128959489278385312q^{77} + 2029949680340770544q^{78} + 3026712362616316816q^{79} + 3057120301750869696q^{80} + 280284797908382295q^{81} - 2418785089528546244q^{82} - 215922401337864868q^{83} + 141493317659087296q^{84} + 380436178852939660q^{85} + 1332333723895598620q^{86} - 14591356949907369192q^{87} - 3097741184498130608q^{88} + 9250198136648847142q^{89} - 288100905697646824q^{90} - 11324095727937747120q^{91} - 2965990502259463392q^{92} + 7763117681805520000q^{93} - 3533635112749329696q^{94} + 7889689928181619768q^{95} - 2990158800543707584q^{96} + 23827823451554462862q^{97} + 8841667592735478822q^{98} - 30782609165566913612q^{99} + O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.20.a \(\chi_{8}(1, \cdot)\) 8.20.a.a 2 1
8.20.a.b 3
8.20.b \(\chi_{8}(5, \cdot)\) 8.20.b.a 18 1

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 458 T + 310936 T^{2} + 168155968 T^{3} + 245186104320 T^{4} + 86379767824384 T^{5} + 97310951121879040 T^{6} - 18463126694689505280 T^{7} + \)\(21\!\cdots\!72\)\( T^{8} + \)\(27\!\cdots\!16\)\( T^{9} + \)\(11\!\cdots\!36\)\( T^{10} - \)\(50\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!80\)\( T^{12} + \)\(65\!\cdots\!24\)\( T^{13} + \)\(97\!\cdots\!60\)\( T^{14} + \)\(34\!\cdots\!12\)\( T^{15} + \)\(33\!\cdots\!12\)\( T^{16} + \)\(26\!\cdots\!68\)\( T^{17} + \)\(29\!\cdots\!48\)\( T^{18} \))
$3$ (\( 1 + 27912 T + 1180208070 T^{2} + 32441042066904 T^{3} + 1350851717672992089 T^{4} \))(\( 1 - 23732 T + 1701619137 T^{2} - 48097327472568 T^{3} + 1977726354444893979 T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(15\!\cdots\!63\)\( T^{6} \))(\( 1 - 7360989290 T^{2} + 28442776776606351153 T^{4} - \)\(76\!\cdots\!48\)\( T^{6} + \)\(15\!\cdots\!04\)\( T^{8} - \)\(27\!\cdots\!96\)\( T^{10} + \)\(39\!\cdots\!60\)\( T^{12} - \)\(51\!\cdots\!64\)\( T^{14} + \)\(62\!\cdots\!30\)\( T^{16} - \)\(72\!\cdots\!04\)\( T^{18} + \)\(84\!\cdots\!70\)\( T^{20} - \)\(94\!\cdots\!44\)\( T^{22} + \)\(98\!\cdots\!40\)\( T^{24} - \)\(90\!\cdots\!36\)\( T^{26} + \)\(71\!\cdots\!96\)\( T^{28} - \)\(46\!\cdots\!28\)\( T^{30} + \)\(23\!\cdots\!37\)\( T^{32} - \)\(81\!\cdots\!90\)\( T^{34} + \)\(14\!\cdots\!09\)\( T^{36} \))
$5$ (\( 1 - 1226620 T + 35930653639550 T^{2} - 23395919799804687500 T^{3} + \)\(36\!\cdots\!25\)\( T^{4} \))(\( 1 - 2140218 T - 5269684108605 T^{2} + 77660066501171142500 T^{3} - \)\(10\!\cdots\!25\)\( T^{4} - \)\(77\!\cdots\!50\)\( T^{5} + \)\(69\!\cdots\!25\)\( T^{6} \))(\( 1 - 149454139847258 T^{2} + \)\(11\!\cdots\!25\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{8} - \)\(83\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(61\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(28\!\cdots\!00\)\( T^{18} + \)\(50\!\cdots\!50\)\( T^{20} - \)\(81\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{24} - \)\(14\!\cdots\!00\)\( T^{26} + \)\(15\!\cdots\!00\)\( T^{28} - \)\(14\!\cdots\!00\)\( T^{30} + \)\(97\!\cdots\!25\)\( T^{32} - \)\(45\!\cdots\!50\)\( T^{34} + \)\(11\!\cdots\!25\)\( T^{36} \))
$7$ (\( 1 - 88510512 T + 11129657221091822 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \))(\( 1 - 55851720 T - 1162511437121979 T^{2} + \)\(11\!\cdots\!68\)\( T^{3} - \)\(13\!\cdots\!97\)\( T^{4} - \)\(72\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!07\)\( T^{6} \))(\( ( 1 + 40353608 T + 44216711939405711 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(86\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(20\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!02\)\( T^{8} - \)\(52\!\cdots\!24\)\( T^{9} + \)\(12\!\cdots\!86\)\( T^{10} - \)\(26\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!60\)\( T^{12} + \)\(26\!\cdots\!84\)\( T^{13} + \)\(16\!\cdots\!00\)\( T^{14} + \)\(34\!\cdots\!12\)\( T^{15} + \)\(11\!\cdots\!77\)\( T^{16} + \)\(11\!\cdots\!08\)\( T^{17} + \)\(32\!\cdots\!43\)\( T^{18} )^{2} \))
$11$ (\( 1 + 7163787608 T + 68277596800784135798 T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(37\!\cdots\!81\)\( T^{4} \))(\( 1 + 297392964 T + \)\(16\!\cdots\!17\)\( T^{2} + \)\(48\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(11\!\cdots\!84\)\( T^{5} + \)\(22\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(57\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!01\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!96\)\( T^{8} - \)\(52\!\cdots\!52\)\( T^{10} + \)\(51\!\cdots\!36\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(31\!\cdots\!66\)\( T^{16} - \)\(20\!\cdots\!12\)\( T^{18} + \)\(11\!\cdots\!46\)\( T^{20} - \)\(59\!\cdots\!20\)\( T^{22} + \)\(26\!\cdots\!76\)\( T^{24} - \)\(10\!\cdots\!92\)\( T^{26} + \)\(33\!\cdots\!96\)\( T^{28} - \)\(86\!\cdots\!80\)\( T^{30} + \)\(16\!\cdots\!61\)\( T^{32} - \)\(22\!\cdots\!62\)\( T^{34} + \)\(14\!\cdots\!21\)\( T^{36} \))
$13$ (\( 1 + 10126923604 T + \)\(29\!\cdots\!58\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!29\)\( T^{4} \))(\( 1 + 14862401022 T + \)\(61\!\cdots\!51\)\( T^{2} + \)\(91\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!27\)\( T^{4} + \)\(31\!\cdots\!38\)\( T^{5} + \)\(31\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!18\)\( T^{2} + \)\(98\!\cdots\!65\)\( T^{4} - \)\(46\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!52\)\( T^{8} - \)\(46\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!60\)\( T^{14} + \)\(38\!\cdots\!78\)\( T^{16} - \)\(60\!\cdots\!08\)\( T^{18} + \)\(83\!\cdots\!62\)\( T^{20} - \)\(10\!\cdots\!60\)\( T^{22} + \)\(10\!\cdots\!84\)\( T^{24} - \)\(97\!\cdots\!08\)\( T^{26} + \)\(73\!\cdots\!48\)\( T^{28} - \)\(44\!\cdots\!40\)\( T^{30} + \)\(20\!\cdots\!85\)\( T^{32} - \)\(60\!\cdots\!98\)\( T^{34} + \)\(93\!\cdots\!69\)\( T^{36} \))
$17$ (\( 1 + 72045078940 T + \)\(76\!\cdots\!06\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!09\)\( T^{4} \))(\( 1 - 803332464534 T + \)\(91\!\cdots\!99\)\( T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!47\)\( T^{4} - \)\(45\!\cdots\!06\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} \))(\( ( 1 - 7060713346 T + \)\(10\!\cdots\!61\)\( T^{2} - \)\(20\!\cdots\!68\)\( T^{3} + \)\(50\!\cdots\!40\)\( T^{4} - \)\(20\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!64\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(43\!\cdots\!74\)\( T^{8} - \)\(29\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!22\)\( T^{10} - \)\(57\!\cdots\!24\)\( T^{11} + \)\(22\!\cdots\!28\)\( T^{12} - \)\(67\!\cdots\!16\)\( T^{13} + \)\(39\!\cdots\!20\)\( T^{14} - \)\(38\!\cdots\!72\)\( T^{15} + \)\(45\!\cdots\!57\)\( T^{16} - \)\(75\!\cdots\!06\)\( T^{17} + \)\(25\!\cdots\!33\)\( T^{18} )^{2} \))
$19$ (\( 1 + 3120480472232 T + \)\(54\!\cdots\!14\)\( T^{2} + \)\(61\!\cdots\!28\)\( T^{3} + \)\(39\!\cdots\!41\)\( T^{4} \))(\( 1 - 3212269666884 T + \)\(86\!\cdots\!81\)\( T^{2} - \)\(13\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(77\!\cdots\!39\)\( T^{6} \))(\( 1 - \)\(20\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!73\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{6} + \)\(60\!\cdots\!32\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(65\!\cdots\!80\)\( T^{12} - \)\(16\!\cdots\!08\)\( T^{14} + \)\(37\!\cdots\!10\)\( T^{16} - \)\(76\!\cdots\!24\)\( T^{18} + \)\(14\!\cdots\!10\)\( T^{20} - \)\(25\!\cdots\!48\)\( T^{22} + \)\(39\!\cdots\!80\)\( T^{24} - \)\(51\!\cdots\!00\)\( T^{26} + \)\(55\!\cdots\!32\)\( T^{28} - \)\(46\!\cdots\!36\)\( T^{30} + \)\(28\!\cdots\!13\)\( T^{32} - \)\(11\!\cdots\!90\)\( T^{34} + \)\(21\!\cdots\!61\)\( T^{36} \))
$23$ (\( 1 + 14759207090288 T + \)\(15\!\cdots\!10\)\( T^{2} + \)\(11\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!69\)\( T^{4} \))(\( 1 - 24948509305560 T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(41\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!91\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!03\)\( T^{6} \))(\( ( 1 - 1088560791976 T + \)\(30\!\cdots\!15\)\( T^{2} - \)\(83\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!52\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(73\!\cdots\!88\)\( T^{6} - \)\(22\!\cdots\!32\)\( T^{7} + \)\(71\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!56\)\( T^{9} + \)\(53\!\cdots\!06\)\( T^{10} - \)\(12\!\cdots\!08\)\( T^{11} + \)\(30\!\cdots\!64\)\( T^{12} - \)\(51\!\cdots\!16\)\( T^{13} + \)\(13\!\cdots\!64\)\( T^{14} - \)\(14\!\cdots\!84\)\( T^{15} + \)\(39\!\cdots\!45\)\( T^{16} - \)\(10\!\cdots\!96\)\( T^{17} + \)\(71\!\cdots\!27\)\( T^{18} )^{2} \))
$29$ (\( 1 + 30249539245044 T + \)\(11\!\cdots\!22\)\( T^{2} + \)\(18\!\cdots\!36\)\( T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 - 77667139511058 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(90\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!47\)\( T^{4} - \)\(28\!\cdots\!38\)\( T^{5} + \)\(22\!\cdots\!09\)\( T^{6} \))(\( 1 - \)\(49\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{4} - \)\(21\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(28\!\cdots\!92\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{12} - \)\(18\!\cdots\!12\)\( T^{14} + \)\(12\!\cdots\!54\)\( T^{16} - \)\(81\!\cdots\!20\)\( T^{18} + \)\(48\!\cdots\!94\)\( T^{20} - \)\(26\!\cdots\!52\)\( T^{22} + \)\(12\!\cdots\!88\)\( T^{24} - \)\(54\!\cdots\!72\)\( T^{26} + \)\(19\!\cdots\!00\)\( T^{28} - \)\(56\!\cdots\!08\)\( T^{30} + \)\(12\!\cdots\!57\)\( T^{32} - \)\(18\!\cdots\!78\)\( T^{34} + \)\(13\!\cdots\!41\)\( T^{36} \))
$31$ (\( 1 + 123389562777920 T + \)\(27\!\cdots\!42\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!41\)\( T^{4} \))(\( 1 + 248431735193568 T + \)\(40\!\cdots\!53\)\( T^{2} + \)\(38\!\cdots\!56\)\( T^{3} + \)\(87\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!11\)\( T^{6} \))(\( ( 1 - 214252885130208 T + \)\(11\!\cdots\!47\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(67\!\cdots\!64\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(41\!\cdots\!96\)\( T^{7} + \)\(68\!\cdots\!54\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{9} + \)\(14\!\cdots\!34\)\( T^{10} - \)\(19\!\cdots\!36\)\( T^{11} + \)\(25\!\cdots\!92\)\( T^{12} - \)\(26\!\cdots\!96\)\( T^{13} + \)\(32\!\cdots\!64\)\( T^{14} - \)\(23\!\cdots\!84\)\( T^{15} + \)\(26\!\cdots\!77\)\( T^{16} - \)\(10\!\cdots\!88\)\( T^{17} + \)\(10\!\cdots\!31\)\( T^{18} )^{2} \))
$37$ (\( 1 - 2015393170174524 T + \)\(22\!\cdots\!90\)\( T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \))(\( 1 + 414866302559142 T + \)\(16\!\cdots\!15\)\( T^{2} + \)\(51\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!18\)\( T^{5} + \)\(24\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(65\!\cdots\!22\)\( T^{2} + \)\(21\!\cdots\!25\)\( T^{4} - \)\(47\!\cdots\!12\)\( T^{6} + \)\(78\!\cdots\!84\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{12} - \)\(10\!\cdots\!08\)\( T^{14} + \)\(81\!\cdots\!26\)\( T^{16} - \)\(54\!\cdots\!20\)\( T^{18} + \)\(31\!\cdots\!54\)\( T^{20} - \)\(15\!\cdots\!28\)\( T^{22} + \)\(67\!\cdots\!00\)\( T^{24} - \)\(24\!\cdots\!28\)\( T^{26} + \)\(71\!\cdots\!16\)\( T^{28} - \)\(16\!\cdots\!52\)\( T^{30} + \)\(29\!\cdots\!25\)\( T^{32} - \)\(35\!\cdots\!42\)\( T^{34} + \)\(21\!\cdots\!69\)\( T^{36} \))
$41$ (\( 1 - 2540784959504244 T + \)\(74\!\cdots\!06\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 - 2818737880869678 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(54\!\cdots\!38\)\( T^{5} + \)\(84\!\cdots\!81\)\( T^{6} \))(\( ( 1 - 600130010570762 T + \)\(20\!\cdots\!69\)\( T^{2} - \)\(37\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!88\)\( T^{4} - \)\(20\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!84\)\( T^{6} + \)\(23\!\cdots\!24\)\( T^{7} + \)\(95\!\cdots\!86\)\( T^{8} + \)\(14\!\cdots\!64\)\( T^{9} + \)\(42\!\cdots\!46\)\( T^{10} + \)\(44\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!04\)\( T^{12} - \)\(77\!\cdots\!84\)\( T^{13} + \)\(36\!\cdots\!88\)\( T^{14} - \)\(27\!\cdots\!68\)\( T^{15} + \)\(63\!\cdots\!49\)\( T^{16} - \)\(83\!\cdots\!22\)\( T^{17} + \)\(61\!\cdots\!41\)\( T^{18} )^{2} \))
$43$ (\( 1 + 5633655093389464 T + \)\(29\!\cdots\!38\)\( T^{2} + \)\(61\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))(\( 1 + 6663230715469860 T + \)\(43\!\cdots\!69\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(78\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(11\!\cdots\!10\)\( T^{2} + \)\(68\!\cdots\!69\)\( T^{4} - \)\(27\!\cdots\!64\)\( T^{6} + \)\(80\!\cdots\!92\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(36\!\cdots\!32\)\( T^{12} - \)\(59\!\cdots\!28\)\( T^{14} + \)\(81\!\cdots\!02\)\( T^{16} - \)\(96\!\cdots\!28\)\( T^{18} + \)\(96\!\cdots\!98\)\( T^{20} - \)\(82\!\cdots\!28\)\( T^{22} + \)\(60\!\cdots\!68\)\( T^{24} - \)\(36\!\cdots\!84\)\( T^{26} + \)\(18\!\cdots\!08\)\( T^{28} - \)\(72\!\cdots\!64\)\( T^{30} + \)\(21\!\cdots\!81\)\( T^{32} - \)\(43\!\cdots\!10\)\( T^{34} + \)\(44\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 + 21948339587130336 T + \)\(23\!\cdots\!90\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(34\!\cdots\!89\)\( T^{4} \))(\( 1 - 1500497644728624 T + \)\(13\!\cdots\!41\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(81\!\cdots\!03\)\( T^{4} - \)\(51\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!87\)\( T^{6} \))(\( ( 1 + 567080479996464 T + \)\(16\!\cdots\!87\)\( T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!28\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!36\)\( T^{6} - \)\(54\!\cdots\!48\)\( T^{7} + \)\(34\!\cdots\!90\)\( T^{8} + \)\(33\!\cdots\!44\)\( T^{9} + \)\(20\!\cdots\!70\)\( T^{10} - \)\(18\!\cdots\!72\)\( T^{11} + \)\(14\!\cdots\!32\)\( T^{12} - \)\(16\!\cdots\!76\)\( T^{13} + \)\(89\!\cdots\!04\)\( T^{14} - \)\(46\!\cdots\!36\)\( T^{15} + \)\(41\!\cdots\!49\)\( T^{16} + \)\(81\!\cdots\!24\)\( T^{17} + \)\(84\!\cdots\!03\)\( T^{18} )^{2} \))
$53$ (\( 1 + 9418125066904676 T + \)\(29\!\cdots\!78\)\( T^{2} + \)\(54\!\cdots\!92\)\( T^{3} + \)\(33\!\cdots\!89\)\( T^{4} \))(\( 1 - 56067344774978154 T + \)\(25\!\cdots\!23\)\( T^{2} - \)\(66\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!91\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(19\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(65\!\cdots\!74\)\( T^{2} + \)\(21\!\cdots\!45\)\( T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(79\!\cdots\!52\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{12} - \)\(94\!\cdots\!68\)\( T^{14} + \)\(69\!\cdots\!22\)\( T^{16} - \)\(43\!\cdots\!92\)\( T^{18} + \)\(23\!\cdots\!58\)\( T^{20} - \)\(10\!\cdots\!28\)\( T^{22} + \)\(39\!\cdots\!12\)\( T^{24} - \)\(12\!\cdots\!08\)\( T^{26} + \)\(32\!\cdots\!48\)\( T^{28} - \)\(65\!\cdots\!44\)\( T^{30} + \)\(99\!\cdots\!05\)\( T^{32} - \)\(99\!\cdots\!94\)\( T^{34} + \)\(50\!\cdots\!09\)\( T^{36} \))
$59$ (\( 1 - 98542449590407624 T + \)\(95\!\cdots\!22\)\( T^{2} - \)\(43\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 + 154317270851496852 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!83\)\( T^{4} + \)\(30\!\cdots\!92\)\( T^{5} + \)\(86\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(45\!\cdots\!86\)\( T^{2} + \)\(10\!\cdots\!73\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{8} - \)\(18\!\cdots\!24\)\( T^{10} + \)\(14\!\cdots\!08\)\( T^{12} - \)\(92\!\cdots\!00\)\( T^{14} + \)\(51\!\cdots\!06\)\( T^{16} - \)\(24\!\cdots\!76\)\( T^{18} + \)\(10\!\cdots\!26\)\( T^{20} - \)\(35\!\cdots\!00\)\( T^{22} + \)\(10\!\cdots\!88\)\( T^{24} - \)\(27\!\cdots\!44\)\( T^{26} + \)\(56\!\cdots\!88\)\( T^{28} - \)\(94\!\cdots\!68\)\( T^{30} + \)\(11\!\cdots\!93\)\( T^{32} - \)\(99\!\cdots\!46\)\( T^{34} + \)\(42\!\cdots\!81\)\( T^{36} \))
$61$ (\( 1 - 10292145377839820 T + \)\(11\!\cdots\!82\)\( T^{2} - \)\(85\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!81\)\( T^{4} \))(\( 1 + 134994376571654862 T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(93\!\cdots\!22\)\( T^{5} + \)\(58\!\cdots\!21\)\( T^{6} \))(\( 1 - \)\(87\!\cdots\!86\)\( T^{2} + \)\(38\!\cdots\!13\)\( T^{4} - \)\(11\!\cdots\!68\)\( T^{6} + \)\(24\!\cdots\!08\)\( T^{8} - \)\(41\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!08\)\( T^{12} - \)\(70\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!46\)\( T^{16} - \)\(65\!\cdots\!16\)\( T^{18} + \)\(50\!\cdots\!26\)\( T^{20} - \)\(34\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!28\)\( T^{24} - \)\(97\!\cdots\!64\)\( T^{26} + \)\(39\!\cdots\!08\)\( T^{28} - \)\(12\!\cdots\!08\)\( T^{30} + \)\(30\!\cdots\!93\)\( T^{32} - \)\(48\!\cdots\!26\)\( T^{34} + \)\(38\!\cdots\!21\)\( T^{36} \))
$67$ (\( 1 - 75753628003984504 T + \)\(44\!\cdots\!10\)\( T^{2} - \)\(37\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!09\)\( T^{4} \))(\( 1 - 151032181904450292 T + \)\(12\!\cdots\!49\)\( T^{2} - \)\(11\!\cdots\!96\)\( T^{3} + \)\(62\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \))(\( 1 - \)\(58\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(32\!\cdots\!76\)\( T^{6} + \)\(45\!\cdots\!32\)\( T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(44\!\cdots\!28\)\( T^{12} - \)\(33\!\cdots\!52\)\( T^{14} + \)\(20\!\cdots\!22\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{18} + \)\(51\!\cdots\!98\)\( T^{20} - \)\(20\!\cdots\!12\)\( T^{22} + \)\(66\!\cdots\!12\)\( T^{24} - \)\(18\!\cdots\!12\)\( T^{26} + \)\(40\!\cdots\!68\)\( T^{28} - \)\(71\!\cdots\!16\)\( T^{30} + \)\(92\!\cdots\!25\)\( T^{32} - \)\(78\!\cdots\!66\)\( T^{34} + \)\(32\!\cdots\!89\)\( T^{36} \))
$71$ (\( 1 - 17407052566713776 T + \)\(27\!\cdots\!06\)\( T^{2} - \)\(25\!\cdots\!56\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 - 1210541848845584136 T + \)\(85\!\cdots\!77\)\( T^{2} - \)\(39\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!87\)\( T^{4} - \)\(26\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!91\)\( T^{6} \))(\( ( 1 + 799738994599899528 T + \)\(77\!\cdots\!91\)\( T^{2} + \)\(47\!\cdots\!52\)\( T^{3} + \)\(27\!\cdots\!80\)\( T^{4} + \)\(13\!\cdots\!68\)\( T^{5} + \)\(64\!\cdots\!44\)\( T^{6} + \)\(28\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!54\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(17\!\cdots\!74\)\( T^{10} + \)\(62\!\cdots\!72\)\( T^{11} + \)\(21\!\cdots\!04\)\( T^{12} + \)\(69\!\cdots\!28\)\( T^{13} + \)\(20\!\cdots\!80\)\( T^{14} + \)\(52\!\cdots\!12\)\( T^{15} + \)\(12\!\cdots\!01\)\( T^{16} + \)\(19\!\cdots\!48\)\( T^{17} + \)\(36\!\cdots\!71\)\( T^{18} )^{2} \))
$73$ (\( 1 + 857508255059832268 T + \)\(53\!\cdots\!30\)\( T^{2} + \)\(21\!\cdots\!16\)\( T^{3} + \)\(64\!\cdots\!69\)\( T^{4} \))(\( 1 + 81876123599662770 T + \)\(39\!\cdots\!23\)\( T^{2} + \)\(38\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!51\)\( T^{4} + \)\(52\!\cdots\!30\)\( T^{5} + \)\(16\!\cdots\!53\)\( T^{6} \))(\( ( 1 + 46502149494904534 T + \)\(10\!\cdots\!01\)\( T^{2} + \)\(15\!\cdots\!08\)\( T^{3} + \)\(59\!\cdots\!96\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!36\)\( T^{6} + \)\(49\!\cdots\!92\)\( T^{7} + \)\(77\!\cdots\!02\)\( T^{8} + \)\(13\!\cdots\!04\)\( T^{9} + \)\(19\!\cdots\!74\)\( T^{10} + \)\(31\!\cdots\!48\)\( T^{11} + \)\(38\!\cdots\!08\)\( T^{12} + \)\(50\!\cdots\!84\)\( T^{13} + \)\(62\!\cdots\!72\)\( T^{14} + \)\(40\!\cdots\!72\)\( T^{15} + \)\(72\!\cdots\!33\)\( T^{16} + \)\(78\!\cdots\!14\)\( T^{17} + \)\(42\!\cdots\!77\)\( T^{18} )^{2} \))
$79$ (\( 1 + 226291921444855072 T + \)\(21\!\cdots\!34\)\( T^{2} + \)\(25\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} \))(\( 1 - 1439028483035907408 T + \)\(10\!\cdots\!53\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!59\)\( T^{6} \))(\( ( 1 - 906987900512632240 T + \)\(41\!\cdots\!87\)\( T^{2} - \)\(30\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!40\)\( T^{4} - \)\(82\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!88\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!58\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(35\!\cdots\!02\)\( T^{10} - \)\(17\!\cdots\!20\)\( T^{11} + \)\(31\!\cdots\!92\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{13} + \)\(21\!\cdots\!60\)\( T^{14} - \)\(65\!\cdots\!84\)\( T^{15} + \)\(10\!\cdots\!93\)\( T^{16} - \)\(24\!\cdots\!40\)\( T^{17} + \)\(31\!\cdots\!79\)\( T^{18} )^{2} \))
$83$ (\( 1 - 767515701460985048 T + \)\(59\!\cdots\!70\)\( T^{2} - \)\(22\!\cdots\!56\)\( T^{3} + \)\(84\!\cdots\!09\)\( T^{4} \))(\( 1 + 983438102798849916 T + \)\(41\!\cdots\!01\)\( T^{2} + \)\(28\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!47\)\( T^{4} + \)\(82\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!23\)\( T^{6} \))(\( 1 - \)\(27\!\cdots\!10\)\( T^{2} + \)\(37\!\cdots\!69\)\( T^{4} - \)\(34\!\cdots\!04\)\( T^{6} + \)\(24\!\cdots\!12\)\( T^{8} - \)\(13\!\cdots\!24\)\( T^{10} + \)\(61\!\cdots\!92\)\( T^{12} - \)\(24\!\cdots\!88\)\( T^{14} + \)\(84\!\cdots\!82\)\( T^{16} - \)\(25\!\cdots\!08\)\( T^{18} + \)\(71\!\cdots\!38\)\( T^{20} - \)\(17\!\cdots\!28\)\( T^{22} + \)\(36\!\cdots\!68\)\( T^{24} - \)\(67\!\cdots\!64\)\( T^{26} + \)\(10\!\cdots\!88\)\( T^{28} - \)\(12\!\cdots\!64\)\( T^{30} + \)\(11\!\cdots\!61\)\( T^{32} - \)\(68\!\cdots\!10\)\( T^{34} + \)\(21\!\cdots\!89\)\( T^{36} \))
$89$ (\( 1 - 6092545894435174548 T + \)\(31\!\cdots\!94\)\( T^{2} - \)\(66\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))(\( 1 + 1312857528832070946 T + \)\(25\!\cdots\!31\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!79\)\( T^{4} + \)\(15\!\cdots\!26\)\( T^{5} + \)\(13\!\cdots\!29\)\( T^{6} \))(\( ( 1 - 2235254885522871770 T + \)\(41\!\cdots\!77\)\( T^{2} - \)\(60\!\cdots\!24\)\( T^{3} + \)\(86\!\cdots\!00\)\( T^{4} - \)\(68\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} - \)\(38\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(17\!\cdots\!82\)\( T^{10} - \)\(45\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!32\)\( T^{12} - \)\(96\!\cdots\!16\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} - \)\(10\!\cdots\!84\)\( T^{15} + \)\(77\!\cdots\!13\)\( T^{16} - \)\(45\!\cdots\!70\)\( T^{17} + \)\(22\!\cdots\!89\)\( T^{18} )^{2} \))
$97$ (\( 1 - 1548148249522347076 T + \)\(66\!\cdots\!10\)\( T^{2} - \)\(86\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \))(\( 1 - 14033245412567998566 T + \)\(22\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!74\)\( T^{5} + \)\(17\!\cdots\!37\)\( T^{6} \))(\( ( 1 - 4123214894732058610 T + \)\(14\!\cdots\!05\)\( T^{2} - \)\(82\!\cdots\!32\)\( T^{3} + \)\(90\!\cdots\!72\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(42\!\cdots\!92\)\( T^{6} - \)\(76\!\cdots\!44\)\( T^{7} + \)\(29\!\cdots\!38\)\( T^{8} - \)\(14\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!54\)\( T^{10} - \)\(24\!\cdots\!16\)\( T^{11} + \)\(75\!\cdots\!04\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(50\!\cdots\!96\)\( T^{14} - \)\(25\!\cdots\!08\)\( T^{15} + \)\(26\!\cdots\!85\)\( T^{16} - \)\(40\!\cdots\!10\)\( T^{17} + \)\(54\!\cdots\!53\)\( T^{18} )^{2} \))
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