Properties

Label 8.14
Level 8
Weight 14
Dimension 15
Nonzero newspaces 2
Newform subspaces 4
Sturm bound 56
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 4 \)
Sturm bound: \(56\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{14}(\Gamma_1(8))\).

Total New Old
Modular forms 29 17 12
Cusp forms 23 15 8
Eisenstein series 6 2 4

Trace form

\( 15q - 2q^{2} + 860q^{3} - 8556q^{4} + 14146q^{5} + 58444q^{6} + 206232q^{7} + 1076872q^{8} - 3319093q^{9} + O(q^{10}) \) \( 15q - 2q^{2} + 860q^{3} - 8556q^{4} + 14146q^{5} + 58444q^{6} + 206232q^{7} + 1076872q^{8} - 3319093q^{9} - 7752648q^{10} + 9989716q^{11} - 8536664q^{12} - 3843462q^{13} + 21101872q^{14} + 42509064q^{15} - 62847216q^{16} - 227240498q^{17} + 262778834q^{18} + 206597196q^{19} + 381237904q^{20} - 1370300832q^{21} + 196775484q^{22} + 1490018760q^{23} + 1739438480q^{24} - 4205723223q^{25} - 685452248q^{26} + 5792047640q^{27} - 978617568q^{28} - 3556537878q^{29} + 539961392q^{30} - 7367666592q^{31} - 8956909792q^{32} + 14055767168q^{33} + 8857451100q^{34} - 15413225904q^{35} + 18075363660q^{36} + 6628831266q^{37} - 23710846420q^{38} + 20630243496q^{39} - 6505554528q^{40} + 47965956118q^{41} - 3443304224q^{42} - 46788121548q^{43} - 28647791928q^{44} + 107010747322q^{45} - 51061300848q^{46} - 201450140016q^{47} - 161587694304q^{48} + 101164221495q^{49} + 246819442102q^{50} + 177530164600q^{51} + 441779820720q^{52} - 125287225454q^{53} - 639572716168q^{54} + 217008668376q^{55} - 699100837568q^{56} - 1007795333472q^{57} + 992195908104q^{58} + 793560430372q^{59} + 1614284565792q^{60} - 698520953142q^{61} - 1019628353344q^{62} - 426693206408q^{63} - 1464077986752q^{64} - 357709159492q^{65} + 2946347707608q^{66} + 1645513736988q^{67} + 2850998874984q^{68} + 1015234563872q^{69} - 5505591528768q^{70} - 4320352976296q^{71} - 8538615449032q^{72} + 3510289845750q^{73} + 9399657781240q^{74} + 844579119460q^{75} + 10892359157736q^{76} + 1067266262304q^{77} - 16235130104944q^{78} - 1836704085264q^{79} - 16787669526720q^{80} + 3452732525415q^{81} + 14910525285612q^{82} + 3890337600524q^{83} + 26784826987072q^{84} + 1613607983652q^{85} - 24718490913284q^{86} - 14659545283608q^{87} - 26046521626416q^{88} - 745298334170q^{89} + 43218339424520q^{90} + 19047242532624q^{91} + 34108342295904q^{92} - 6946080702080q^{93} - 37727942061792q^{94} - 25493775576920q^{95} - 62733552180160q^{96} - 15392053442370q^{97} + 61714328751438q^{98} + 42866396476420q^{99} + O(q^{100}) \)

Decomposition of \(S_{14}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.14.a \(\chi_{8}(1, \cdot)\) 8.14.a.a 1 1
8.14.a.b 2
8.14.b \(\chi_{8}(5, \cdot)\) 8.14.b.a 2 1
8.14.b.b 10

Decomposition of \(S_{14}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{14}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{14}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{14}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( 1 + 112 T + 8192 T^{2} \))(\( 1 - 110 T + 8408 T^{2} - 390976 T^{3} - 935936 T^{4} + 3612475392 T^{5} - 7667187712 T^{6} - 26237955211264 T^{7} + 4622346883170304 T^{8} - 495395959010754560 T^{9} + 36893488147419103232 T^{10} \))
$3$ (\( 1 + 12 T + 1594323 T^{2} \))(\( 1 - 872 T + 179766 T^{2} - 1390249656 T^{3} + 2541865828329 T^{4} \))(\( 1 + 2069910 T^{2} + 2541865828329 T^{4} \))(\( 1 - 8978642 T^{2} + 41923886773365 T^{4} - \)\(13\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!22\)\( T^{8} - \)\(56\!\cdots\!92\)\( T^{10} + \)\(78\!\cdots\!38\)\( T^{12} - \)\(85\!\cdots\!12\)\( T^{14} + \)\(68\!\cdots\!85\)\( T^{16} - \)\(37\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))
$5$ (\( 1 + 4330 T + 1220703125 T^{2} \))(\( 1 - 18476 T + 2066094350 T^{2} - 22553710937500 T^{3} + 1490116119384765625 T^{4} \))(\( 1 - 1931729850 T^{2} + 1490116119384765625 T^{4} \))(\( 1 - 4565314338 T^{2} + 10147469641592691461 T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!50\)\( T^{12} - \)\(32\!\cdots\!00\)\( T^{14} + \)\(33\!\cdots\!25\)\( T^{16} - \)\(22\!\cdots\!50\)\( T^{18} + \)\(73\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 + 139992 T + 96889010407 T^{2} \))(\( 1 - 110928 T + 39195942926 T^{2} - 10747704146427696 T^{3} + \)\(93\!\cdots\!49\)\( T^{4} \))(\( ( 1 + 175832 T + 96889010407 T^{2} )^{2} \))(\( ( 1 - 293480 T + 239610643203 T^{2} - 95417056996483936 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!28\)\( T^{5} + \)\(31\!\cdots\!18\)\( T^{6} - \)\(89\!\cdots\!64\)\( T^{7} + \)\(21\!\cdots\!29\)\( T^{8} - \)\(25\!\cdots\!80\)\( T^{9} + \)\(85\!\cdots\!07\)\( T^{10} )^{2} \))
$11$ (\( 1 + 6484324 T + 34522712143931 T^{2} \))(\( 1 - 16474040 T + 135991936296326 T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - 62100824999962 T^{2} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 - 179340783809858 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!78\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{10} + \)\(60\!\cdots\!58\)\( T^{12} - \)\(14\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!21\)\( T^{16} - \)\(36\!\cdots\!78\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 + 22588034 T + 302875106592253 T^{2} \))(\( 1 - 18744572 T + 344120051417598 T^{2} - \)\(56\!\cdots\!16\)\( T^{3} + \)\(91\!\cdots\!09\)\( T^{4} \))(\( 1 + 360392330876150 T^{2} + \)\(91\!\cdots\!09\)\( T^{4} \))(\( 1 - 1881700477395890 T^{2} + \)\(17\!\cdots\!13\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(48\!\cdots\!02\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!18\)\( T^{12} - \)\(91\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!77\)\( T^{16} - \)\(13\!\cdots\!90\)\( T^{18} + \)\(64\!\cdots\!49\)\( T^{20} \))
$17$ (\( 1 + 23732270 T + 9904578032905937 T^{2} \))(\( 1 + 153793628 T + 21058111940247014 T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(98\!\cdots\!69\)\( T^{4} \))(\( ( 1 + 133520302 T + 9904578032905937 T^{2} )^{2} \))(\( ( 1 - 108663002 T + 32276400633367229 T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!42\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!54\)\( T^{6} - \)\(23\!\cdots\!96\)\( T^{7} + \)\(31\!\cdots\!37\)\( T^{8} - \)\(10\!\cdots\!22\)\( T^{9} + \)\(95\!\cdots\!57\)\( T^{10} )^{2} \))
$19$ (\( 1 - 325344836 T + 42052983462257059 T^{2} \))(\( 1 + 118747640 T + 59473815443581782 T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 82896428399326282 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \))(\( 1 - 162780863064290354 T^{2} + \)\(13\!\cdots\!97\)\( T^{4} - \)\(80\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!66\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(69\!\cdots\!46\)\( T^{12} - \)\(25\!\cdots\!96\)\( T^{14} + \)\(75\!\cdots\!77\)\( T^{16} - \)\(15\!\cdots\!34\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \))
$23$ (\( 1 - 921600632 T + 504036361936467383 T^{2} \))(\( 1 - 718268912 T + 1097306915486653358 T^{2} - \)\(36\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!89\)\( T^{4} \))(\( ( 1 + 35585416 T + 504036361936467383 T^{2} )^{2} \))(\( ( 1 + 39339976 T + 1242751119340586771 T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(76\!\cdots\!02\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(38\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(15\!\cdots\!77\)\( T^{8} + \)\(25\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!43\)\( T^{10} )^{2} \))
$29$ (\( 1 + 3865879218 T + 10260628712958602189 T^{2} \))(\( 1 - 309341340 T + 2504177667132675934 T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 18002148940349036842 T^{2} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 60047671171855484498 T^{2} + \)\(18\!\cdots\!41\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(58\!\cdots\!78\)\( T^{8} - \)\(67\!\cdots\!84\)\( T^{10} + \)\(61\!\cdots\!38\)\( T^{12} - \)\(42\!\cdots\!60\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} - \)\(73\!\cdots\!38\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$31$ (\( 1 + 2253401440 T + 24417546297445042591 T^{2} \))(\( 1 - 5767504192 T + 50116253247327696702 T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(59\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2} \))(\( ( 1 - 324116896 T + 94759416869201467419 T^{2} - \)\(76\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(32\!\cdots\!48\)\( T^{5} + \)\(98\!\cdots\!78\)\( T^{6} - \)\(45\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!49\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(86\!\cdots\!51\)\( T^{10} )^{2} \))
$37$ (\( 1 - 18250384566 T + \)\(24\!\cdots\!97\)\( T^{2} \))(\( 1 + 11621553300 T + \)\(18\!\cdots\!38\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(31\!\cdots\!70\)\( T^{2} + \)\(59\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(85\!\cdots\!14\)\( T^{2} + \)\(35\!\cdots\!29\)\( T^{4} - \)\(83\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!64\)\( T^{10} + \)\(67\!\cdots\!02\)\( T^{12} - \)\(29\!\cdots\!44\)\( T^{14} + \)\(73\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!54\)\( T^{18} + \)\(73\!\cdots\!49\)\( T^{20} \))
$41$ (\( 1 - 34422845322 T + \)\(92\!\cdots\!21\)\( T^{2} \))(\( 1 - 1311168276 T + \)\(16\!\cdots\!30\)\( T^{2} - \)\(12\!\cdots\!96\)\( T^{3} + \)\(85\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 23546348918 T + \)\(92\!\cdots\!21\)\( T^{2} )^{2} \))(\( ( 1 - 29662320178 T + \)\(14\!\cdots\!89\)\( T^{2} - \)\(15\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(97\!\cdots\!02\)\( T^{6} - \)\(13\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!29\)\( T^{8} - \)\(21\!\cdots\!18\)\( T^{9} + \)\(67\!\cdots\!01\)\( T^{10} )^{2} \))
$43$ (\( 1 + 17192501444 T + \)\(17\!\cdots\!43\)\( T^{2} \))(\( 1 + 29595620104 T + \)\(33\!\cdots\!74\)\( T^{2} + \)\(50\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(32\!\cdots\!30\)\( T^{2} + \)\(29\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(61\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!05\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{6} + \)\(52\!\cdots\!62\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!38\)\( T^{12} - \)\(22\!\cdots\!04\)\( T^{14} + \)\(39\!\cdots\!45\)\( T^{16} - \)\(46\!\cdots\!14\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \))
$47$ (\( 1 + 67371749904 T + \)\(54\!\cdots\!27\)\( T^{2} \))(\( 1 - 12313617888 T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(67\!\cdots\!76\)\( T^{3} + \)\(29\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 68107736592 T + \)\(54\!\cdots\!27\)\( T^{2} )^{2} \))(\( ( 1 + 5088267408 T + \)\(10\!\cdots\!43\)\( T^{2} + \)\(16\!\cdots\!28\)\( T^{3} + \)\(79\!\cdots\!70\)\( T^{4} + \)\(90\!\cdots\!04\)\( T^{5} + \)\(43\!\cdots\!90\)\( T^{6} + \)\(48\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!69\)\( T^{8} + \)\(45\!\cdots\!28\)\( T^{9} + \)\(48\!\cdots\!07\)\( T^{10} )^{2} \))
$53$ (\( 1 + 87281218426 T + \)\(26\!\cdots\!73\)\( T^{2} \))(\( 1 + 38006007028 T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(98\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(15\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!49\)\( T^{4} - \)\(63\!\cdots\!52\)\( T^{6} + \)\(24\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!20\)\( T^{10} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{14} + \)\(37\!\cdots\!61\)\( T^{16} - \)\(32\!\cdots\!38\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} \))
$59$ (\( 1 - 540214518668 T + \)\(10\!\cdots\!79\)\( T^{2} \))(\( 1 - 253345911704 T + \)\(16\!\cdots\!06\)\( T^{2} - \)\(26\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!22\)\( T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(54\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!77\)\( T^{4} - \)\(32\!\cdots\!52\)\( T^{6} + \)\(49\!\cdots\!86\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} + \)\(54\!\cdots\!26\)\( T^{12} - \)\(39\!\cdots\!12\)\( T^{14} + \)\(21\!\cdots\!17\)\( T^{16} - \)\(80\!\cdots\!02\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 + 51276568850 T + \)\(16\!\cdots\!81\)\( T^{2} \))(\( 1 + 647244384292 T + \)\(42\!\cdots\!62\)\( T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(14\!\cdots\!62\)\( T^{2} + \)\(26\!\cdots\!61\)\( T^{4} \))(\( 1 - \)\(10\!\cdots\!62\)\( T^{2} + \)\(47\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{6} + \)\(29\!\cdots\!66\)\( T^{8} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(77\!\cdots\!26\)\( T^{12} - \)\(96\!\cdots\!52\)\( T^{14} + \)\(86\!\cdots\!97\)\( T^{16} - \)\(48\!\cdots\!42\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \))
$67$ (\( 1 - 25519930676 T + \)\(54\!\cdots\!87\)\( T^{2} \))(\( 1 - 1619993806312 T + \)\(17\!\cdots\!86\)\( T^{2} - \)\(88\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(95\!\cdots\!30\)\( T^{2} + \)\(30\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!42\)\( T^{2} + \)\(23\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(71\!\cdots\!60\)\( T^{10} + \)\(37\!\cdots\!34\)\( T^{12} - \)\(17\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!41\)\( T^{16} - \)\(16\!\cdots\!82\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} \))
$71$ (\( 1 + 1387500699032 T + \)\(11\!\cdots\!11\)\( T^{2} \))(\( 1 + 1040270142512 T + \)\(22\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 1309471657368 T + \)\(11\!\cdots\!11\)\( T^{2} )^{2} \))(\( ( 1 - 363180589992 T + \)\(34\!\cdots\!95\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(61\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!34\)\( T^{6} - \)\(20\!\cdots\!36\)\( T^{7} + \)\(53\!\cdots\!45\)\( T^{8} - \)\(66\!\cdots\!72\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} )^{2} \))
$73$ (\( 1 + 819049441238 T + \)\(16\!\cdots\!33\)\( T^{2} \))(\( 1 - 4005283908692 T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))(\( ( 1 - 478647871914 T + \)\(16\!\cdots\!33\)\( T^{2} )^{2} \))(\( ( 1 + 316620182766 T + \)\(64\!\cdots\!53\)\( T^{2} + \)\(24\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(60\!\cdots\!52\)\( T^{5} + \)\(31\!\cdots\!82\)\( T^{6} + \)\(67\!\cdots\!76\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} + \)\(24\!\cdots\!86\)\( T^{9} + \)\(13\!\cdots\!93\)\( T^{10} )^{2} \))
$79$ (\( 1 + 4030935615344 T + \)\(46\!\cdots\!39\)\( T^{2} \))(\( 1 + 2521777572064 T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!21\)\( T^{4} \))(\( ( 1 + 364547231600 T + \)\(46\!\cdots\!39\)\( T^{2} )^{2} \))(\( ( 1 - 2722551782672 T + \)\(12\!\cdots\!19\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(55\!\cdots\!14\)\( T^{4} - \)\(94\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!46\)\( T^{6} - \)\(49\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!99\)\( T^{10} )^{2} \))
$83$ (\( 1 - 4180823831428 T + \)\(88\!\cdots\!63\)\( T^{2} \))(\( 1 + 290486230904 T + \)\(14\!\cdots\!54\)\( T^{2} + \)\(25\!\cdots\!52\)\( T^{3} + \)\(78\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!70\)\( T^{2} + \)\(78\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(43\!\cdots\!74\)\( T^{2} + \)\(87\!\cdots\!65\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(76\!\cdots\!04\)\( T^{14} + \)\(42\!\cdots\!85\)\( T^{16} - \)\(16\!\cdots\!54\)\( T^{18} + \)\(30\!\cdots\!49\)\( T^{20} \))
$89$ (\( 1 - 2677027798266 T + \)\(21\!\cdots\!69\)\( T^{2} \))(\( 1 + 8723755657740 T + \)\(61\!\cdots\!42\)\( T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + \)\(48\!\cdots\!61\)\( T^{4} \))(\( ( 1 + 102457641350 T + \)\(21\!\cdots\!69\)\( T^{2} )^{2} \))(\( ( 1 - 2753172404002 T + \)\(62\!\cdots\!29\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!48\)\( T^{5} + \)\(47\!\cdots\!26\)\( T^{6} - \)\(70\!\cdots\!24\)\( T^{7} + \)\(66\!\cdots\!61\)\( T^{8} - \)\(64\!\cdots\!42\)\( T^{9} + \)\(51\!\cdots\!49\)\( T^{10} )^{2} \))
$97$ (\( 1 + 14039464316446 T + \)\(67\!\cdots\!77\)\( T^{2} \))(\( 1 - 9601712299972 T + \)\(81\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(45\!\cdots\!29\)\( T^{4} \))(\( ( 1 + 6157717373342 T + \)\(67\!\cdots\!77\)\( T^{2} )^{2} \))(\( ( 1 - 680566660394 T + \)\(17\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!90\)\( T^{6} + \)\(53\!\cdots\!32\)\( T^{7} + \)\(54\!\cdots\!49\)\( T^{8} - \)\(13\!\cdots\!54\)\( T^{9} + \)\(13\!\cdots\!57\)\( T^{10} )^{2} \))
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