Properties

Label 3.42
Level 3
Weight 42
Dimension 7
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 28
Trace bound 0

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 42 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_1(3))\).

Total New Old
Modular forms 15 7 8
Cusp forms 13 7 6
Eisenstein series 2 0 2

Trace form

\( 7q - 359202q^{2} + 3486784401q^{3} + 3098681410132q^{4} + 157187509968306q^{5} + 765551409514758q^{6} + 105654053587518368q^{7} + 8547659196809395080q^{8} + 85103658213398501607q^{9} + O(q^{10}) \) \( 7q - 359202q^{2} + 3486784401q^{3} + 3098681410132q^{4} + 157187509968306q^{5} + 765551409514758q^{6} + 105654053587518368q^{7} + 8547659196809395080q^{8} + 85103658213398501607q^{9} + 1029435805124780542164q^{10} + 2887214225248717473660q^{11} + 26524714300400374748532q^{12} + 35871439840874147686034q^{13} + 814121134383105838377552q^{14} + 278546714684548950421854q^{15} - 11421429587339909471968496q^{16} - 49221515452774894676004402q^{17} - 4367057748224166939176802q^{18} + 458239816960683610422662276q^{19} + 1317078708802896919455800184q^{20} + 679429495183093137017433504q^{21} - 10835653338027404099619293304q^{22} - 20060723551274129388995858616q^{23} + 13987560939606541875202583688q^{24} + 112741912197127321489742871001q^{25} + 59951728496322544602361807524q^{26} + 42391158275216203514294433201q^{27} + 84612644924529015441375525536q^{28} - 1977654451176175768473033328278q^{29} + 1471653955084285604978002090116q^{30} + 2800527890869229757891015241496q^{31} + 10451255461999993483707442222368q^{32} - 5012576606051926011540924758748q^{33} + 42009179670420411943699963793244q^{34} + 54061742342062084106633253973440q^{35} + 37672731948583633247947804011732q^{36} - 242065375197179473114939693545142q^{37} + 314930001138451392992297251383480q^{38} - 186750060884734904297998103261250q^{39} + 1734682904709656035688674311540528q^{40} + 1919772488651818449733020167096406q^{41} + 602440311893806808992036727951184q^{42} + 2632359378218766285475830966693884q^{43} - 19937525714249582008307420905778832q^{44} + 1911033160536840537417125208581106q^{45} - 36254649124121789242053962151968304q^{46} + 55633854121325526194894129703603648q^{47} - 1753714286036360588361388333995504q^{48} + 138035826290957567679458976119440143q^{49} - 187990375985029453423588111219375326q^{50} - 94985755416285397758254198036972574q^{51} - 617218879754977585961579952999168232q^{52} + 130398746597587280084701480383481074q^{53} + 9307317928589919211191457164745158q^{54} + 1391092265182485973540701545396386056q^{55} + 1372448310706724961467332343944230720q^{56} + 228194964739280558656530161545568220q^{57} + 1118358566329856274942833707260398820q^{58} - 6391974776969397351480135095996322756q^{59} + 884963814320884755837398974113114936q^{60} - 3312712827218201776253073157251557422q^{61} + 14374498612437137001029858555251499616q^{62} + 1284506637910321854753178081255716768q^{63} + 16591897231730805285648730656448281664q^{64} - 7033288095370619074610943489877063332q^{65} - 6306358621743532720405111555196485848q^{66} - 40405345193781585575277868138739047612q^{67} - 55757692275676972944877351967679897624q^{68} - 37414675978671770942781643564589712648q^{69} - 28830724918545442617573855638025077280q^{70} + 65352753330602401579129433901404850744q^{71} + 103919580972839873560231523607439699080q^{72} - 38087455622379499723731802436503585786q^{73} - 285201647615202678069885610769237365068q^{74} + 425247987466981253017161984441433632399q^{75} + 591482750285183039994389345638940386064q^{76} - 121632284063118375562702456886125352064q^{77} - 647972920533404340562713236608161798828q^{78} + 303763181148476262286751933509169683880q^{79} - 883861409260176376420569508380870292896q^{80} + 1034661805900421463212582471444683083207q^{81} + 506031307509525931610314373513550403116q^{82} + 338569766654854122754573119661314323844q^{83} + 4477946200719041862634183758838962173088q^{84} + 5867714862762767359034463946267521425892q^{85} - 10044657135880119837887847414531534010680q^{86} - 329652723836651682867696900815988461850q^{87} - 1871964150689809834871464807801300340640q^{88} - 42501185633754390669462796933591813434474q^{89} + 12515536130282004128514779324929526465364q^{90} + 10902563805559929629531821715171473391296q^{91} - 54939669480377726604465653835946284263712q^{92} + 54954157779576051546191591009674393431912q^{93} + 133210973073415598318228698663050409590240q^{94} - 76709837386263260643038774052059049311496q^{95} + 100557259542361618704666346381235459514144q^{96} - 15729678699391001866619231092467803918962q^{97} - 336568502755298601082628274470658413757714q^{98} + 35101784659204143757639477722503212881660q^{99} + O(q^{100}) \)

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_1(3))\)

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
3.42.a \(\chi_{3}(1, \cdot)\) 3.42.a.a 3 1
3.42.a.b 4

Decomposition of \(S_{42}^{\mathrm{old}}(\Gamma_1(3))\) into lower level spaces

\( S_{42}^{\mathrm{old}}(\Gamma_1(3)) \cong \) \(S_{42}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 289380 T + 4467538364928 T^{2} + 316609849852428288 T^{3} + \)\(98\!\cdots\!56\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} \))(\( 1 + 69822 T + 1724010272480 T^{2} - 2024128434727084032 T^{3} + \)\(57\!\cdots\!56\)\( T^{4} - \)\(44\!\cdots\!64\)\( T^{5} + \)\(83\!\cdots\!20\)\( T^{6} + \)\(74\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!16\)\( T^{8} \))
$3$ (\( ( 1 + 3486784401 T )^{3} \))(\( ( 1 - 3486784401 T )^{4} \))
$5$ (\( 1 - 38650546192026 T + \)\(71\!\cdots\!75\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!75\)\( T^{4} - \)\(79\!\cdots\!50\)\( T^{5} + \)\(94\!\cdots\!25\)\( T^{6} \))(\( 1 - 118536963776280 T + \)\(39\!\cdots\!00\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!50\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + 44602211301408768 T - \)\(17\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} - \)\(78\!\cdots\!73\)\( T^{4} + \)\(88\!\cdots\!32\)\( T^{5} + \)\(88\!\cdots\!43\)\( T^{6} \))(\( 1 - 150256264888927136 T + \)\(11\!\cdots\!52\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!30\)\( T^{4} - \)\(59\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} - \)\(13\!\cdots\!48\)\( T^{7} + \)\(39\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 - \)\(21\!\cdots\!04\)\( T + \)\(13\!\cdots\!93\)\( T^{2} - \)\(21\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!23\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(72\!\cdots\!56\)\( T + \)\(21\!\cdots\!72\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!26\)\( T^{4} - \)\(50\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!12\)\( T^{6} - \)\(89\!\cdots\!36\)\( T^{7} + \)\(61\!\cdots\!41\)\( T^{8} \))
$13$ (\( 1 - \)\(44\!\cdots\!42\)\( T + \)\(12\!\cdots\!59\)\( T^{2} - \)\(38\!\cdots\!76\)\( T^{3} + \)\(59\!\cdots\!67\)\( T^{4} - \)\(98\!\cdots\!98\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} \))(\( 1 + \)\(88\!\cdots\!08\)\( T + \)\(84\!\cdots\!44\)\( T^{2} - \)\(83\!\cdots\!84\)\( T^{3} + \)\(45\!\cdots\!50\)\( T^{4} - \)\(39\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!36\)\( T^{6} + \)\(91\!\cdots\!76\)\( T^{7} + \)\(48\!\cdots\!61\)\( T^{8} \))
$17$ (\( 1 + \)\(10\!\cdots\!14\)\( T + \)\(12\!\cdots\!83\)\( T^{2} - \)\(39\!\cdots\!52\)\( T^{3} + \)\(34\!\cdots\!11\)\( T^{4} + \)\(86\!\cdots\!46\)\( T^{5} + \)\(22\!\cdots\!13\)\( T^{6} \))(\( 1 + \)\(38\!\cdots\!88\)\( T + \)\(11\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} + \)\(67\!\cdots\!60\)\( T^{5} + \)\(91\!\cdots\!08\)\( T^{6} + \)\(84\!\cdots\!44\)\( T^{7} + \)\(62\!\cdots\!21\)\( T^{8} \))
$19$ (\( 1 - \)\(19\!\cdots\!28\)\( T + \)\(12\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!16\)\( T^{3} + \)\(34\!\cdots\!27\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{5} + \)\(19\!\cdots\!59\)\( T^{6} \))(\( 1 - \)\(26\!\cdots\!48\)\( T + \)\(75\!\cdots\!12\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(25\!\cdots\!34\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(54\!\cdots\!32\)\( T^{6} - \)\(50\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!21\)\( T^{8} \))
$23$ (\( 1 + \)\(46\!\cdots\!84\)\( T + \)\(19\!\cdots\!89\)\( T^{2} + \)\(61\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} + \)\(21\!\cdots\!36\)\( T^{5} + \)\(31\!\cdots\!67\)\( T^{6} \))(\( 1 + \)\(15\!\cdots\!32\)\( T + \)\(19\!\cdots\!48\)\( T^{2} + \)\(15\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} + \)\(10\!\cdots\!84\)\( T^{5} + \)\(91\!\cdots\!92\)\( T^{6} + \)\(47\!\cdots\!44\)\( T^{7} + \)\(21\!\cdots\!41\)\( T^{8} \))
$29$ (\( 1 + \)\(94\!\cdots\!14\)\( T - \)\(50\!\cdots\!89\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} - \)\(45\!\cdots\!81\)\( T^{4} + \)\(77\!\cdots\!74\)\( T^{5} + \)\(74\!\cdots\!89\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!64\)\( T + \)\(27\!\cdots\!60\)\( T^{2} + \)\(26\!\cdots\!08\)\( T^{3} + \)\(32\!\cdots\!98\)\( T^{4} + \)\(23\!\cdots\!32\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} + \)\(77\!\cdots\!96\)\( T^{7} + \)\(68\!\cdots\!81\)\( T^{8} \))
$31$ (\( 1 + \)\(64\!\cdots\!08\)\( T + \)\(51\!\cdots\!93\)\( T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(72\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(92\!\cdots\!04\)\( T + \)\(77\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!72\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(55\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \))
$37$ (\( 1 + \)\(44\!\cdots\!98\)\( T + \)\(12\!\cdots\!71\)\( T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(24\!\cdots\!27\)\( T^{4} + \)\(17\!\cdots\!62\)\( T^{5} + \)\(77\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(20\!\cdots\!56\)\( T + \)\(55\!\cdots\!92\)\( T^{2} - \)\(86\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!70\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \))
$41$ (\( 1 + \)\(43\!\cdots\!98\)\( T + \)\(40\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} + \)\(53\!\cdots\!63\)\( T^{4} + \)\(77\!\cdots\!38\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} \))(\( 1 - \)\(23\!\cdots\!04\)\( T + \)\(51\!\cdots\!08\)\( T^{2} - \)\(70\!\cdots\!72\)\( T^{3} + \)\(99\!\cdots\!94\)\( T^{4} - \)\(94\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!48\)\( T^{6} - \)\(55\!\cdots\!84\)\( T^{7} + \)\(31\!\cdots\!61\)\( T^{8} \))
$43$ (\( 1 + \)\(13\!\cdots\!76\)\( T + \)\(27\!\cdots\!33\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(11\!\cdots\!24\)\( T^{5} + \)\(82\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(39\!\cdots\!60\)\( T + \)\(29\!\cdots\!00\)\( T^{2} - \)\(85\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!98\)\( T^{4} - \)\(79\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} - \)\(32\!\cdots\!20\)\( T^{7} + \)\(77\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - \)\(64\!\cdots\!68\)\( T + \)\(22\!\cdots\!37\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!39\)\( T^{4} - \)\(83\!\cdots\!12\)\( T^{5} + \)\(46\!\cdots\!23\)\( T^{6} \))(\( 1 + \)\(88\!\cdots\!20\)\( T + \)\(10\!\cdots\!20\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(41\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))
$53$ (\( 1 - \)\(34\!\cdots\!46\)\( T + \)\(10\!\cdots\!59\)\( T^{2} - \)\(28\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!27\)\( T^{4} - \)\(85\!\cdots\!14\)\( T^{5} + \)\(12\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(95\!\cdots\!28\)\( T + \)\(13\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(85\!\cdots\!10\)\( T^{4} - \)\(54\!\cdots\!16\)\( T^{5} + \)\(33\!\cdots\!92\)\( T^{6} - \)\(11\!\cdots\!56\)\( T^{7} + \)\(60\!\cdots\!81\)\( T^{8} \))
$59$ (\( 1 + \)\(45\!\cdots\!48\)\( T + \)\(15\!\cdots\!33\)\( T^{2} + \)\(37\!\cdots\!84\)\( T^{3} + \)\(63\!\cdots\!47\)\( T^{4} + \)\(74\!\cdots\!88\)\( T^{5} + \)\(65\!\cdots\!79\)\( T^{6} \))(\( 1 + \)\(18\!\cdots\!08\)\( T + \)\(66\!\cdots\!52\)\( T^{2} + \)\(24\!\cdots\!16\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} + \)\(99\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!61\)\( T^{8} \))
$61$ (\( 1 + \)\(86\!\cdots\!62\)\( T + \)\(51\!\cdots\!19\)\( T^{2} + \)\(21\!\cdots\!96\)\( T^{3} + \)\(81\!\cdots\!59\)\( T^{4} + \)\(21\!\cdots\!02\)\( T^{5} + \)\(39\!\cdots\!81\)\( T^{6} \))(\( 1 - \)\(53\!\cdots\!40\)\( T + \)\(67\!\cdots\!56\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!76\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!41\)\( T^{8} \))
$67$ (\( 1 - \)\(32\!\cdots\!16\)\( T + \)\(19\!\cdots\!53\)\( T^{2} - \)\(47\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!63\)\( T^{6} \))(\( 1 + \)\(73\!\cdots\!28\)\( T + \)\(45\!\cdots\!12\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!06\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} + \)\(29\!\cdots\!64\)\( T^{7} + \)\(29\!\cdots\!21\)\( T^{8} \))
$71$ (\( 1 - \)\(14\!\cdots\!96\)\( T + \)\(21\!\cdots\!85\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(95\!\cdots\!36\)\( T^{5} + \)\(50\!\cdots\!11\)\( T^{6} \))(\( 1 + \)\(84\!\cdots\!52\)\( T + \)\(22\!\cdots\!48\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} + \)\(42\!\cdots\!72\)\( T^{7} + \)\(40\!\cdots\!81\)\( T^{8} \))
$73$ (\( 1 - \)\(66\!\cdots\!46\)\( T + \)\(29\!\cdots\!99\)\( T^{2} + \)\(27\!\cdots\!84\)\( T^{3} + \)\(74\!\cdots\!27\)\( T^{4} - \)\(41\!\cdots\!34\)\( T^{5} + \)\(15\!\cdots\!17\)\( T^{6} \))(\( 1 + \)\(44\!\cdots\!32\)\( T + \)\(51\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!72\)\( T^{6} + \)\(69\!\cdots\!44\)\( T^{7} + \)\(38\!\cdots\!41\)\( T^{8} \))
$79$ (\( 1 + \)\(11\!\cdots\!00\)\( T + \)\(20\!\cdots\!37\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!39\)\( T^{6} \))(\( 1 - \)\(14\!\cdots\!80\)\( T + \)\(22\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(91\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))
$83$ (\( 1 + \)\(11\!\cdots\!60\)\( T + \)\(12\!\cdots\!37\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!71\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!87\)\( T^{6} \))(\( 1 - \)\(15\!\cdots\!04\)\( T + \)\(99\!\cdots\!00\)\( T^{2} - \)\(72\!\cdots\!24\)\( T^{3} + \)\(39\!\cdots\!46\)\( T^{4} - \)\(34\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!00\)\( T^{6} - \)\(16\!\cdots\!48\)\( T^{7} + \)\(53\!\cdots\!21\)\( T^{8} \))
$89$ (\( 1 + \)\(29\!\cdots\!02\)\( T + \)\(77\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(65\!\cdots\!07\)\( T^{4} + \)\(20\!\cdots\!42\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \))(\( 1 + \)\(39\!\cdots\!72\)\( T + \)\(90\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!94\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} + \)\(23\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!41\)\( T^{8} \))
$97$ (\( 1 + \)\(52\!\cdots\!14\)\( T + \)\(46\!\cdots\!23\)\( T^{2} + \)\(30\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!31\)\( T^{4} + \)\(43\!\cdots\!26\)\( T^{5} + \)\(23\!\cdots\!73\)\( T^{6} \))(\( 1 - \)\(36\!\cdots\!52\)\( T + \)\(23\!\cdots\!52\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} - \)\(59\!\cdots\!94\)\( T^{4} + \)\(91\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(86\!\cdots\!96\)\( T^{7} + \)\(67\!\cdots\!81\)\( T^{8} \))
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