Properties

Label 2.76.a
Level 2
Weight 76
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 2
Sturm bound 19
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(19\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{76}(\Gamma_0(2))\).

Total New Old
Modular forms 20 6 14
Cusp forms 18 6 12
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(3\)

Trace form

\( 6q + 1705498902863007720q^{3} + 113336795588871485128704q^{4} - 30476827565682466961243340q^{5} - 108441336830676229092525735936q^{6} + 12122201978496638924869575734160q^{7} + 82361916372728383069660311762464382q^{9} + O(q^{10}) \) \( 6q + 1705498902863007720q^{3} + 113336795588871485128704q^{4} - 30476827565682466961243340q^{5} - 108441336830676229092525735936q^{6} + 12122201978496638924869575734160q^{7} + 82361916372728383069660311762464382q^{9} - 49499862517581316311125669645364756480q^{10} + 1465550801636051753338937183456765880312q^{11} + 32215963421804881809895339739490790932480q^{12} - 1125845527689880288320852450008843291510460q^{13} - 11854765018430542049380300142291471415902208q^{14} + 531682141721515337891142522137765144294806320q^{15} + 2140871539058939821587428954174242704574119936q^{16} - 2858889474468025925628869183894767589187643860q^{17} - 84906596595279405116071608834024501078644490240q^{18} - 888971623839919293994237817579267004997881014520q^{19} - 575690996001506250615928057273283216656627138560q^{20} + 45041848467490506542324387616415633517027774542272q^{21} + 98061939135380581799248808021795190845288050851840q^{22} + 388185115742187941653436201968293251774654337919920q^{23} - 2048398937627052092809160626895625171745769546317824q^{24} + 18070897297784288584351872928181869115179992908111850q^{25} + 99546303021743005094547437366409957888747960149213184q^{26} + 680734682425435069671863065380018333629297430414863760q^{27} + 228981921287314509354065536892774721213529717148221440q^{28} - 6877232017518564637732794749543722232025970665240801180q^{29} - 10563106827841239918850343130757583889594367360326696960q^{30} - 64665307498524090137269025681077778105916247568551985088q^{31} - 1738389044322888107810703320646334694022370511632650493920q^{33} - 3445639243383333645073254694727612280728361904936266498048q^{34} - 21532243462114279631730583516480850942482596809187388619040q^{35} + 1555772613373940725850374333095645734209769775503914303488q^{36} - 112408022660764207843478517252625221853389811204182104235660q^{37} - 459199122966556825368030582314147000908588240754160634429440q^{38} - 3249018265371653484547819023439503172123566105460893478749456q^{39} - 935025966638725849496497481614084824265194550179666403000320q^{40} - 4569430998671436099055216867854419985860955311220740904029988q^{41} - 12643356129211913793904090514100858239148953072140750128414720q^{42} - 49458776477914861548275480473460327027376042429737800006877640q^{43} + 27683471938355323211425027496439486390925568089379883196612608q^{44} + 440416419373008234527850892661087944105637316568242458315614020q^{45} - 21230534415495266402222312416399389344741990676759073360183296q^{46} + 1491929616135495185777393291100411415876754623169608730717643360q^{47} + 608542343505943440789597096565425515289428267944242514828984320q^{48} + 6725936829579826549585444762508254778952416835808810923771155958q^{49} - 6243302576022018573431009072181591415289545765835834494078156800q^{50} + 13552397707089728377161712703662875947963669271236343314013636432q^{51} - 21266620739405518954701766000571151941281508550948440037277040640q^{52} - 60108358164229615923644712516524935326680580703660063831212394860q^{53} - 235592548052616779200261004283520407207143310081168543193277399040q^{54} - 544456734370049344521834827202628380707716663181530992892228861680q^{55} - 223930179941327774707416049528494080762133418862371229475192963072q^{56} + 866256274620564586983962921420474108418213263746961917604562363360q^{57} - 342614301266754269662398756788455363273579957958836824474680033280q^{58} + 4792977580662723835151784946431384189296876826180380913167915946840q^{59} + 10043191702424130558647160698278437553823104287647533429517325434880q^{60} + 19686326720834781156328022804498489908022651023228048060027254696612q^{61} + 14332866880775621962660149787097017662771276552468848217481417850880q^{62} - 13273853878431110988386994849728412307979535806379796700537090487600q^{63} + 40439920000725959692000522630529446010455716633810858721655415373824q^{64} - 414834079011781749625000789087311646313182077414197224151804606563080q^{65} - 196961733872555387767658170063634154282305123224812945891277638991872q^{66} - 549361113070829377579138973968485083103627995638937741530254051175320q^{67} - 54002895329826479839415152590400523291898677018769775214689769226240q^{68} + 1240604490939895940074393936404147800729678425716003464452118266630464q^{69} - 50643430004919912435115859590639294644431090240894494760795772026880q^{70} + 5055228439700391598733278928515185578509384266316962142211216648477712q^{71} - 1603840263744325589941350837426425020036269381358358823587111145308160q^{72} + 17833116798913518065803534553264839397010603100058748892693013675078940q^{73} + 2821111879286599592685083524515320528734419709763880516865160298627072q^{74} + 31515098194005543136537074885930516708850247069098328297237975589466200q^{75} - 16792199202575347701309757253576693115167544415895309814157706715463680q^{76} - 176876168686196791515608000007831191146787002360180888497317885941613760q^{77} - 13175854004838032990557527630462288981363900186842150185753334104719360q^{78} - 414618165038650389130367983432524144413302854751155708293936851388208480q^{79} - 10874495456029424234650973991381989853684746449312786081950416806871040q^{80} + 227129858616070136029242242688526106661001655175232324447902535043807286q^{81} + 727633575313169965948927583434694974684386362391165800034948178740510720q^{82} - 365380980296849434089399689634077461850327884677693467309544186132526200q^{83} + 850816462117482650737792136487152155355351393795796095894741768101429248q^{84} + 7603577266494656826847033469689752557609725576902484781363233968805598760q^{85} - 1092009338675897638695112358617199909717769451613164476962470588638560256q^{86} + 12685326993689075935764534600181093911441164976495452293801457032395857520q^{87} + 1852337658472497663269056061254699490163875586984145575538414846539202560q^{88} - 48795344253743362658377973278924634941596646329873302000071783183839916740q^{89} - 57792277165760092642954117797017236971112447154085230124973232258348482560q^{90} - 38206057527202705843659578772472464167479972093140577729234253104415268768q^{91} + 7332609518919128867767921225848989537443334653139583244233368898140897280q^{92} + 53270972557617800782916214519538202114490801927186497537919515301412273920q^{93} + 190873389458004810315822070205614896201393532677578497450462710504181727232q^{94} - 25879185658575995138262977871649687206746178148088326274797415832024637200q^{95} - 38693161946383119005589081837630218167243003233592020141066804089654870016q^{96} + 907693164349022563062701567553478512929793318388504102355145154694807249100q^{97} + 683969170889206584645706074418954041679008199336237535718616135899115683840q^{98} - 270180457059752090850417884218708645593970737076320028036193624197746106536q^{99} + O(q^{100}) \)

Decomposition of \(S_{76}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.76.a.a \(3\) \(71.246\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-412316860416\) \(12\!\cdots\!04\) \(16\!\cdots\!50\) \(49\!\cdots\!12\) \(+\) \(q-2^{37}q^{2}+(415752245544732668+\cdots)q^{3}+\cdots\)
2.76.a.b \(3\) \(71.246\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(412316860416\) \(45\!\cdots\!16\) \(-1\!\cdots\!90\) \(-3\!\cdots\!52\) \(-\) \(q+2^{37}q^{2}+(152747388742936572+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{76}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{76}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{76}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 137438953472 T )^{3} \))(\( ( 1 - 137438953472 T )^{3} \))
$3$ (\( 1 - 1247256736634198004 T + \)\(15\!\cdots\!93\)\( T^{2} - \)\(13\!\cdots\!88\)\( T^{3} + \)\(92\!\cdots\!51\)\( T^{4} - \)\(46\!\cdots\!96\)\( T^{5} + \)\(22\!\cdots\!43\)\( T^{6} \))(\( 1 - 458242166228809716 T + \)\(11\!\cdots\!73\)\( T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(70\!\cdots\!11\)\( T^{4} - \)\(16\!\cdots\!84\)\( T^{5} + \)\(22\!\cdots\!43\)\( T^{6} \))
$5$ (\( 1 - \)\(16\!\cdots\!50\)\( T + \)\(37\!\cdots\!75\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!75\)\( T^{4} - \)\(11\!\cdots\!50\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!90\)\( T + \)\(65\!\cdots\!75\)\( T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!75\)\( T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - \)\(49\!\cdots\!12\)\( T + \)\(43\!\cdots\!77\)\( T^{2} - \)\(15\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} - \)\(28\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(37\!\cdots\!52\)\( T + \)\(13\!\cdots\!97\)\( T^{2} + \)\(82\!\cdots\!76\)\( T^{3} + \)\(33\!\cdots\!71\)\( T^{4} + \)\(21\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 - \)\(37\!\cdots\!96\)\( T + \)\(27\!\cdots\!25\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!75\)\( T^{4} - \)\(60\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!51\)\( T^{6} \))(\( 1 - \)\(10\!\cdots\!16\)\( T - \)\(37\!\cdots\!95\)\( T^{2} + \)\(23\!\cdots\!20\)\( T^{3} - \)\(47\!\cdots\!45\)\( T^{4} - \)\(17\!\cdots\!16\)\( T^{5} + \)\(20\!\cdots\!51\)\( T^{6} \))
$13$ (\( 1 + \)\(92\!\cdots\!66\)\( T + \)\(85\!\cdots\!23\)\( T^{2} + \)\(46\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!34\)\( T^{5} + \)\(43\!\cdots\!93\)\( T^{6} \))(\( 1 + \)\(20\!\cdots\!94\)\( T + \)\(15\!\cdots\!83\)\( T^{2} - \)\(15\!\cdots\!92\)\( T^{3} + \)\(53\!\cdots\!31\)\( T^{4} + \)\(24\!\cdots\!06\)\( T^{5} + \)\(43\!\cdots\!93\)\( T^{6} \))
$17$ (\( 1 - \)\(11\!\cdots\!62\)\( T + \)\(17\!\cdots\!27\)\( T^{2} - \)\(18\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!11\)\( T^{4} - \)\(41\!\cdots\!38\)\( T^{5} + \)\(70\!\cdots\!57\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!22\)\( T + \)\(46\!\cdots\!07\)\( T^{2} + \)\(54\!\cdots\!16\)\( T^{3} + \)\(89\!\cdots\!51\)\( T^{4} + \)\(51\!\cdots\!78\)\( T^{5} + \)\(70\!\cdots\!57\)\( T^{6} \))
$19$ (\( 1 - \)\(12\!\cdots\!00\)\( T + \)\(15\!\cdots\!97\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} - \)\(79\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!99\)\( T^{6} \))(\( 1 + \)\(21\!\cdots\!20\)\( T + \)\(33\!\cdots\!97\)\( T^{2} + \)\(32\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(52\!\cdots\!99\)\( T^{6} \))
$23$ (\( 1 - \)\(27\!\cdots\!44\)\( T + \)\(21\!\cdots\!33\)\( T^{2} + \)\(38\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!31\)\( T^{4} - \)\(49\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(11\!\cdots\!76\)\( T + \)\(32\!\cdots\!13\)\( T^{2} - \)\(43\!\cdots\!52\)\( T^{3} + \)\(44\!\cdots\!91\)\( T^{4} - \)\(21\!\cdots\!24\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \))
$29$ (\( 1 + \)\(21\!\cdots\!70\)\( T + \)\(13\!\cdots\!47\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!03\)\( T^{4} + \)\(50\!\cdots\!70\)\( T^{5} + \)\(10\!\cdots\!49\)\( T^{6} \))(\( 1 + \)\(46\!\cdots\!10\)\( T + \)\(97\!\cdots\!47\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!03\)\( T^{4} + \)\(10\!\cdots\!10\)\( T^{5} + \)\(10\!\cdots\!49\)\( T^{6} \))
$31$ (\( 1 + \)\(84\!\cdots\!64\)\( T + \)\(15\!\cdots\!85\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} + \)\(42\!\cdots\!64\)\( T^{5} + \)\(36\!\cdots\!51\)\( T^{6} \))(\( 1 - \)\(19\!\cdots\!76\)\( T + \)\(95\!\cdots\!45\)\( T^{2} - \)\(52\!\cdots\!40\)\( T^{3} + \)\(68\!\cdots\!95\)\( T^{4} - \)\(10\!\cdots\!76\)\( T^{5} + \)\(36\!\cdots\!51\)\( T^{6} \))
$37$ (\( 1 + \)\(66\!\cdots\!18\)\( T + \)\(89\!\cdots\!87\)\( T^{2} + \)\(49\!\cdots\!64\)\( T^{3} + \)\(36\!\cdots\!91\)\( T^{4} + \)\(11\!\cdots\!82\)\( T^{5} + \)\(70\!\cdots\!57\)\( T^{6} \))(\( 1 + \)\(45\!\cdots\!42\)\( T + \)\(98\!\cdots\!67\)\( T^{2} + \)\(28\!\cdots\!56\)\( T^{3} + \)\(40\!\cdots\!31\)\( T^{4} + \)\(78\!\cdots\!58\)\( T^{5} + \)\(70\!\cdots\!57\)\( T^{6} \))
$41$ (\( 1 + \)\(49\!\cdots\!74\)\( T + \)\(25\!\cdots\!95\)\( T^{2} + \)\(82\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!95\)\( T^{4} + \)\(40\!\cdots\!74\)\( T^{5} + \)\(75\!\cdots\!01\)\( T^{6} \))(\( 1 - \)\(36\!\cdots\!86\)\( T + \)\(18\!\cdots\!35\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(29\!\cdots\!86\)\( T^{5} + \)\(75\!\cdots\!01\)\( T^{6} \))
$43$ (\( 1 + \)\(20\!\cdots\!96\)\( T + \)\(10\!\cdots\!93\)\( T^{2} + \)\(13\!\cdots\!12\)\( T^{3} + \)\(34\!\cdots\!51\)\( T^{4} + \)\(21\!\cdots\!04\)\( T^{5} + \)\(33\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(28\!\cdots\!44\)\( T + \)\(12\!\cdots\!33\)\( T^{2} + \)\(19\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!31\)\( T^{4} + \)\(30\!\cdots\!56\)\( T^{5} + \)\(33\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 - \)\(51\!\cdots\!52\)\( T + \)\(55\!\cdots\!97\)\( T^{2} - \)\(29\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(33\!\cdots\!48\)\( T^{5} + \)\(16\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(14\!\cdots\!08\)\( T + \)\(14\!\cdots\!17\)\( T^{2} - \)\(84\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!31\)\( T^{4} - \)\(94\!\cdots\!92\)\( T^{5} + \)\(16\!\cdots\!07\)\( T^{6} \))
$53$ (\( 1 - \)\(63\!\cdots\!14\)\( T + \)\(41\!\cdots\!03\)\( T^{2} - \)\(12\!\cdots\!68\)\( T^{3} + \)\(87\!\cdots\!71\)\( T^{4} - \)\(27\!\cdots\!86\)\( T^{5} + \)\(91\!\cdots\!93\)\( T^{6} \))(\( 1 + \)\(66\!\cdots\!74\)\( T + \)\(66\!\cdots\!63\)\( T^{2} + \)\(25\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!91\)\( T^{4} + \)\(29\!\cdots\!26\)\( T^{5} + \)\(91\!\cdots\!93\)\( T^{6} \))
$59$ (\( 1 - \)\(51\!\cdots\!60\)\( T + \)\(25\!\cdots\!97\)\( T^{2} - \)\(66\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(27\!\cdots\!99\)\( T^{6} \))(\( 1 + \)\(37\!\cdots\!20\)\( T + \)\(10\!\cdots\!97\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(67\!\cdots\!03\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(27\!\cdots\!99\)\( T^{6} \))
$61$ (\( 1 - \)\(61\!\cdots\!66\)\( T + \)\(21\!\cdots\!55\)\( T^{2} - \)\(85\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!55\)\( T^{4} - \)\(38\!\cdots\!66\)\( T^{5} + \)\(50\!\cdots\!01\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!46\)\( T + \)\(17\!\cdots\!75\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} - \)\(85\!\cdots\!46\)\( T^{5} + \)\(50\!\cdots\!01\)\( T^{6} \))
$67$ (\( 1 + \)\(13\!\cdots\!68\)\( T + \)\(25\!\cdots\!37\)\( T^{2} + \)\(24\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!91\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(73\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(41\!\cdots\!52\)\( T + \)\(74\!\cdots\!97\)\( T^{2} - \)\(40\!\cdots\!24\)\( T^{3} + \)\(67\!\cdots\!71\)\( T^{4} + \)\(33\!\cdots\!48\)\( T^{5} + \)\(73\!\cdots\!07\)\( T^{6} \))
$71$ (\( 1 - \)\(10\!\cdots\!16\)\( T + \)\(55\!\cdots\!05\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!55\)\( T^{4} - \)\(50\!\cdots\!16\)\( T^{5} + \)\(34\!\cdots\!51\)\( T^{6} \))(\( 1 + \)\(53\!\cdots\!04\)\( T + \)\(18\!\cdots\!25\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!51\)\( T^{6} \))
$73$ (\( 1 - \)\(23\!\cdots\!54\)\( T + \)\(30\!\cdots\!43\)\( T^{2} - \)\(26\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!51\)\( T^{4} - \)\(75\!\cdots\!46\)\( T^{5} + \)\(17\!\cdots\!93\)\( T^{6} \))(\( 1 + \)\(61\!\cdots\!14\)\( T + \)\(13\!\cdots\!03\)\( T^{2} + \)\(56\!\cdots\!68\)\( T^{3} + \)\(77\!\cdots\!71\)\( T^{4} + \)\(19\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!93\)\( T^{6} \))
$79$ (\( 1 - \)\(20\!\cdots\!20\)\( T + \)\(53\!\cdots\!97\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(91\!\cdots\!20\)\( T^{5} + \)\(92\!\cdots\!99\)\( T^{6} \))(\( 1 + \)\(62\!\cdots\!00\)\( T + \)\(19\!\cdots\!97\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!03\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{5} + \)\(92\!\cdots\!99\)\( T^{6} \))
$83$ (\( 1 - \)\(11\!\cdots\!04\)\( T + \)\(11\!\cdots\!93\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(96\!\cdots\!51\)\( T^{4} - \)\(84\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(15\!\cdots\!04\)\( T + \)\(30\!\cdots\!93\)\( T^{2} + \)\(25\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!51\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(62\!\cdots\!43\)\( T^{6} \))
$89$ (\( 1 + \)\(39\!\cdots\!70\)\( T + \)\(94\!\cdots\!47\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!03\)\( T^{4} + \)\(10\!\cdots\!70\)\( T^{5} + \)\(40\!\cdots\!49\)\( T^{6} \))(\( 1 + \)\(92\!\cdots\!70\)\( T + \)\(75\!\cdots\!47\)\( T^{2} - \)\(26\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} + \)\(23\!\cdots\!70\)\( T^{5} + \)\(40\!\cdots\!49\)\( T^{6} \))
$97$ (\( 1 + \)\(39\!\cdots\!78\)\( T - \)\(34\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!16\)\( T^{3} - \)\(35\!\cdots\!49\)\( T^{4} + \)\(40\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!57\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!78\)\( T + \)\(85\!\cdots\!07\)\( T^{2} - \)\(33\!\cdots\!84\)\( T^{3} + \)\(86\!\cdots\!51\)\( T^{4} - \)\(13\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!57\)\( T^{6} \))
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