Properties

Label 1.80.a
Level 1
Weight 80
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 80 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{80}(\Gamma_0(1))\).

Total New Old
Modular forms 7 7 0
Cusp forms 6 6 0
Eisenstein series 1 1 0

Trace form

\( 6q - 16086577320q^{2} + 1942174928281180680q^{3} + 1549737362258405903681088q^{4} + 6092984858773763455723827540q^{5} - 2227106590325939088404927087328q^{6} - 204133057837186288371299604423600q^{7} + 542570064661378776298915601390568960q^{8} + 98496326492153316059886441015309696222q^{9} + O(q^{10}) \) \( 6q - 16086577320q^{2} + 1942174928281180680q^{3} + 1549737362258405903681088q^{4} + 6092984858773763455723827540q^{5} - 2227106590325939088404927087328q^{6} - 204133057837186288371299604423600q^{7} + 542570064661378776298915601390568960q^{8} + 98496326492153316059886441015309696222q^{9} + 2720626696366658228219065452729623532240q^{10} + 326182707503353434196853598345012484552152q^{11} + 3662741799314627362069151990938777789858560q^{12} - 249286867791526696387831019648013646893106140q^{13} + 1866634067474695788912841823339193553138295616q^{14} + 53939724989186019570794541522119517534771796080q^{15} - 681627631644510368706137880459644753465449181184q^{16} + 829554518710324966978506506427078451803483469740q^{17} + 20783419681388619276512545895581772264264550807480q^{18} - 50727940848110344442361931427855575819428647228760q^{19} - 1467030065046629207751264576643706806587735144958080q^{20} + 11440846038310308150487768012211089482880511198926272q^{21} + 36817507512313772519930542191150104222671820963556960q^{22} - 1046924887201563380993656359631631933258453525094738960q^{23} + 4647998047182641817986339484942489649049309172966328320q^{24} + 10836995626543537474051791335153252813979338793331307850q^{25} - 165657588130323625805363235243378873928973276983585168368q^{26} + 452244485424938477892504860024752019079362449583874943440q^{27} + 1004680642640065837930850364413184139977872155405489282560q^{28} - 7120822866262655112931363073018274881104708247497947521340q^{29} - 1093089132342530488188112356341250721972394415479943059520q^{30} + 134345039550847531062964315804153164343444264343589333172032q^{31} - 367680407387361126658464447414461787802285956837455133573120q^{32} + 694937277679484430969386562214930563011269531321798125315360q^{33} + 3154306959722239121253159757854043021326472297974364404980016q^{34} + 9191056770903655504272784196241783249847740561191433159558240q^{35} + 19880287065959074647932230862843896929437344679866234828597056q^{36} + 200218543091040103155222783880989961642213560574664429550198420q^{37} + 866365883974611161836395390925862263352921297032838009795060640q^{38} + 1846928010477803855453412626793191219191424716227436821180804144q^{39} + 9116669264447274408945009430906339427674431802850195585057510400q^{40} + 19216804901837266991242147464997667315402616992351385711235458972q^{41} + 112315174478490788723912174839938916243710780267342081630678531840q^{42} + 108832931910908864051094641347526853298463718485111641638401400600q^{43} + 495514650695382135238363336424200091081783038631873261666636656896q^{44} + 1089311173064980984859644147008913582964905804089164802662670706980q^{45} + 2339423217798161589056888023635021837153755261863446723443331974592q^{46} + 766732507168785220421816169748850175208159379232549271592508353760q^{47} - 1227537570851109157458973684400646232156423736031048093315900456960q^{48} + 3801016219812200810689187746878210669081675204402752707309154018358q^{49} - 50723521315108078882353551947661611720805543300108849711322864485400q^{50} - 139865466375558864334882784640098531626186142446280524443365433199728q^{51} - 279692249004488191323071540832092417841619330373410063262542264310400q^{52} - 312193237964710957330604375297544855480450344577449082153802391971020q^{53} - 996221038973994559873624494431619704350011373216284617130312268666560q^{54} - 861958222068919393142591787960630586871784969446963670182827020790320q^{55} + 1123629643438673418216699268533992018915072902064015953839024305295360q^{56} + 7795598779799587371741461301046795526293245245773998812837273820000480q^{57} + 16584702356914156466182151363602691995360915944935914137385742159510160q^{58} + 24289171784532714189639095038219795042285164994033400426178461238907320q^{59} + 73025956047571458013400417312791668535879728806786116560639243534051840q^{60} + 80560570425826852880081093564684385117241348668836108242495308815618052q^{61} + 59777568713863815377707698536571620658579394379949737090862924510332160q^{62} - 276708678056196744256455115102015383166989187390931876478468368856762480q^{63} - 719203226636338274682481908236724651863342420498670194455988567598170112q^{64} - 1007462919483461647300380829039677154191109543250817689988654733030592520q^{65} - 3347562853423538991999694027669961627986534585541581689746575315495214976q^{66} - 1851521876973893932513297183455714100127257334040085257904842903903517560q^{67} - 5899582938115809421316043854246769288699297144820927766700932855979726720q^{68} - 1427672735602453311144050658084524391331260046448660152415370050225171136q^{69} + 31159148342952686307596185437939815723517531766889921556924090292621741440q^{70} + 38507401093845789827495210229167852070939724868887506542264571604541264592q^{71} + 87904411000641647434917578390462554629488250565124366592415232306933521920q^{72} + 61123654206386061400310438862981106227383729069503080628162601940159787740q^{73} + 136513352063977853606694381822259496818965955504365628093794971441921163216q^{74} + 113188706706427317048365034543759604727098826077328333023878456326260918200q^{75} - 249903768766885839664688587580825575582546834454828556400015599314274074880q^{76} - 1195099252106099653800449972868194220235362175583181442486256041652274244800q^{77} - 3022231856978250782599441544561070134871217051487771942772538597436408443200q^{78} - 188234723479363903277484397100590307076741783512190126828886315155446684640q^{79} - 2750248132852209015421484068233717972689944891189322073844459755369097666560q^{80} - 4202996734890713231427597758184507891641102417282671988811370042535012566794q^{81} + 3834792966214046709977047126406184257851543195738078433637414913324683717360q^{82} + 14506845760765784842493952470927474286413727909244314121162679395439622145320q^{83} + 48942789227023150657016145376829988370028655394086238910889917134273615865856q^{84} + 38946066828627954832778050727572362568084538101228336437758435671222134267240q^{85} + 66363633323077708072616297459993363454350669286030390599179655949306835458912q^{86} - 12188467007600920252987005731487987332580129454574096199578444638747687628880q^{87} - 68086553439664359269830684533538140843829745277114000504065479672707840583680q^{88} + 88790169947368466670673875396871954428202115924137373286785364937730407494780q^{89} - 678121360108159288271315537216087664710338266311311066150921870820864233179120q^{90} - 1105665330428280782729905276805137770501582996765091919549546053816493845198368q^{91} - 581852119074779563003978084378550280950528927013853021971971412987670782543360q^{92} - 724063571132401087495277238269757837179269218308363767014177350097338957840640q^{93} - 342394004508386413823498343959337212855421843329465021317054652742559913721984q^{94} + 4133010515441143062055572903340407460351722715065934937506390604234109634359600q^{95} + 4500277093535091993817003185543220432802869972147132801867826645045331188908032q^{96} + 2053881285944963905800936410853289740689894385863240159901124624782493905690060q^{97} + 21591608457226642856925919189308208713047358263642522360137695155926522503830040q^{98} + 12210721611676619111412804269965201185505307365258823385190308559518832844781624q^{99} + O(q^{100}) \)

Decomposition of \(S_{80}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.80.a.a \(6\) \(39.524\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-16086577320\) \(19\!\cdots\!80\) \(60\!\cdots\!40\) \(-2\!\cdots\!00\) \(+\) \(q+(-2681096220+\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16086577320 T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(94\!\cdots\!32\)\( T^{4} - \)\(36\!\cdots\!40\)\( T^{5} + \)\(67\!\cdots\!60\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!08\)\( T^{8} - \)\(36\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!20\)\( T^{10} + \)\(12\!\cdots\!60\)\( T^{11} + \)\(48\!\cdots\!84\)\( T^{12} \)
$3$ \( 1 - 1942174928281180680 T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(75\!\cdots\!67\)\( T^{4} - \)\(25\!\cdots\!40\)\( T^{5} + \)\(44\!\cdots\!20\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!63\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{9} + \)\(59\!\cdots\!90\)\( T^{10} - \)\(56\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!69\)\( T^{12} \)
$5$ \( 1 - \)\(60\!\cdots\!40\)\( T + \)\(62\!\cdots\!50\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!75\)\( T^{4} - \)\(71\!\cdots\!00\)\( T^{5} + \)\(39\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(47\!\cdots\!50\)\( T^{10} - \)\(75\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!00\)\( T^{6} - \)\(61\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!03\)\( T^{8} + \)\(60\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 - \)\(32\!\cdots\!52\)\( T + \)\(88\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!95\)\( T^{4} - \)\(46\!\cdots\!92\)\( T^{5} + \)\(70\!\cdots\!04\)\( T^{6} - \)\(86\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} - \)\(73\!\cdots\!52\)\( T^{11} + \)\(41\!\cdots\!41\)\( T^{12} \)
$13$ \( 1 + \)\(24\!\cdots\!40\)\( T + \)\(46\!\cdots\!30\)\( T^{2} + \)\(63\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!87\)\( T^{4} + \)\(97\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!40\)\( T^{6} + \)\(97\!\cdots\!40\)\( T^{7} + \)\(82\!\cdots\!23\)\( T^{8} + \)\(63\!\cdots\!40\)\( T^{9} + \)\(46\!\cdots\!30\)\( T^{10} + \)\(25\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \)
$17$ \( 1 - \)\(82\!\cdots\!40\)\( T + \)\(44\!\cdots\!10\)\( T^{2} - \)\(95\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!27\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} - \)\(35\!\cdots\!60\)\( T^{7} + \)\(32\!\cdots\!43\)\( T^{8} - \)\(39\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!10\)\( T^{10} - \)\(88\!\cdots\!20\)\( T^{11} + \)\(17\!\cdots\!29\)\( T^{12} \)
$19$ \( 1 + \)\(50\!\cdots\!60\)\( T + \)\(40\!\cdots\!74\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!15\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!15\)\( T^{8} + \)\(66\!\cdots\!00\)\( T^{9} + \)\(49\!\cdots\!94\)\( T^{10} + \)\(65\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!21\)\( T^{12} \)
$23$ \( 1 + \)\(10\!\cdots\!60\)\( T + \)\(18\!\cdots\!70\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!07\)\( T^{4} + \)\(97\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!83\)\( T^{8} + \)\(79\!\cdots\!60\)\( T^{9} + \)\(37\!\cdots\!70\)\( T^{10} + \)\(79\!\cdots\!20\)\( T^{11} + \)\(28\!\cdots\!09\)\( T^{12} \)
$29$ \( 1 + \)\(71\!\cdots\!40\)\( T + \)\(76\!\cdots\!14\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!80\)\( T^{6} + \)\(63\!\cdots\!00\)\( T^{7} + \)\(46\!\cdots\!15\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!94\)\( T^{10} + \)\(31\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!81\)\( T^{12} \)
$31$ \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(37\!\cdots\!86\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!95\)\( T^{4} - \)\(50\!\cdots\!92\)\( T^{5} + \)\(51\!\cdots\!64\)\( T^{6} - \)\(33\!\cdots\!32\)\( T^{7} + \)\(25\!\cdots\!95\)\( T^{8} - \)\(11\!\cdots\!20\)\( T^{9} + \)\(70\!\cdots\!66\)\( T^{10} - \)\(16\!\cdots\!32\)\( T^{11} + \)\(80\!\cdots\!21\)\( T^{12} \)
$37$ \( 1 - \)\(20\!\cdots\!20\)\( T + \)\(51\!\cdots\!30\)\( T^{2} - \)\(68\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(99\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!40\)\( T^{6} - \)\(76\!\cdots\!80\)\( T^{7} + \)\(60\!\cdots\!23\)\( T^{8} - \)\(31\!\cdots\!20\)\( T^{9} + \)\(18\!\cdots\!30\)\( T^{10} - \)\(55\!\cdots\!60\)\( T^{11} + \)\(21\!\cdots\!89\)\( T^{12} \)
$41$ \( 1 - \)\(19\!\cdots\!72\)\( T + \)\(25\!\cdots\!26\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(66\!\cdots\!44\)\( T^{6} - \)\(31\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(40\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!66\)\( T^{10} - \)\(21\!\cdots\!72\)\( T^{11} + \)\(28\!\cdots\!61\)\( T^{12} \)
$43$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(79\!\cdots\!50\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(91\!\cdots\!00\)\( T^{5} + \)\(33\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!03\)\( T^{8} - \)\(62\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!50\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(76\!\cdots\!60\)\( T + \)\(65\!\cdots\!90\)\( T^{2} - \)\(43\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!67\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!20\)\( T^{6} - \)\(12\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!63\)\( T^{8} - \)\(84\!\cdots\!60\)\( T^{9} + \)\(15\!\cdots\!90\)\( T^{10} - \)\(23\!\cdots\!80\)\( T^{11} + \)\(37\!\cdots\!69\)\( T^{12} \)
$53$ \( 1 + \)\(31\!\cdots\!20\)\( T + \)\(95\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!67\)\( T^{4} + \)\(48\!\cdots\!60\)\( T^{5} + \)\(72\!\cdots\!20\)\( T^{6} + \)\(79\!\cdots\!20\)\( T^{7} + \)\(91\!\cdots\!63\)\( T^{8} + \)\(77\!\cdots\!20\)\( T^{9} + \)\(70\!\cdots\!90\)\( T^{10} + \)\(38\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!69\)\( T^{12} \)
$59$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(42\!\cdots\!34\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!15\)\( T^{4} - \)\(58\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!80\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!15\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!94\)\( T^{10} - \)\(74\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!61\)\( T^{12} \)
$61$ \( 1 - \)\(80\!\cdots\!52\)\( T + \)\(69\!\cdots\!06\)\( T^{2} - \)\(31\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!95\)\( T^{4} - \)\(49\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!04\)\( T^{6} - \)\(54\!\cdots\!72\)\( T^{7} + \)\(18\!\cdots\!95\)\( T^{8} - \)\(41\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} - \)\(12\!\cdots\!52\)\( T^{11} + \)\(17\!\cdots\!41\)\( T^{12} \)
$67$ \( 1 + \)\(18\!\cdots\!60\)\( T + \)\(82\!\cdots\!10\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!27\)\( T^{4} + \)\(39\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!80\)\( T^{6} + \)\(71\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!43\)\( T^{8} + \)\(76\!\cdots\!60\)\( T^{9} + \)\(90\!\cdots\!10\)\( T^{10} + \)\(36\!\cdots\!80\)\( T^{11} + \)\(36\!\cdots\!29\)\( T^{12} \)
$71$ \( 1 - \)\(38\!\cdots\!92\)\( T + \)\(15\!\cdots\!46\)\( T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} - \)\(21\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!95\)\( T^{8} - \)\(19\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!66\)\( T^{10} - \)\(68\!\cdots\!92\)\( T^{11} + \)\(31\!\cdots\!81\)\( T^{12} \)
$73$ \( 1 - \)\(61\!\cdots\!40\)\( T + \)\(69\!\cdots\!70\)\( T^{2} - \)\(30\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!07\)\( T^{4} - \)\(68\!\cdots\!20\)\( T^{5} + \)\(34\!\cdots\!60\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} + \)\(48\!\cdots\!83\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{9} + \)\(44\!\cdots\!70\)\( T^{10} - \)\(62\!\cdots\!80\)\( T^{11} + \)\(16\!\cdots\!09\)\( T^{12} \)
$79$ \( 1 + \)\(18\!\cdots\!40\)\( T + \)\(10\!\cdots\!14\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!15\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(78\!\cdots\!80\)\( T^{6} + \)\(96\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!15\)\( T^{8} + \)\(58\!\cdots\!00\)\( T^{9} + \)\(47\!\cdots\!94\)\( T^{10} + \)\(68\!\cdots\!60\)\( T^{11} + \)\(29\!\cdots\!81\)\( T^{12} \)
$83$ \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(28\!\cdots\!10\)\( T^{2} - \)\(27\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!27\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(85\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!43\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(76\!\cdots\!10\)\( T^{10} - \)\(15\!\cdots\!40\)\( T^{11} + \)\(43\!\cdots\!29\)\( T^{12} \)
$89$ \( 1 - \)\(88\!\cdots\!80\)\( T + \)\(39\!\cdots\!54\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!00\)\( T^{7} + \)\(55\!\cdots\!15\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(40\!\cdots\!94\)\( T^{10} - \)\(90\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \)
$97$ \( 1 - \)\(20\!\cdots\!60\)\( T + \)\(21\!\cdots\!90\)\( T^{2} - \)\(53\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!67\)\( T^{4} - \)\(73\!\cdots\!80\)\( T^{5} + \)\(38\!\cdots\!20\)\( T^{6} - \)\(66\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!63\)\( T^{8} - \)\(39\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!90\)\( T^{10} - \)\(12\!\cdots\!80\)\( T^{11} + \)\(53\!\cdots\!69\)\( T^{12} \)
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