Properties

Label 1.118.a
Level 1
Weight 118
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 118 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{118}(\Gamma_0(1))\).

Total New Old
Modular forms 10 10 0
Cusp forms 9 9 0
Eisenstein series 1 1 0

Trace form

\( 9q + 40417036685158752q^{2} + 10302905323878152966875623636q^{3} + 679626474486553930371683615220261888q^{4} + 38453607070720263635688515692568185914150q^{5} - 4850209982284354841792124153951223028275738752q^{6} - 33936637323452994043236135663221508618375503117208q^{7} + 84807426568136809852447254974619690045330359678238720q^{8} + 228322388396621838147039528942639942618778890580268310597q^{9} + O(q^{10}) \) \( 9q + 40417036685158752q^{2} + 10302905323878152966875623636q^{3} + 679626474486553930371683615220261888q^{4} + 38453607070720263635688515692568185914150q^{5} - 4850209982284354841792124153951223028275738752q^{6} - 33936637323452994043236135663221508618375503117208q^{7} + 84807426568136809852447254974619690045330359678238720q^{8} + 228322388396621838147039528942639942618778890580268310597q^{9} - 116795550914172753602500302406542633798241819198212925771200q^{10} - 18850526893652982349030077823189856392436130547985552802563652q^{11} + 3227106826098906050636969251983952779002958811977260223984586752q^{12} - 98574936372230476445754809314443731682386957493373954090363278834q^{13} + 35258972517405629268130631622450754971434643685593795240367372736256q^{14} - 915629885182684846104756133008435145438889653925288537510093151318600q^{15} + 61702437097071514242829543900966314104457520182055990902400499737165824q^{16} - 680002505828907486847031510246007941006426153324907295268216079884418078q^{17} - 41733213100034654524842848551561180884346520214587575830257187744066967584q^{18} - 160689510555833725295714486201244011986205134154473703462344729085218574940q^{19} - 2238340134838118789188928650249499474871086998961353735072744834498868275200q^{20} - 325231066595782226476051172214780538694781261136602948155806527762690471674592q^{21} - 13112988820785042404307735263630112517712749663800581417200667121255782067843456q^{22} - 58153519408904797009281866377912077673911181772550205922884346377262357403061704q^{23} - 3038238801938958809634055377152977696212679179273938957003033953876165693929553920q^{24} + 5528673600140129203782645127829908484096231088932064383194690227200862517083894375q^{25} + 56682227661992970696514878061752865405652252745109723390003965691218585268522989888q^{26} + 1827774475813800388315702586366166077051345533885606497916848989661362329232290194120q^{27} - 6468486093767359175378792181544485287676854914825022016113590703460551157550283513856q^{28} - 48386553564988730519261358531155605067882474797775404905351553453290924873675961170210q^{29} - 933061727033306076487821089990705907034760165404132848486598428012292996783024527379200q^{30} - 621957331647921924682092054631242956130629654943098483596990494053452820493965659370592q^{31} + 5001840478035603540410109619911952695496967661900989837684340698950683681301601905016832q^{32} + 121811126973553577276247861343842508235672706892636830990043974788533005362046408289426992q^{33} - 739829614062010739581951319533134982709290713812079124382329752841795383755334475547959104q^{34} - 4738290388179207473702786515865081056559422688400941664804502829786561878838282647304413200q^{35} + 45099229254587225493722800395751937761885426855646977453280381634636333644471918552752149504q^{36} - 34964212885413714773919749832666304501127253457648317866689087418325240609603845719432098618q^{37} + 721325600351264057185221862351496532134128096990026176343425219143447708368813195315354449280q^{38} - 5256181041389152420659051237224453246783624348532029963454064662671494105217119495256720274216q^{39} - 3971440606190340326438464214808272022762933594464202348081080358302577556396774721686560768000q^{40} + 10020257933125498318762169522473576122868342311015731314988395755981139260089136547294206867738q^{41} + 252970816117377205232172452762620886935079264765511703102710428456975064426237455951343632231424q^{42} + 50615459145440588569921800707258109935625533244654510855325763075886611210775326812060053355356q^{43} - 4005524110904096730553185857622707642025685539352197636568303886000422137318429894399607389425664q^{44} + 5473276137456663664850729608899955954082083355607725618741940940860469618299772760360253073375950q^{45} + 1478376152522396457093466079811724410922025958141013781241190503228240084745237352373452997174528q^{46} + 16685567512808747231645711437000622991513185216117548515196412601564484313290777292415784238201712q^{47} - 216224080066172714935125828338658772066778469688001789861117569570442266773620007884590970572898304q^{48} + 169974231907168786505678866260391714519211137411643020101217953968783780867359385345616236202809313q^{49} - 2469575139897634839277498689855848621914551651256318080857839988659447958716938283028005975429980000q^{50} + 22542359242748231968428303678420614483799317368695374408674201779527797470892277868165393834070628328q^{51} - 3641374884516561660680840591520895881667064402106525036999341733309771062927915917819681324515084288q^{52} - 213846929196927043507342255756620254016567403070821501560181236631146130543117495693232522403826884714q^{53} + 70368920588017409616903857310987095626556658223669990059448426800295554432360247812557938014199560960q^{54} + 148096007032108307991537648756429447360412191714871283455777374803615140028070019602170536767368773800q^{55} + 4912754841589119447020005683504898647317892135202904974904770520342259478270647905318907285925688770560q^{56} + 4725267742102384458969291922476649493908095234626754028721076337022388939733447041972841854747117257040q^{57} - 61691450689404126451745332477746935968828799400881738634159938087335342513296447359674633686816855103680q^{58} - 7764769731115259168900983283133365701920906050369948450392029627233204947390357923079821008349713252820q^{59} + 242167860317000066114168168973610808827817432218852606538275645995232520514013583121633628591005079756800q^{60} - 222929606503067710799398508192199729986190593058228065549044227683974026321115005709568884316645733270402q^{61} + 1827141305370492398064568813635835649207511770956919266173894646721039854569012442466628721293062130637824q^{62} - 6149430428052066771901587011857473350643630395143732809634023246996722252441575455955324150064952541470264q^{63} + 1713574251954623356255571860783147114735206198681534690227478857137685402094507991899117111923895957454848q^{64} + 12869386968955442746581655138229434721016903840553054939904074922997747298934225413734850521793432682742900q^{65} - 19801437990698690478552825543250876645747446443556478809261957366009300212043482490741679345707416428383744q^{66} + 107684064602401021489570577643707663917970871646716750467969519975118177172469237075867827266943410721303572q^{67} - 251444068481959286401052998092041093692640408644327131004605869607770327086902833194726609492888262105757696q^{68} - 636551002833346124243317331997139947630296871486667493370629456953598473495382187316694584320495561912206496q^{69} + 2445861734303677525943378629352934810152324980084761531508197481523725822715645101402879849637685872106969600q^{70} + 1502419505224176558531294169685503912214338337110158873961577642731392413889415756979097582494411807941436328q^{71} - 14190460126457652958916758648699640896837300874267064764308762389188142317932680769710235258136844562310922240q^{72} - 3680626529239964132564912872164428299172360038443306444645976164259015478054371367373032231078544242855946054q^{73} + 18995030812524563839649408435663136474226268660391755695671253594362128606663820760673799433403903696813952576q^{74} - 5954144082365263500038634316183618316282448578954739961163265060170444211914471312789969527239069427687252500q^{75} + 75521635038824380969585182658956176528723379098044694676717523275104397685022044564398827919896405931443589120q^{76} - 240771158437798357271369248931142348166973346297260790267580673145058666334232888264118432356615568402288764576q^{77} - 1477784208875012842343990522137891883864267581783960725138689920617516099527473962194409706068146624342516014848q^{78} - 2285355792246583755429666361022638813506260919808216029514856597259553412365830319339899046809462718803389829360q^{79} + 731147255618903270849902761914980040425478725108304832815559424615479093332325975905808837098967573706466918400q^{80} - 3532367670468406933076216472986157092840312801374884126730791036696135691770424326872669760546172592566341977631q^{81} - 25860102286736164243905760962310812415991288634659729695499047849150555031014924251023990367180912563898028431936q^{82} - 26117732855991809313933081725190177898263990806499246105780578093346337131589033812952875783820862626814896023324q^{83} - 303841612882842424652673770557081077080232195176517190811304259946807947644403425630681019724069771858331742470144q^{84} - 271784820026506038543716236739677268739946655832804497399888502506585385926171661627342672703869328144134352799700q^{85} - 159145514743719971775182317155196763266785145636471136796587486856422371505286708178643130744777323230578900292992q^{86} - 1355100613113558654481164729536860328574661378469179428445142384013367192233258069363369980837305546640235173931240q^{87} - 5330018914167959490555492640418721613119677246727591406115888524964039124470757411539029636129427474678220869468160q^{88} - 4395897084707064164421802509143573377632162722728879042427722619171467149720889139563855226466565875308146508794230q^{89} - 23411936176011976544246884681770302612989489885021236648740047507452430500242447452152815918439699794497873950801600q^{90} - 17606324029223302595865761669828668634914557724735380270289077831587010762982273184351868955067224006438510779110352q^{91} - 55772404313727512066859492963963844589319577918377808030728482638027588994319973465872892384865446342595251431088128q^{92} - 116072789390555091884134123921208861440848184837899694086563014238934971188422968399738169216529674194608920356995968q^{93} - 215603007291864892759964647694822697504579441834466411893750448780856116572052774856258753465664728162623916745168384q^{94} - 363311345030338353402396700470528946025867499164153062121513673791488768341279059764913540333394357951818599838481000q^{95} - 701671877550835095764080820751918498721447481909785429122092358416515475269177135171212459682250091038670024691679232q^{96} - 298117805360022175643266169567709680719740267556363080426954452630296899863133621017987146964797348637751058662202638q^{97} - 2126339804064355593061344246003551033537506779034582996212286410224706625784123846976709615330292659973382337208735136q^{98} - 1520953441055457260978457109220493407200764700486006435781344466290947949395060067028631646046107176946794789660596916q^{99} + O(q^{100}) \)

Decomposition of \(S_{118}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.118.a.a \(9\) \(86.689\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(40\!\cdots\!52\) \(10\!\cdots\!36\) \(38\!\cdots\!50\) \(-3\!\cdots\!08\) \(+\) \(q+(4490781853906528-\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 40417036685158752 T + \)\(40\!\cdots\!32\)\( T^{2} - \)\(42\!\cdots\!20\)\( T^{3} + \)\(97\!\cdots\!56\)\( T^{4} - \)\(12\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!12\)\( T^{6} - \)\(39\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!76\)\( T^{8} - \)\(80\!\cdots\!72\)\( T^{9} + \)\(55\!\cdots\!72\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(85\!\cdots\!76\)\( T^{12} - \)\(96\!\cdots\!32\)\( T^{13} + \)\(12\!\cdots\!92\)\( T^{14} - \)\(89\!\cdots\!80\)\( T^{15} + \)\(14\!\cdots\!16\)\( T^{16} - \)\(23\!\cdots\!72\)\( T^{17} + \)\(96\!\cdots\!92\)\( T^{18} \)
$3$ \( 1 - \)\(10\!\cdots\!36\)\( T + \)\(23\!\cdots\!83\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!36\)\( T^{4} - \)\(34\!\cdots\!36\)\( T^{5} + \)\(40\!\cdots\!28\)\( T^{6} - \)\(33\!\cdots\!40\)\( T^{7} + \)\(34\!\cdots\!86\)\( T^{8} - \)\(25\!\cdots\!76\)\( T^{9} + \)\(22\!\cdots\!18\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!16\)\( T^{12} - \)\(67\!\cdots\!96\)\( T^{13} + \)\(47\!\cdots\!48\)\( T^{14} - \)\(21\!\cdots\!80\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} - \)\(39\!\cdots\!56\)\( T^{17} + \)\(25\!\cdots\!23\)\( T^{18} \)
$5$ \( 1 - \)\(38\!\cdots\!50\)\( T + \)\(25\!\cdots\!25\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!00\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!50\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!50\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(75\!\cdots\!00\)\( T^{12} - \)\(24\!\cdots\!00\)\( T^{13} + \)\(28\!\cdots\!00\)\( T^{14} - \)\(57\!\cdots\!00\)\( T^{15} + \)\(71\!\cdots\!25\)\( T^{16} - \)\(66\!\cdots\!50\)\( T^{17} + \)\(10\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 + \)\(33\!\cdots\!08\)\( T + \)\(38\!\cdots\!07\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!96\)\( T^{4} + \)\(17\!\cdots\!68\)\( T^{5} + \)\(88\!\cdots\!72\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(79\!\cdots\!06\)\( T^{8} + \)\(14\!\cdots\!48\)\( T^{9} + \)\(59\!\cdots\!42\)\( T^{10} + \)\(98\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!96\)\( T^{12} + \)\(55\!\cdots\!68\)\( T^{13} + \)\(17\!\cdots\!72\)\( T^{14} + \)\(20\!\cdots\!00\)\( T^{15} + \)\(52\!\cdots\!01\)\( T^{16} + \)\(34\!\cdots\!08\)\( T^{17} + \)\(77\!\cdots\!07\)\( T^{18} \)
$11$ \( 1 + \)\(18\!\cdots\!52\)\( T + \)\(53\!\cdots\!63\)\( T^{2} + \)\(80\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!04\)\( T^{6} + \)\(20\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!98\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!58\)\( T^{10} + \)\(99\!\cdots\!36\)\( T^{11} + \)\(66\!\cdots\!44\)\( T^{12} + \)\(38\!\cdots\!92\)\( T^{13} + \)\(21\!\cdots\!60\)\( T^{14} + \)\(92\!\cdots\!84\)\( T^{15} + \)\(42\!\cdots\!33\)\( T^{16} + \)\(10\!\cdots\!72\)\( T^{17} + \)\(38\!\cdots\!31\)\( T^{18} \)
$13$ \( 1 + \)\(98\!\cdots\!34\)\( T + \)\(70\!\cdots\!73\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{5} + \)\(45\!\cdots\!28\)\( T^{6} - \)\(35\!\cdots\!80\)\( T^{7} + \)\(72\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!16\)\( T^{9} + \)\(15\!\cdots\!78\)\( T^{10} - \)\(16\!\cdots\!20\)\( T^{11} + \)\(44\!\cdots\!36\)\( T^{12} - \)\(21\!\cdots\!16\)\( T^{13} + \)\(10\!\cdots\!68\)\( T^{14} + \)\(32\!\cdots\!40\)\( T^{15} + \)\(14\!\cdots\!21\)\( T^{16} + \)\(44\!\cdots\!94\)\( T^{17} + \)\(96\!\cdots\!53\)\( T^{18} \)
$17$ \( 1 + \)\(68\!\cdots\!78\)\( T + \)\(50\!\cdots\!57\)\( T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!76\)\( T^{4} + \)\(13\!\cdots\!48\)\( T^{5} + \)\(23\!\cdots\!92\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!66\)\( T^{8} + \)\(22\!\cdots\!88\)\( T^{9} + \)\(27\!\cdots\!82\)\( T^{10} + \)\(17\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!36\)\( T^{12} + \)\(92\!\cdots\!68\)\( T^{13} + \)\(85\!\cdots\!32\)\( T^{14} + \)\(28\!\cdots\!60\)\( T^{15} + \)\(27\!\cdots\!21\)\( T^{16} + \)\(34\!\cdots\!18\)\( T^{17} + \)\(45\!\cdots\!37\)\( T^{18} \)
$19$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(24\!\cdots\!51\)\( T^{2} + \)\(61\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!56\)\( T^{4} + \)\(88\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!96\)\( T^{6} + \)\(70\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!66\)\( T^{8} + \)\(36\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!74\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!24\)\( T^{12} + \)\(25\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!44\)\( T^{14} + \)\(29\!\cdots\!80\)\( T^{15} + \)\(49\!\cdots\!29\)\( T^{16} + \)\(13\!\cdots\!40\)\( T^{17} + \)\(33\!\cdots\!59\)\( T^{18} \)
$23$ \( 1 + \)\(58\!\cdots\!04\)\( T + \)\(10\!\cdots\!63\)\( T^{2} + \)\(50\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!16\)\( T^{4} + \)\(22\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} + \)\(71\!\cdots\!80\)\( T^{7} + \)\(46\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!44\)\( T^{9} + \)\(97\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!56\)\( T^{12} + \)\(44\!\cdots\!64\)\( T^{13} + \)\(21\!\cdots\!88\)\( T^{14} + \)\(43\!\cdots\!60\)\( T^{15} + \)\(18\!\cdots\!81\)\( T^{16} + \)\(21\!\cdots\!44\)\( T^{17} + \)\(79\!\cdots\!83\)\( T^{18} \)
$29$ \( 1 + \)\(48\!\cdots\!10\)\( T + \)\(49\!\cdots\!81\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!16\)\( T^{4} + \)\(28\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!36\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} + \)\(19\!\cdots\!00\)\( T^{9} + \)\(94\!\cdots\!74\)\( T^{10} + \)\(25\!\cdots\!40\)\( T^{11} + \)\(24\!\cdots\!44\)\( T^{12} + \)\(71\!\cdots\!80\)\( T^{13} + \)\(34\!\cdots\!84\)\( T^{14} + \)\(73\!\cdots\!20\)\( T^{15} + \)\(25\!\cdots\!89\)\( T^{16} + \)\(30\!\cdots\!10\)\( T^{17} + \)\(80\!\cdots\!89\)\( T^{18} \)
$31$ \( 1 + \)\(62\!\cdots\!92\)\( T + \)\(97\!\cdots\!83\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{3} + \)\(55\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(27\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} + \)\(15\!\cdots\!80\)\( T^{9} + \)\(35\!\cdots\!58\)\( T^{10} + \)\(43\!\cdots\!56\)\( T^{11} + \)\(79\!\cdots\!04\)\( T^{12} + \)\(95\!\cdots\!12\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(25\!\cdots\!93\)\( T^{16} + \)\(51\!\cdots\!52\)\( T^{17} + \)\(25\!\cdots\!91\)\( T^{18} \)
$37$ \( 1 + \)\(34\!\cdots\!18\)\( T + \)\(16\!\cdots\!57\)\( T^{2} + \)\(49\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!36\)\( T^{4} + \)\(37\!\cdots\!08\)\( T^{5} + \)\(67\!\cdots\!32\)\( T^{6} + \)\(19\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!86\)\( T^{8} + \)\(68\!\cdots\!68\)\( T^{9} + \)\(79\!\cdots\!62\)\( T^{10} + \)\(17\!\cdots\!60\)\( T^{11} + \)\(18\!\cdots\!16\)\( T^{12} + \)\(31\!\cdots\!68\)\( T^{13} + \)\(32\!\cdots\!52\)\( T^{14} + \)\(37\!\cdots\!80\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} + \)\(24\!\cdots\!38\)\( T^{17} + \)\(20\!\cdots\!97\)\( T^{18} \)
$41$ \( 1 - \)\(10\!\cdots\!38\)\( T + \)\(34\!\cdots\!93\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!60\)\( T^{4} - \)\(35\!\cdots\!68\)\( T^{5} + \)\(57\!\cdots\!24\)\( T^{6} - \)\(29\!\cdots\!44\)\( T^{7} + \)\(40\!\cdots\!18\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(20\!\cdots\!58\)\( T^{10} - \)\(72\!\cdots\!84\)\( T^{11} + \)\(70\!\cdots\!84\)\( T^{12} - \)\(21\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!60\)\( T^{14} - \)\(40\!\cdots\!16\)\( T^{15} + \)\(25\!\cdots\!73\)\( T^{16} - \)\(36\!\cdots\!58\)\( T^{17} + \)\(18\!\cdots\!21\)\( T^{18} \)
$43$ \( 1 - \)\(50\!\cdots\!56\)\( T + \)\(59\!\cdots\!43\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!96\)\( T^{4} - \)\(55\!\cdots\!76\)\( T^{5} + \)\(43\!\cdots\!28\)\( T^{6} - \)\(92\!\cdots\!00\)\( T^{7} + \)\(72\!\cdots\!06\)\( T^{8} - \)\(12\!\cdots\!36\)\( T^{9} + \)\(94\!\cdots\!58\)\( T^{10} - \)\(15\!\cdots\!00\)\( T^{11} + \)\(96\!\cdots\!96\)\( T^{12} - \)\(16\!\cdots\!76\)\( T^{13} + \)\(73\!\cdots\!28\)\( T^{14} - \)\(12\!\cdots\!00\)\( T^{15} + \)\(38\!\cdots\!01\)\( T^{16} - \)\(42\!\cdots\!56\)\( T^{17} + \)\(11\!\cdots\!43\)\( T^{18} \)
$47$ \( 1 - \)\(16\!\cdots\!12\)\( T + \)\(18\!\cdots\!07\)\( T^{2} - \)\(84\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!16\)\( T^{4} - \)\(12\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(93\!\cdots\!80\)\( T^{7} + \)\(63\!\cdots\!46\)\( T^{8} - \)\(48\!\cdots\!92\)\( T^{9} + \)\(27\!\cdots\!02\)\( T^{10} - \)\(17\!\cdots\!20\)\( T^{11} + \)\(93\!\cdots\!56\)\( T^{12} - \)\(42\!\cdots\!32\)\( T^{13} + \)\(25\!\cdots\!12\)\( T^{14} - \)\(54\!\cdots\!60\)\( T^{15} + \)\(51\!\cdots\!81\)\( T^{16} - \)\(20\!\cdots\!52\)\( T^{17} + \)\(52\!\cdots\!27\)\( T^{18} \)
$53$ \( 1 + \)\(21\!\cdots\!14\)\( T + \)\(40\!\cdots\!33\)\( T^{2} + \)\(48\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!36\)\( T^{4} + \)\(50\!\cdots\!64\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{7} + \)\(32\!\cdots\!86\)\( T^{8} + \)\(23\!\cdots\!24\)\( T^{9} + \)\(17\!\cdots\!18\)\( T^{10} + \)\(11\!\cdots\!40\)\( T^{11} + \)\(80\!\cdots\!16\)\( T^{12} + \)\(46\!\cdots\!04\)\( T^{13} + \)\(28\!\cdots\!48\)\( T^{14} + \)\(13\!\cdots\!20\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} + \)\(17\!\cdots\!94\)\( T^{17} + \)\(45\!\cdots\!73\)\( T^{18} \)
$59$ \( 1 + \)\(77\!\cdots\!20\)\( T + \)\(87\!\cdots\!71\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!96\)\( T^{4} + \)\(70\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!56\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!46\)\( T^{8} + \)\(44\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!74\)\( T^{10} + \)\(54\!\cdots\!80\)\( T^{11} + \)\(38\!\cdots\!04\)\( T^{12} + \)\(40\!\cdots\!60\)\( T^{13} + \)\(33\!\cdots\!04\)\( T^{14} + \)\(16\!\cdots\!40\)\( T^{15} + \)\(18\!\cdots\!69\)\( T^{16} + \)\(25\!\cdots\!20\)\( T^{17} + \)\(50\!\cdots\!79\)\( T^{18} \)
$61$ \( 1 + \)\(22\!\cdots\!02\)\( T + \)\(27\!\cdots\!13\)\( T^{2} + \)\(63\!\cdots\!04\)\( T^{3} + \)\(43\!\cdots\!60\)\( T^{4} + \)\(93\!\cdots\!32\)\( T^{5} + \)\(51\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(49\!\cdots\!98\)\( T^{8} + \)\(91\!\cdots\!80\)\( T^{9} + \)\(38\!\cdots\!58\)\( T^{10} + \)\(60\!\cdots\!36\)\( T^{11} + \)\(23\!\cdots\!44\)\( T^{12} + \)\(31\!\cdots\!92\)\( T^{13} + \)\(11\!\cdots\!60\)\( T^{14} + \)\(12\!\cdots\!84\)\( T^{15} + \)\(42\!\cdots\!33\)\( T^{16} + \)\(26\!\cdots\!22\)\( T^{17} + \)\(89\!\cdots\!81\)\( T^{18} \)
$67$ \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(26\!\cdots\!07\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!76\)\( T^{4} - \)\(36\!\cdots\!52\)\( T^{5} + \)\(35\!\cdots\!92\)\( T^{6} - \)\(28\!\cdots\!20\)\( T^{7} + \)\(22\!\cdots\!66\)\( T^{8} - \)\(15\!\cdots\!12\)\( T^{9} + \)\(10\!\cdots\!82\)\( T^{10} - \)\(56\!\cdots\!80\)\( T^{11} + \)\(32\!\cdots\!36\)\( T^{12} - \)\(14\!\cdots\!32\)\( T^{13} + \)\(68\!\cdots\!32\)\( T^{14} - \)\(23\!\cdots\!40\)\( T^{15} + \)\(95\!\cdots\!21\)\( T^{16} - \)\(17\!\cdots\!32\)\( T^{17} + \)\(71\!\cdots\!87\)\( T^{18} \)
$71$ \( 1 - \)\(15\!\cdots\!28\)\( T + \)\(16\!\cdots\!23\)\( T^{2} - \)\(20\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(11\!\cdots\!84\)\( T^{7} + \)\(55\!\cdots\!38\)\( T^{8} - \)\(53\!\cdots\!20\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} - \)\(17\!\cdots\!04\)\( T^{11} + \)\(66\!\cdots\!24\)\( T^{12} - \)\(46\!\cdots\!48\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(80\!\cdots\!16\)\( T^{15} + \)\(24\!\cdots\!13\)\( T^{16} - \)\(90\!\cdots\!88\)\( T^{17} + \)\(23\!\cdots\!11\)\( T^{18} \)
$73$ \( 1 + \)\(36\!\cdots\!54\)\( T + \)\(66\!\cdots\!13\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(21\!\cdots\!16\)\( T^{4} + \)\(73\!\cdots\!44\)\( T^{5} + \)\(43\!\cdots\!28\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!46\)\( T^{8} + \)\(17\!\cdots\!44\)\( T^{9} + \)\(63\!\cdots\!38\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(46\!\cdots\!56\)\( T^{12} + \)\(79\!\cdots\!64\)\( T^{13} + \)\(23\!\cdots\!88\)\( T^{14} + \)\(26\!\cdots\!60\)\( T^{15} + \)\(77\!\cdots\!81\)\( T^{16} + \)\(43\!\cdots\!94\)\( T^{17} + \)\(11\!\cdots\!33\)\( T^{18} \)
$79$ \( 1 + \)\(22\!\cdots\!60\)\( T + \)\(85\!\cdots\!31\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!16\)\( T^{4} + \)\(48\!\cdots\!80\)\( T^{5} + \)\(76\!\cdots\!36\)\( T^{6} + \)\(93\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!74\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(88\!\cdots\!44\)\( T^{12} + \)\(60\!\cdots\!80\)\( T^{13} + \)\(42\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!89\)\( T^{16} + \)\(34\!\cdots\!60\)\( T^{17} + \)\(15\!\cdots\!39\)\( T^{18} \)
$83$ \( 1 + \)\(26\!\cdots\!24\)\( T + \)\(19\!\cdots\!03\)\( T^{2} + \)\(55\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} + \)\(53\!\cdots\!84\)\( T^{5} + \)\(14\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!40\)\( T^{7} + \)\(69\!\cdots\!26\)\( T^{8} + \)\(12\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!98\)\( T^{10} + \)\(36\!\cdots\!60\)\( T^{11} + \)\(55\!\cdots\!76\)\( T^{12} + \)\(72\!\cdots\!44\)\( T^{13} + \)\(92\!\cdots\!08\)\( T^{14} + \)\(87\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} + \)\(47\!\cdots\!44\)\( T^{17} + \)\(61\!\cdots\!63\)\( T^{18} \)
$89$ \( 1 + \)\(43\!\cdots\!30\)\( T + \)\(14\!\cdots\!61\)\( T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + \)\(73\!\cdots\!76\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!76\)\( T^{6} + \)\(28\!\cdots\!20\)\( T^{7} + \)\(36\!\cdots\!06\)\( T^{8} + \)\(41\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!74\)\( T^{10} + \)\(41\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!64\)\( T^{12} + \)\(26\!\cdots\!40\)\( T^{13} + \)\(18\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!60\)\( T^{15} + \)\(52\!\cdots\!49\)\( T^{16} + \)\(18\!\cdots\!30\)\( T^{17} + \)\(51\!\cdots\!69\)\( T^{18} \)
$97$ \( 1 + \)\(29\!\cdots\!38\)\( T + \)\(15\!\cdots\!57\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} + \)\(27\!\cdots\!88\)\( T^{5} + \)\(70\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!46\)\( T^{8} + \)\(42\!\cdots\!08\)\( T^{9} + \)\(76\!\cdots\!02\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(16\!\cdots\!56\)\( T^{12} + \)\(17\!\cdots\!68\)\( T^{13} + \)\(23\!\cdots\!12\)\( T^{14} + \)\(20\!\cdots\!40\)\( T^{15} + \)\(22\!\cdots\!81\)\( T^{16} + \)\(12\!\cdots\!98\)\( T^{17} + \)\(11\!\cdots\!77\)\( T^{18} \)
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