Properties

Label 1.116.a
Level 1
Weight 116
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: \( k \) \(=\) \( 116 \)
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 \(M_{116}(\Gamma_0(1))\).

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

Trace form

\( 9q - 116180770089455544q^{2} + 344843759214607763941669692q^{3} + 104670059680921154232978400984552512q^{4} - 9416265734776133053944609096571667575890q^{5} + 256991120598421923060797165626503750361356768q^{6} + 1823100316304157772850498188908022382936745628056q^{7} - 219342532030837467786318171319354148041440220761600q^{8} + 26072637303355254030743970272505117374453788742960594013q^{9} + O(q^{10}) \) \( 9q - 116180770089455544q^{2} + 344843759214607763941669692q^{3} + 104670059680921154232978400984552512q^{4} - 9416265734776133053944609096571667575890q^{5} + 256991120598421923060797165626503750361356768q^{6} + 1823100316304157772850498188908022382936745628056q^{7} - 219342532030837467786318171319354148041440220761600q^{8} + 26072637303355254030743970272505117374453788742960594013q^{9} + 6889731598544239023205889874968132433435838108852679867760q^{10} - 445517152984004554452882006139810856593102366136026223610732q^{11} - 42821766403658150616877740531461298756313417540328083617855744q^{12} + 19120191369167112279983663882072248970668936436622013715636775942q^{13} - 874484410352602036165423857414222226528651993964136670010092090176q^{14} + 57599849217209390109884779325066884761013420104089343277210728579720q^{15} - 1898660452996213931642847506744088625997939708105388690720226275618816q^{16} + 33419179892190498379140269229352347311321292371418143383895702694253506q^{17} - 2890446730701337146666584437647276674841816394744930484862532014864739608q^{18} + 56323667191172324184321794810244934428905386312408168014796206412959893100q^{19} - 120777388922455433799357146114685194143652980138750550147727102823046920320q^{20} + 28176272485874451401581069312956900459966003840123920247504389614132269060768q^{21} + 210984989629936406579159416842818263081445382036693993098790665852739866289312q^{22} + 5043025461154544652592399598502520927437665517917355864726536202832817581701192q^{23} - 7479406270402029067027148815379966966654468567388028134173932380834142711449600q^{24} + 121685992880890163904850134332941584046982669818587694611007334128142016185838975q^{25} - 5914878085893066644910373209258644573936085196716403877449593179133594373009735632q^{26} - 5437898471824326237019158372464459291991052787593142902719608572344381893810183400q^{27} + 134945777488992574285149234278139874324546574917240128067914181134023530915243857408q^{28} + 2439895086110265548692574276791512440826974071718418782816151860616339046059592317750q^{29} - 1116585539518521415480577570330322838097236714153628065703800981349351376364839660480q^{30} - 115077428025700644556762203100792633275869803062860787032812289936236157369971844291232q^{31} + 182002942689670990671600876052552389114866685587533572692282985590686349099701286240256q^{32} + 369141873981337522256423851297810174364779891698538054811380159251387321031750289147184q^{33} + 16133584300638755367079868159687776681138177363676896002357157762964033939697234321079824q^{34} - 52628449929177569411912369226649064291208149114238009380905357827049459818370441573185840q^{35} - 343110691203669342080842230156946587735541070043733921826769094626380948288641172661825216q^{36} + 287193704739393607505131056484688448302333303851532744432838755870145511209962195474555406q^{37} + 1887285676044348998237316198233063024983501152361943246105012398202088272208210660325749600q^{38} + 57495405068562246295455616507616293122146350008991254547433422285102442790821436306805531176q^{39} - 377538131908105786834217342693439591573182569233316142689856344096115044105226591101231948800q^{40} + 450036474709634029239457226906183730406636229485540132536579898525082307383544109285527636618q^{41} - 3614527346318500983114558586611383169745609085282334175950975443992847582191151261402533817088q^{42} + 11238467180897591407367974873375005136437267786775173672510908822395481702707823219190726952692q^{43} + 8699999971697013733855125470633863112208674234411673595635316509240061153413462554031896481024q^{44} - 354262889245566735270191791009177826124944074544236912619488508381128369609584375919402912590330q^{45} + 211097623274700628625262911429039098879374408053211152239698326314586272783546202145178950255168q^{46} - 316976878807259053071085359106141878112628948954523917108980455021512269952447135287167972348144q^{47} + 4410525927214080719418968105605950188534122907207757808623172340024505114953341149753389138952192q^{48} - 5402924848400424806022935001474260663230730313005626183937801432100886221124285978670797398474063q^{49} - 107358804410623226001357232665018218738784300597386966398329608024306023910080977639712704319931400q^{50} + 392230202654118035824948006166789320012418877782515914171168122937227894930092554264584283119946168q^{51} + 186847646489930730643602674069595365244797337372298185295802970592172751947352587676622924463312256q^{52} + 334496168286075107324733466358006346326362975568659741057867434494885827999701558072502824894138942q^{53} + 815273172520443779763028707379283569820579781203093260711879845468622355550376013747519262940305600q^{54} - 23563721346912912517453864325899810796736448452180735731334842246903047272785132546150581553652938280q^{55} + 50501579633901923276218857218558899542358162690763518272752547626356309717401481440460372264435200000q^{56} + 19326182159286888425606068195996356732391193445128832852476856331643144551112440129841029596191125200q^{57} - 276537143978921040553065169225197607521839521198309506889392716069666566548578247536870903521071531600q^{58} + 26559202104122447477824144844523074937734984825144901102500175836351773839109503993732582002313425700q^{59} - 1442907433717223203332783871119423031323933407363314628882895297513470975585089853716965229400414768640q^{60} + 466761553545603468366117849047093343161086980381148338979195572768003416466078862368134301594309428118q^{61} + 28949531649262422331932844636470226821904033422254832199912074503071191433543806733834531878575668538112q^{62} - 26525953947162098838236213455266609374207032849533466856093515022060397668973168836773250206710710550408q^{63} - 46169280478215780445338819251740629236256446911950466222769056677599024448485239500558186236965165989888q^{64} + 30622590124662743929353875752733901757221609411050393019261768040290807869865915951670063818998022920820q^{65} + 58885521825482117580183439615438901701040523699794385163331031210597262925198785331149493283518615857536q^{66} + 522391041991614299569962418777982996178060317974008251288477038656645322955247899200108470501056740725756q^{67} - 963970789349034201430960171930241226642674599765083803055821765339886224671560293508121902691671583492992q^{68} - 4768744134714834748105886788969216829087468736062331961872791186003972694205687719806877940995404594939424q^{69} + 648102273330646244918318234903390383158022917116167184817069407742837923825354916186559362576551723218560q^{70} + 32114907384466138287517273839727096331385168126590004271972827521185198833360861938868844865182495384954968q^{71} + 134036350375106083011376263362911564045625075783489246636869772536858707114389024728586916900414866675980800q^{72} - 111532806797088262693446813432847569894350723689665928769373202408064805330900544787546502644693976667521558q^{73} + 146367501244421643149579339910137613783225032865117034030178763862717664956463164057696425085332997488423024q^{74} + 191011236006610982766017086665378142663007338029810651608018025472800438314119281980403153054303538075361700q^{75} + 1898752275573850608619854563706181213417282281074664303825494148687676736613874275716737807304301723964896000q^{76} + 8547320523119144259127272245019592075910296652673224064175151890175172791298973939747111362021472266254476512q^{77} + 8439857258376544014998782670363733435136214517799335731701302157574158558998894356877778911194771366104305984q^{78} - 2698128344097500588934092059818825897117643397087596829234810281027633187083900600989280771432492738133389200q^{79} + 117204555830492291605465104654149864678123037973909556311887751421858566469073727335484870898703213772212264960q^{80} + 243844251998840861544064848215817562747898683224158368441742398035262682571557660719126722958354994801550865809q^{81} + 473149820338187161445671999094492312190196735403799620250132055162721259658633637951105669203590924984388498512q^{82} + 976812723170338476169372114700471197638723418120402034771092851931233919858674637831041045969218865325658875692q^{83} + 2539321467869080915143952852614384190778021160799093130626725907993426541237386266518480321379169708798426343424q^{84} + 3552505762114082405879304975670471704857686108558874865895506783571862811153977851101780608475304927035932222460q^{85} + 11838276402773622473531002179261101730801162542213640979598425595359582429205612657343793928842614227749715441568q^{86} + 27897493284886904362947680243525683713496258927093426130335092322232937971279756139421073573801725541246460925800q^{87} + 52038964178994366220097267685544117931455067665556497129662971665594737875310678877513435089254305724995593676800q^{88} + 59446666639398225412533241825642117887211504765823780420331081870088511796669962981765187912981764097798200324250q^{89} + 259316827454272274338207071834855070930759935980768037766603430323600593416394106984590468235314680478785223708720q^{90} + 245939937247192661534362402615346617600082162228156200993853962356431783555050183953904527617619011036872688337168q^{91} + 613988446074365886926604878609711354412545763779069644197907853873975238992484853992399764939548985670950928032256q^{92} + 1223125501788227369263854591107492905645880857959405771691286774974787266171291063996752221071041873907531275398784q^{93} + 1499086835290066702674923853658203297973424744123451952620504314323336222430358359629023729105486706372629303556224q^{94} + 799041738816013640930530159507467835290369458135599646151456080273195210563093613872275650940535406365162133749800q^{95} + 3260013267518849074696485824622037660135078104167613123139522044708136137474372602480504246720813695045123797024768q^{96} + 2523963432996685893503346240622778846413532532685755052598994124525775010632417867867345710038197597044049732141106q^{97} + 585262872930561598617358018346679967180230161387125148358962641400172126269413925011919715802450101152197028464008q^{98} - 8131297049226938077737975811278315123841467115358913895092773707388454338324871486984094251614634664934679647525724q^{99} + O(q^{100}) \)

Decomposition of \(S_{116}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.116.a.a \(9\) \(83.750\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-1\!\cdots\!44\) \(34\!\cdots\!92\) \(-9\!\cdots\!90\) \(18\!\cdots\!56\) \(+\) \(q+(-12908974454383949-\beta _{1}+\cdots)q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 116180770089455544 T + \)\(14\!\cdots\!68\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!96\)\( T^{4} + \)\(10\!\cdots\!24\)\( T^{5} + \)\(65\!\cdots\!28\)\( T^{6} + \)\(58\!\cdots\!00\)\( T^{7} + \)\(31\!\cdots\!56\)\( T^{8} + \)\(27\!\cdots\!64\)\( T^{9} + \)\(13\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(47\!\cdots\!96\)\( T^{12} + \)\(30\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(73\!\cdots\!00\)\( T^{15} + \)\(30\!\cdots\!76\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{17} + \)\(36\!\cdots\!68\)\( T^{18} \)
$3$ \( 1 - \)\(34\!\cdots\!92\)\( T + \)\(20\!\cdots\!07\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!96\)\( T^{4} - \)\(28\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} + \)\(75\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} + \)\(12\!\cdots\!48\)\( T^{9} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(65\!\cdots\!96\)\( T^{12} - \)\(84\!\cdots\!32\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} - \)\(70\!\cdots\!00\)\( T^{15} + \)\(24\!\cdots\!01\)\( T^{16} - \)\(30\!\cdots\!92\)\( T^{17} + \)\(66\!\cdots\!07\)\( T^{18} \)
$5$ \( 1 + \)\(94\!\cdots\!90\)\( T + \)\(10\!\cdots\!25\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!00\)\( T^{4} + \)\(85\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!00\)\( T^{7} + \)\(89\!\cdots\!50\)\( T^{8} + \)\(95\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!50\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!00\)\( T^{12} + \)\(28\!\cdots\!00\)\( T^{13} + \)\(52\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!00\)\( T^{15} + \)\(49\!\cdots\!25\)\( T^{16} + \)\(10\!\cdots\!50\)\( T^{17} + \)\(27\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 - \)\(18\!\cdots\!56\)\( T + \)\(73\!\cdots\!43\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(81\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(38\!\cdots\!36\)\( T^{9} + \)\(25\!\cdots\!58\)\( T^{10} - \)\(49\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!96\)\( T^{12} - \)\(45\!\cdots\!76\)\( T^{13} + \)\(25\!\cdots\!28\)\( T^{14} - \)\(24\!\cdots\!00\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} - \)\(56\!\cdots\!56\)\( T^{17} + \)\(47\!\cdots\!43\)\( T^{18} \)
$11$ \( 1 + \)\(44\!\cdots\!32\)\( T + \)\(31\!\cdots\!03\)\( T^{2} + \)\(12\!\cdots\!84\)\( T^{3} + \)\(48\!\cdots\!60\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(49\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!76\)\( T^{7} + \)\(36\!\cdots\!58\)\( T^{8} + \)\(92\!\cdots\!80\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(47\!\cdots\!76\)\( T^{11} + \)\(93\!\cdots\!64\)\( T^{12} + \)\(17\!\cdots\!32\)\( T^{13} + \)\(30\!\cdots\!60\)\( T^{14} + \)\(44\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!53\)\( T^{16} + \)\(53\!\cdots\!32\)\( T^{17} + \)\(69\!\cdots\!51\)\( T^{18} \)
$13$ \( 1 - \)\(19\!\cdots\!42\)\( T + \)\(73\!\cdots\!57\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(66\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} - \)\(42\!\cdots\!94\)\( T^{8} + \)\(45\!\cdots\!48\)\( T^{9} - \)\(53\!\cdots\!58\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{11} + \)\(31\!\cdots\!96\)\( T^{12} - \)\(17\!\cdots\!32\)\( T^{13} + \)\(59\!\cdots\!72\)\( T^{14} - \)\(33\!\cdots\!00\)\( T^{15} + \)\(39\!\cdots\!01\)\( T^{16} - \)\(12\!\cdots\!42\)\( T^{17} + \)\(85\!\cdots\!57\)\( T^{18} \)
$17$ \( 1 - \)\(33\!\cdots\!06\)\( T + \)\(12\!\cdots\!93\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!96\)\( T^{4} - \)\(28\!\cdots\!76\)\( T^{5} + \)\(30\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} + \)\(14\!\cdots\!64\)\( T^{9} + \)\(33\!\cdots\!58\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{11} + \)\(97\!\cdots\!96\)\( T^{12} - \)\(29\!\cdots\!76\)\( T^{13} + \)\(23\!\cdots\!28\)\( T^{14} - \)\(22\!\cdots\!00\)\( T^{15} + \)\(40\!\cdots\!01\)\( T^{16} - \)\(34\!\cdots\!06\)\( T^{17} + \)\(32\!\cdots\!93\)\( T^{18} \)
$19$ \( 1 - \)\(56\!\cdots\!00\)\( T + \)\(55\!\cdots\!91\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!36\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{5} + \)\(30\!\cdots\!16\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(50\!\cdots\!74\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!84\)\( T^{12} - \)\(10\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!64\)\( T^{14} - \)\(54\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!09\)\( T^{16} - \)\(15\!\cdots\!00\)\( T^{17} + \)\(32\!\cdots\!99\)\( T^{18} \)
$23$ \( 1 - \)\(50\!\cdots\!92\)\( T + \)\(23\!\cdots\!07\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!96\)\( T^{4} - \)\(63\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} - \)\(37\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!06\)\( T^{8} - \)\(16\!\cdots\!52\)\( T^{9} + \)\(32\!\cdots\!42\)\( T^{10} - \)\(59\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!32\)\( T^{13} + \)\(23\!\cdots\!72\)\( T^{14} - \)\(29\!\cdots\!00\)\( T^{15} + \)\(36\!\cdots\!01\)\( T^{16} - \)\(31\!\cdots\!92\)\( T^{17} + \)\(24\!\cdots\!07\)\( T^{18} \)
$29$ \( 1 - \)\(24\!\cdots\!50\)\( T + \)\(79\!\cdots\!41\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!36\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!16\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!26\)\( T^{8} - \)\(41\!\cdots\!00\)\( T^{9} + \)\(80\!\cdots\!74\)\( T^{10} - \)\(68\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!84\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!64\)\( T^{14} - \)\(13\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!09\)\( T^{16} - \)\(62\!\cdots\!50\)\( T^{17} + \)\(38\!\cdots\!49\)\( T^{18} \)
$31$ \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(16\!\cdots\!03\)\( T^{2} + \)\(85\!\cdots\!84\)\( T^{3} + \)\(58\!\cdots\!60\)\( T^{4} + \)\(20\!\cdots\!32\)\( T^{5} + \)\(31\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!24\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{9} + \)\(37\!\cdots\!58\)\( T^{10} - \)\(71\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(21\!\cdots\!32\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!84\)\( T^{15} + \)\(56\!\cdots\!53\)\( T^{16} + \)\(12\!\cdots\!32\)\( T^{17} + \)\(36\!\cdots\!51\)\( T^{18} \)
$37$ \( 1 - \)\(28\!\cdots\!06\)\( T + \)\(16\!\cdots\!93\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!96\)\( T^{4} - \)\(48\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} - \)\(62\!\cdots\!36\)\( T^{9} + \)\(40\!\cdots\!58\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(62\!\cdots\!96\)\( T^{12} - \)\(11\!\cdots\!76\)\( T^{13} + \)\(65\!\cdots\!28\)\( T^{14} - \)\(67\!\cdots\!00\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} - \)\(15\!\cdots\!06\)\( T^{17} + \)\(12\!\cdots\!93\)\( T^{18} \)
$41$ \( 1 - \)\(45\!\cdots\!18\)\( T + \)\(83\!\cdots\!53\)\( T^{2} - \)\(41\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!64\)\( T^{6} - \)\(59\!\cdots\!24\)\( T^{7} + \)\(34\!\cdots\!58\)\( T^{8} - \)\(16\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} - \)\(51\!\cdots\!24\)\( T^{11} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{13} + \)\(87\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!53\)\( T^{16} - \)\(25\!\cdots\!18\)\( T^{17} + \)\(17\!\cdots\!01\)\( T^{18} \)
$43$ \( 1 - \)\(11\!\cdots\!92\)\( T + \)\(39\!\cdots\!07\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!96\)\( T^{4} - \)\(89\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(87\!\cdots\!52\)\( T^{9} + \)\(71\!\cdots\!42\)\( T^{10} - \)\(52\!\cdots\!00\)\( T^{11} + \)\(37\!\cdots\!96\)\( T^{12} - \)\(22\!\cdots\!32\)\( T^{13} + \)\(13\!\cdots\!72\)\( T^{14} - \)\(58\!\cdots\!00\)\( T^{15} + \)\(34\!\cdots\!01\)\( T^{16} - \)\(69\!\cdots\!92\)\( T^{17} + \)\(43\!\cdots\!07\)\( T^{18} \)
$47$ \( 1 + \)\(31\!\cdots\!44\)\( T + \)\(58\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(83\!\cdots\!76\)\( T^{5} + \)\(59\!\cdots\!28\)\( T^{6} - \)\(87\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!36\)\( T^{9} + \)\(26\!\cdots\!58\)\( T^{10} - \)\(33\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!96\)\( T^{12} - \)\(12\!\cdots\!76\)\( T^{13} + \)\(61\!\cdots\!28\)\( T^{14} + \)\(74\!\cdots\!00\)\( T^{15} + \)\(63\!\cdots\!01\)\( T^{16} + \)\(67\!\cdots\!44\)\( T^{17} + \)\(41\!\cdots\!43\)\( T^{18} \)
$53$ \( 1 - \)\(33\!\cdots\!42\)\( T + \)\(67\!\cdots\!57\)\( T^{2} - \)\(73\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!96\)\( T^{4} - \)\(32\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(98\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!06\)\( T^{8} - \)\(24\!\cdots\!52\)\( T^{9} + \)\(50\!\cdots\!42\)\( T^{10} - \)\(37\!\cdots\!00\)\( T^{11} + \)\(80\!\cdots\!96\)\( T^{12} - \)\(46\!\cdots\!32\)\( T^{13} + \)\(85\!\cdots\!72\)\( T^{14} - \)\(41\!\cdots\!00\)\( T^{15} + \)\(73\!\cdots\!01\)\( T^{16} - \)\(72\!\cdots\!42\)\( T^{17} + \)\(42\!\cdots\!57\)\( T^{18} \)
$59$ \( 1 - \)\(26\!\cdots\!00\)\( T + \)\(16\!\cdots\!91\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!16\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!26\)\( T^{8} + \)\(83\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!74\)\( T^{10} + \)\(30\!\cdots\!00\)\( T^{11} + \)\(64\!\cdots\!84\)\( T^{12} + \)\(73\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!64\)\( T^{14} + \)\(88\!\cdots\!00\)\( T^{15} + \)\(56\!\cdots\!09\)\( T^{16} - \)\(40\!\cdots\!00\)\( T^{17} + \)\(67\!\cdots\!99\)\( T^{18} \)
$61$ \( 1 - \)\(46\!\cdots\!18\)\( T + \)\(11\!\cdots\!53\)\( T^{2} - \)\(56\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!60\)\( T^{4} - \)\(25\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!24\)\( T^{7} + \)\(70\!\cdots\!58\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(29\!\cdots\!24\)\( T^{11} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!68\)\( T^{13} + \)\(24\!\cdots\!60\)\( T^{14} - \)\(42\!\cdots\!16\)\( T^{15} + \)\(18\!\cdots\!53\)\( T^{16} - \)\(14\!\cdots\!18\)\( T^{17} + \)\(65\!\cdots\!01\)\( T^{18} \)
$67$ \( 1 - \)\(52\!\cdots\!56\)\( T + \)\(67\!\cdots\!43\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(42\!\cdots\!28\)\( T^{6} - \)\(29\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!36\)\( T^{9} + \)\(59\!\cdots\!58\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(42\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!76\)\( T^{13} + \)\(21\!\cdots\!28\)\( T^{14} - \)\(43\!\cdots\!00\)\( T^{15} + \)\(65\!\cdots\!01\)\( T^{16} - \)\(50\!\cdots\!56\)\( T^{17} + \)\(97\!\cdots\!43\)\( T^{18} \)
$71$ \( 1 - \)\(32\!\cdots\!68\)\( T + \)\(52\!\cdots\!03\)\( T^{2} - \)\(14\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} - \)\(32\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!64\)\( T^{6} - \)\(44\!\cdots\!24\)\( T^{7} + \)\(22\!\cdots\!58\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!58\)\( T^{10} - \)\(27\!\cdots\!24\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!68\)\( T^{13} + \)\(39\!\cdots\!60\)\( T^{14} - \)\(34\!\cdots\!16\)\( T^{15} + \)\(95\!\cdots\!53\)\( T^{16} - \)\(46\!\cdots\!68\)\( T^{17} + \)\(11\!\cdots\!51\)\( T^{18} \)
$73$ \( 1 + \)\(11\!\cdots\!58\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!96\)\( T^{4} + \)\(48\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(50\!\cdots\!06\)\( T^{8} + \)\(34\!\cdots\!48\)\( T^{9} + \)\(97\!\cdots\!42\)\( T^{10} + \)\(56\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!96\)\( T^{12} + \)\(65\!\cdots\!68\)\( T^{13} + \)\(14\!\cdots\!72\)\( T^{14} + \)\(51\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!01\)\( T^{16} + \)\(20\!\cdots\!58\)\( T^{17} + \)\(34\!\cdots\!57\)\( T^{18} \)
$79$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(97\!\cdots\!91\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(34\!\cdots\!26\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!74\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{11} + \)\(72\!\cdots\!84\)\( T^{12} + \)\(54\!\cdots\!00\)\( T^{13} + \)\(65\!\cdots\!64\)\( T^{14} + \)\(44\!\cdots\!00\)\( T^{15} + \)\(38\!\cdots\!09\)\( T^{16} + \)\(17\!\cdots\!00\)\( T^{17} + \)\(11\!\cdots\!99\)\( T^{18} \)
$83$ \( 1 - \)\(97\!\cdots\!92\)\( T + \)\(80\!\cdots\!07\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!32\)\( T^{5} + \)\(28\!\cdots\!72\)\( T^{6} - \)\(83\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!06\)\( T^{8} - \)\(51\!\cdots\!52\)\( T^{9} + \)\(11\!\cdots\!42\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!96\)\( T^{12} - \)\(48\!\cdots\!32\)\( T^{13} + \)\(62\!\cdots\!72\)\( T^{14} - \)\(64\!\cdots\!00\)\( T^{15} + \)\(57\!\cdots\!01\)\( T^{16} - \)\(34\!\cdots\!92\)\( T^{17} + \)\(17\!\cdots\!07\)\( T^{18} \)
$89$ \( 1 - \)\(59\!\cdots\!50\)\( T + \)\(23\!\cdots\!41\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!36\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(59\!\cdots\!16\)\( T^{6} - \)\(95\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(17\!\cdots\!00\)\( T^{9} + \)\(20\!\cdots\!74\)\( T^{10} - \)\(21\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!84\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} + \)\(12\!\cdots\!64\)\( T^{14} - \)\(80\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!09\)\( T^{16} - \)\(16\!\cdots\!50\)\( T^{17} + \)\(41\!\cdots\!49\)\( T^{18} \)
$97$ \( 1 - \)\(25\!\cdots\!06\)\( T + \)\(18\!\cdots\!93\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(41\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!36\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!96\)\( T^{12} - \)\(33\!\cdots\!76\)\( T^{13} + \)\(44\!\cdots\!28\)\( T^{14} - \)\(35\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!01\)\( T^{16} - \)\(17\!\cdots\!06\)\( T^{17} + \)\(20\!\cdots\!93\)\( T^{18} \)
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