Properties

Label 1.86.a
Level 1
Weight 86
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 7
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 86 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{86}(\Gamma_0(1))\).

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

Trace form

\( 6q - 3596910688800q^{2} - 158574346104614272200q^{3} + 140991407339885551387640832q^{4} - 937371071374809475998758615100q^{5} - 2163199670798237803919272036318848q^{6} + 376764526445862177226143056468406000q^{7} + 444573639427992844351424894916577689600q^{8} + 57207108743130009711910898489373895669038q^{9} + O(q^{10}) \) \( 6q - 3596910688800q^{2} - 158574346104614272200q^{3} + 140991407339885551387640832q^{4} - 937371071374809475998758615100q^{5} - 2163199670798237803919272036318848q^{6} + 376764526445862177226143056468406000q^{7} + 444573639427992844351424894916577689600q^{8} + 57207108743130009711910898489373895669038q^{9} + 94651165962435813693946284861726959764800q^{10} - 56344805701335918998354740625397619012956888q^{11} - 38113762524253328553901066462492282955462553600q^{12} - 380409468466294979156527797290184755501373645900q^{13} + 7432102291035232990382031969981134697595704847104q^{14} + 144434402836684505734903243186122115181592351080400q^{15} + 2636948568634008151994191667083023793288011486068736q^{16} - 51365945458144838311659694040021464865555671618554900q^{17} + 463507663069864508791700717104909645587637083805192800q^{18} + 2653138201866103222242250000135395331152453781661358360q^{19} - 52474536419792695778976723338965188133980339812187187200q^{20} - 58207920650233869679987953151176418416472072173047654208q^{21} + 3581897081140779531372538339240606543513118132263441718400q^{22} - 2885510207115801266245992081450013398631817932489294263600q^{23} - 68567070931993796242748985500475684139954880414010505297920q^{24} + 669968020810517762245853029111649576216707129701526846916250q^{25} - 145338677717367061512932551412743816475529536844287947743168q^{26} - 2889483981995895623844053829009837579840855941832615148040400q^{27} + 26317831788111975278561322769557791839073407987295193709977600q^{28} - 126005957659709196887223202461409590769523616790635653593969260q^{29} + 136086353401335907485123608360817834511749084976435568495980800q^{30} - 5173343222869174361945421290475955243870433981978751385509914688q^{31} + 18707915462589185913352646999729455863851090555123851448234803200q^{32} + 41065540677287661644797365484571058991908764323686266561418093600q^{33} - 261567067349813682388055634030637558284890219695677173368549801536q^{34} - 54720272714504178732776803597096361295121671934223039039646799200q^{35} + 3399696315710870996405230044380107361656113863069854483749794804736q^{36} - 5097772448818865838142977649516403030637492688710699135052039832700q^{37} - 48700117286552805406633697613393636582143669921611431061746550761600q^{38} + 110170203428145375146126906937514681017464730369816703999974469819536q^{39} - 142216395874814214555238931398128417040750190812408712268526706688000q^{40} - 37443196175066827621993378286362463826782697218192853395514169611588q^{41} - 3196148185952018443143303103341158979894551408731781313074799986918400q^{42} - 1200203503033233321208433978807766368170132198490662545566874292959000q^{43} - 18639061742892206758508411691062975986004422355135358223957061170139136q^{44} - 86500611154071576329235297315912364271843998646281427040496904001252300q^{45} - 191201172378193769791517179835271412887470991983918549920785652631609088q^{46} - 239424902055614885318194585971856964272866563499240512979067317687189600q^{47} - 2011644131471271504966965322061354320977669648116299895717796775041433600q^{48} - 3550036072199457619495079372134003262477445775089732177587676397325457258q^{49} - 11494548593012586974824937998111766287236762210273483206149818153415980000q^{50} - 10328830658918507785729550461151529723270274866387219032022415269933043728q^{51} - 58712069233310142887861446777253810318205781466445594810258784313124096000q^{52} - 50490336878498884627542374457418517146058176491360508941999343448428076700q^{53} - 127995453221656133194026073100343370388415314716409270018601938523610333440q^{54} + 39301939079139487846929886085782264562447760597309570618345636737530354800q^{55} + 587681993552427976427603903951469305598093495610137163751240432058398146560q^{56} + 609918064048630066051419788625468329088455176450389301471226460423851647200q^{57} + 1968773654564353990385431635289950834066592911242318428340655057560215041600q^{58} + 6616590288827867291331513194467053420958403068835561414972249041164074836680q^{59} + 31689950181369930414484676957647463168779067278456560490215008364167133388800q^{60} + 14357623272339506273937696878826484910577784548921932472541795844444520630612q^{61} + 6341655988756470271607502111666483932112595649902706601178034698608656870400q^{62} + 35795966074431291550182082060995682584043172792907267107900820593108358481200q^{63} - 15036793202050903912474653273096694719068286214625123977211579195245857865728q^{64} - 207821434856417257245443053802293031083793273150318306362240037299645012878600q^{65} - 1499689309727229977020921941751495239601349525904665530644129573530308780586496q^{66} - 676342476586532149280756429155984326543089556982073869153059292790609382659400q^{67} - 3163312242521042817824168546059434229649323239544821535981619303392310998579200q^{68} - 745502955929877889390374691705185133399296067237471890306223900077423420825024q^{69} - 6084039126370540917643976340500810012180196811570702685650569892863510961958400q^{70} - 6034633570006156478287618014555933721359683069405109018337202921275711590822288q^{71} + 42205813302686543920197586259094606661421265173628042056899709260796372565196800q^{72} + 35801030708258502256835696755718519936386302900895514245440128302587327554955900q^{73} + 107340388271157498582387358259935699803904453010588240913436902735784529206718784q^{74} + 145624067696916644200840962911775023608813806434012931341647651068546287549085000q^{75} + 161366039234563681095121956921292746446621209914504282248055858395547636251627520q^{76} + 251398792676383195671611646509617352303457127744134928661968949244057263571048000q^{77} + 171841231286923953547335431872475383039854385588725970844304858012728142132768000q^{78} - 1371264739680910465180747804297591457073099822996075412863178473301594188074547360q^{79} - 4133302113696354289363675864719400535772990320893278365909081697225137404615065600q^{80} - 4310060020996155465671846173285816617646939986231807893561334612249450355160008554q^{81} - 1521817925451755348555955193889971681134404529402094946034920762170082779676769600q^{82} - 6003587143963376493630220870754035373224271946820478343977301287860006839810301800q^{83} - 12398071481018763514585592745934914502465484593761365063657562420201819291615461376q^{84} + 17517819416067376478784379475048913828437460883636030543204171384490930355142297800q^{85} + 21492879965867203934779808388379416439833228958041181100627630027642419671434758272q^{86} + 50876914632638320845048890401340559827753840179451919522781964204571716769119542800q^{87} + 272530707230233729788642007241687387343090541246894873629875449706966387853783859200q^{88} + 160168600498492831669414330089333488062247500152820008689246769174592053409073033820q^{89} + 84790751750172409977703550300414634726575284296402896259290449409557083536687630400q^{90} - 25835286532054033379869466839714977397623074555127707439218236615589791891497666528q^{91} - 409168494351117176400001947891440713569740400895293133641599839567086438014810726400q^{92} - 593175476050386740279600068063559704679480457431525655959184215070525871275273350400q^{93} - 1363919087472816080652148503710266001883533236220476628035821611168261836377449504256q^{94} - 5333227198671761899795642368410473530089556162431166206847226445312136951029790006000q^{95} - 8776777977440964474954793317935829637440571398014671264365724282514189936233386344448q^{96} + 1982258923770441792809455137165411551481047984016852591852301644261535493014975149900q^{97} + 4998389537167322966144480371748484731963375313000939154777788888434561052122410226400q^{98} + 14740689076241711017386038088080153281412276754860625822798307085897192961497015786376q^{99} + O(q^{100}) \)

Decomposition of \(S_{86}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.86.a.a \(6\) \(45.755\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-3\!\cdots\!00\) \(-1\!\cdots\!00\) \(-9\!\cdots\!00\) \(37\!\cdots\!00\) \(+\) \(q+(-599485114800+\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3596910688800 T + \)\(52\!\cdots\!80\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!72\)\( T^{4} - \)\(38\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!40\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!28\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!80\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!24\)\( T^{12} \)
$3$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(91\!\cdots\!10\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!47\)\( T^{4} + \)\(60\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(72\!\cdots\!03\)\( T^{8} + \)\(56\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!10\)\( T^{10} + \)\(94\!\cdots\!00\)\( T^{11} + \)\(21\!\cdots\!49\)\( T^{12} \)
$5$ \( 1 + \)\(93\!\cdots\!00\)\( T + \)\(87\!\cdots\!50\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!75\)\( T^{4} + \)\(68\!\cdots\!00\)\( T^{5} + \)\(41\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!75\)\( T^{8} + \)\(76\!\cdots\!00\)\( T^{9} + \)\(39\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(29\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(37\!\cdots\!00\)\( T + \)\(38\!\cdots\!50\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(57\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(29\!\cdots\!03\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{9} + \)\(83\!\cdots\!50\)\( T^{10} - \)\(55\!\cdots\!00\)\( T^{11} + \)\(10\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(56\!\cdots\!88\)\( T + \)\(11\!\cdots\!66\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!95\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(21\!\cdots\!24\)\( T^{6} + \)\(33\!\cdots\!08\)\( T^{7} + \)\(63\!\cdots\!95\)\( T^{8} + \)\(50\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!66\)\( T^{10} + \)\(22\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!01\)\( T^{12} \)
$13$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(21\!\cdots\!70\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!47\)\( T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!03\)\( T^{8} + \)\(60\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!70\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 + \)\(51\!\cdots\!00\)\( T + \)\(19\!\cdots\!90\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!47\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!20\)\( T^{6} + \)\(92\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!03\)\( T^{8} + \)\(28\!\cdots\!00\)\( T^{9} + \)\(42\!\cdots\!90\)\( T^{10} + \)\(44\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!49\)\( T^{12} \)
$19$ \( 1 - \)\(26\!\cdots\!60\)\( T + \)\(16\!\cdots\!94\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!00\)\( T^{5} + \)\(97\!\cdots\!80\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!15\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!94\)\( T^{10} - \)\(78\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!01\)\( T^{12} \)
$23$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(19\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!03\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!49\)\( T^{12} \)
$29$ \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(73\!\cdots\!94\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} + \)\(60\!\cdots\!15\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!94\)\( T^{10} + \)\(41\!\cdots\!40\)\( T^{11} + \)\(66\!\cdots\!01\)\( T^{12} \)
$31$ \( 1 + \)\(51\!\cdots\!88\)\( T + \)\(33\!\cdots\!66\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!95\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{5} + \)\(36\!\cdots\!24\)\( T^{6} + \)\(73\!\cdots\!08\)\( T^{7} + \)\(16\!\cdots\!95\)\( T^{8} + \)\(23\!\cdots\!80\)\( T^{9} + \)\(38\!\cdots\!66\)\( T^{10} + \)\(34\!\cdots\!88\)\( T^{11} + \)\(39\!\cdots\!01\)\( T^{12} \)
$37$ \( 1 + \)\(50\!\cdots\!00\)\( T + \)\(93\!\cdots\!70\)\( T^{2} + \)\(48\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!47\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(98\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!70\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(60\!\cdots\!49\)\( T^{12} \)
$41$ \( 1 + \)\(37\!\cdots\!88\)\( T + \)\(48\!\cdots\!66\)\( T^{2} + \)\(73\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} + \)\(23\!\cdots\!08\)\( T^{5} + \)\(14\!\cdots\!24\)\( T^{6} + \)\(28\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(13\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(33\!\cdots\!01\)\( T^{12} \)
$43$ \( 1 + \)\(12\!\cdots\!00\)\( T + \)\(24\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!47\)\( T^{4} + \)\(47\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} + \)\(32\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} + \)\(13\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(23\!\cdots\!00\)\( T + \)\(51\!\cdots\!10\)\( T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(78\!\cdots\!47\)\( T^{4} + \)\(65\!\cdots\!00\)\( T^{5} + \)\(77\!\cdots\!80\)\( T^{6} + \)\(87\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(15\!\cdots\!00\)\( T^{9} + \)\(16\!\cdots\!10\)\( T^{10} + \)\(10\!\cdots\!00\)\( T^{11} + \)\(58\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 + \)\(50\!\cdots\!00\)\( T + \)\(23\!\cdots\!10\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!47\)\( T^{4} + \)\(34\!\cdots\!00\)\( T^{5} + \)\(76\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!03\)\( T^{8} + \)\(30\!\cdots\!00\)\( T^{9} + \)\(41\!\cdots\!10\)\( T^{10} + \)\(33\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!49\)\( T^{12} \)
$59$ \( 1 - \)\(66\!\cdots\!80\)\( T + \)\(33\!\cdots\!94\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!15\)\( T^{4} - \)\(78\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} - \)\(26\!\cdots\!00\)\( T^{7} + \)\(37\!\cdots\!15\)\( T^{8} - \)\(42\!\cdots\!00\)\( T^{9} + \)\(41\!\cdots\!94\)\( T^{10} - \)\(27\!\cdots\!20\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \)
$61$ \( 1 - \)\(14\!\cdots\!12\)\( T + \)\(25\!\cdots\!66\)\( T^{2} - \)\(25\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!95\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!24\)\( T^{6} - \)\(12\!\cdots\!92\)\( T^{7} + \)\(90\!\cdots\!95\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{9} + \)\(26\!\cdots\!66\)\( T^{10} - \)\(83\!\cdots\!12\)\( T^{11} + \)\(32\!\cdots\!01\)\( T^{12} \)
$67$ \( 1 + \)\(67\!\cdots\!00\)\( T + \)\(80\!\cdots\!90\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!47\)\( T^{4} + \)\(97\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(70\!\cdots\!03\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} + \)\(59\!\cdots\!90\)\( T^{10} + \)\(81\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} \)
$71$ \( 1 + \)\(60\!\cdots\!88\)\( T + \)\(10\!\cdots\!66\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(43\!\cdots\!08\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} + \)\(59\!\cdots\!80\)\( T^{9} + \)\(28\!\cdots\!66\)\( T^{10} + \)\(36\!\cdots\!88\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \)
$73$ \( 1 - \)\(35\!\cdots\!00\)\( T + \)\(14\!\cdots\!30\)\( T^{2} - \)\(33\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!40\)\( T^{6} - \)\(34\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!03\)\( T^{8} - \)\(47\!\cdots\!00\)\( T^{9} + \)\(48\!\cdots\!30\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} \)
$79$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(15\!\cdots\!94\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(77\!\cdots\!15\)\( T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!80\)\( T^{6} + \)\(80\!\cdots\!00\)\( T^{7} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(90\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!94\)\( T^{10} + \)\(42\!\cdots\!40\)\( T^{11} + \)\(61\!\cdots\!01\)\( T^{12} \)
$83$ \( 1 + \)\(60\!\cdots\!00\)\( T + \)\(60\!\cdots\!90\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} + \)\(61\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} + \)\(81\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!03\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!90\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{11} + \)\(53\!\cdots\!49\)\( T^{12} \)
$89$ \( 1 - \)\(16\!\cdots\!20\)\( T + \)\(32\!\cdots\!94\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!15\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!80\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(45\!\cdots\!00\)\( T^{9} + \)\(20\!\cdots\!94\)\( T^{10} - \)\(49\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(29\!\cdots\!10\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!47\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!80\)\( T^{6} - \)\(33\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!03\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(92\!\cdots\!10\)\( T^{10} - \)\(47\!\cdots\!00\)\( T^{11} + \)\(17\!\cdots\!49\)\( T^{12} \)
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