Properties

Label 1.102
Level 1
Weight 102
Dimension 8
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 102 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{102}(\Gamma_1(1))\).

Total New Old
Modular forms 9 9 0
Cusp forms 8 8 0
Eisenstein series 1 1 0

Trace form

\( 8q - 434989091795040q^{2} - 1209375896611771518910560q^{3} + 9008967123164113911511576503296q^{4} + 38238791540232828026421308589217200q^{5} + 2477260046146272006672820535971480695936q^{6} - 5786789720778120168956257648884421874283200q^{7} - 6101688461407407841888405145259998920458731520q^{8} + 5273640617280587177839635242070674204864681879784q^{9} + O(q^{10}) \) \( 8q - 434989091795040q^{2} - 1209375896611771518910560q^{3} + 9008967123164113911511576503296q^{4} + 38238791540232828026421308589217200q^{5} + 2477260046146272006672820535971480695936q^{6} - 5786789720778120168956257648884421874283200q^{7} - 6101688461407407841888405145259998920458731520q^{8} + 5273640617280587177839635242070674204864681879784q^{9} - 376167658847177398692429835067300112512860198593600q^{10} + 4622006661480385115355021397632645270205791478300896q^{11} - 7273290781506553143908216801865849714958844187989422080q^{12} + 250316830865150221699601352256397536965043294984328650480q^{13} - 4856789037716039923353321450215036970203737520869624889088q^{14} - 296460708185002952419818874947342543917521978696100167156800q^{15} - 10662656154906175558835021484824464714661655552497093859868672q^{16} - 397968518694853312322660892941326449007299986932512193696587120q^{17} - 7270968925217398747066352706388703938040714138623386028515252960q^{18} - 21387776045417165942123954421061470474744368808488511816400717280q^{19} + 1745002690168409059221421019241581897927614179506161156200285542400q^{20} + 4030935126807879780648600654579655736182360851489572706466859540736q^{21} + 61357881995862711169740037407631980426480577127973320405539739944320q^{22} - 1366243345208606848425464574347748611753917469593153677393324561698880q^{23} + 10457354719000260635321058863018299052737315743164922361342586480885760q^{24} + 77471873089568593564333770877949420045843582630742632687744142562715000q^{25} - 97549630676937426518340388130407212490384846814009484188479115327886144q^{26} - 5951874968930013487417142282111771144801435961988629201748805497863184320q^{27} + 9217624035174751886516129105576165061663773766415040330328487084880404480q^{28} + 15465799862710125255662370242568323665682807414567501281109138719156160880q^{29} + 1101689156065505631051293671607641382064945196508526428823375711603202118400q^{30} - 6594370386043135339784923725045360443937393326734068279132668478779615001344q^{31} + 12265519409733718324179959581296851912742484455832107241543734281342036213760q^{32} + 43813172582105581329585760338258562815639759809845990212191452867766348215680q^{33} + 9531455444774917507766540524882635666895777953789094347547215472626183995712q^{34} - 1122850631208004175280217132184898888977788452428069383882586628450512616297600q^{35} + 1966242586607976986737003826554029255087710066072863106322127653222050407339008q^{36} + 39729696767983511833540209209183332457877293383323859312175998401968644864715440q^{37} - 70873740526925065519455012686645778106113298999572267198467188879790467046148480q^{38} - 269289195198196884798470270521708844679404215685249767615656452740889673105306432q^{39} - 763642761360299050617237751714968411270996317577517686387843428508400019349504000q^{40} + 5607291724825101314952096252772109505395882457063737303656079003304969203257094736q^{41} + 3008066923831569706716293907931878334959183836053498604235677758281531565332526080q^{42} - 28642921955044714117801359306021921736063108483729021321044095944585527287681223200q^{43} - 207838425393205175674959052733026300855796348257745356716186803699782799277851086848q^{44} + 712633045382921242540931880134776511927082560144290402835715503414563529286220047600q^{45} + 1088094260619851586903932234100844276278561023621942541398958535856528393570339218176q^{46} - 4595399220200007631460417274797048154524720407021496316223236391731855709593438718080q^{47} - 58354182581817955899154335674671016574355963424701638945735582725996041826866298880q^{48} + 12068461406676719302337658501270392621238416595305390197216228129207599916872005856456q^{49} - 40493179837482260394665357435432250163405353367416857829233726008281167215503492420000q^{50} + 117646730044094759803745147399491979694134742922341911657750530682089815486256248611136q^{51} - 736020167800817028850658909331852777746165402605863935107936992324104798352324258252800q^{52} + 131026696145917691346168139299202731343660174597285772716290345361212508775951813313840q^{53} + 6803608957095266771149163140167539716357850176039815317805755392633822625226741503089920q^{54} - 14585157967675450217148723878599628333971210001615112897255333178063634767400530051473600q^{55} - 23665562342657326400415121984960944704012341384928617813498515591030351969339927087022080q^{56} - 66899487754306674554237314096607510056291565049587799982155034523058479948455991635655040q^{57} + 29173417521771002425120779628349939892591335515731485644911210555324770598093888308326080q^{58} + 212583149562022564923669670483574922097682552074321909615339251529169016840977455874187360q^{59} - 3455800135608334392541843481344001808338357411235959344412061781382196941870275476295065600q^{60} - 3390796423636162146406492296389682436951029380958058095674208836333532599894943317230895504q^{61} - 5878859817222970028213836429448832127296537243368792127423103234889716583457526212133370880q^{62} - 20171938924061808002384576493874339158887341995607670819297079343133751793255425885494984640q^{63} - 74870090698021647967203019932764377907062372989513317693678846896652408519412785858115272704q^{64} - 161067075283368767182138254547148962244907084326790507487294907612770451283405321317426536800q^{65} - 745278055170676259198413344675313071673593040227005461838089798951153726758412015071955911168q^{66} - 618774503558862942537948810466095132313460919373052732511642097824837019429997681150398682720q^{67} - 2182412150351293808121583165511486752102068977919329447376923905220322529537335456462676162560q^{68} - 5393950442083315109606578094475120918497802785657891521452512469623636760248087300551286537472q^{69} - 12923520657540473465038555822652190723092643731233782501675276653712048411292540691873715571200q^{70} - 15797470289857673806303846655911576507977904169297655042639265226995032204172380517793831135424q^{71} - 55126715113590704564504537983251285080485074916503283789364546398928839504287032847376708239360q^{72} - 20462651635605668591994589976366754984142669867347912050159975933828495442291952971923777071280q^{73} - 142953411935712165334538916257020870493984381200782662943456748743148514063783881306064429441088q^{74} - 204312291750914950304829914412770260373631281822544572731633720960361618548975517964814619460000q^{75} + 64556217054886497186737063238064337992823205997653378371004031509089422583081356800943896432640q^{76} + 250231448169383959900618348975396989794902211753647918986458150866408618577007907131029306438400q^{77} + 1260981709052401111398092363772739439414079820346824747462408082793911096042108256417770344646400q^{78} + 1474556321278768744579716979675929750918930811984805387023357390321899878561204701118206999166080q^{79} + 6033438037435277941170744030808192554572458649979375320464765273747209092446481051552234550067200q^{80} + 14613214105102112111129469132612601529386527627758931132459526147171392070559192290643105065182408q^{81} + 30083863369214129982421840097969977845328701273986110579467106900080547794123383776302929686086720q^{82} + 33762254529254022439253159341806143692643909652777369725086142424861011401680843403961249425712160q^{83} + 57799153837692447414380568170128797835929379783304235587752969314531310322308849537769529417695232q^{84} + 17314966267116570951109803841412092095679328201414316724042066880549149931729490643535478494114400q^{85} + 67989564188024628582959760637462873453376851598916319417912395832044408164663357850732592560556416q^{86} + 2586434133428712948881772736535019341274237946379240114707154003676967765014837223291966941531840q^{87} - 363099602372059077387783242048689913608334478213970275423791339756517232974574737922754584123146240q^{88} - 620007165949691950044675276484549383308720953973642955804615855186521697520022438525123333812557360q^{89} - 4716676146177962028099989697458384454375488041754925418224025761569034561814581975790426208019028800q^{90} - 3685711793015793225308940042728501386995879977832891226647110522985454537354035281338855172828873344q^{91} - 4618779932089713137770970202224414792688843158843334085630908366025929325979388063902511470793400320q^{92} - 3975022445092224432029728046757042348886229630318622378634497589784647142077875089398179428115102720q^{93} - 17371370943540992303713705925513939673302729250250795188383540356993592041601425149438637190677682688q^{94} - 2642077700832720169247174948884057845373174576960370774287679094739730873649398763127258995555528000q^{95} + 4679875418673156206717240026356824763600126819048403120235276936002238426687957733980747642601209856q^{96} + 64020476116668335744724892375903667651900818166631138467214326399663196857065619648203365824701173520q^{97} + 202971561788470754768820264959327860776947519106194867162872422213964707447467488105939520314595817120q^{98} + 225914901454627435116272921386700923755368861971179328902362103653753469396751969300619190223813523808q^{99} + O(q^{100}) \)

Decomposition of \(S_{102}^{\mathrm{new}}(\Gamma_1(1))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1.102.a \(\chi_{1}(1, \cdot)\) 1.102.a.a 8 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 434989091795040 T + \)\(57\!\cdots\!60\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(90\!\cdots\!20\)\( T^{6} + \)\(52\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} + \)\(13\!\cdots\!20\)\( T^{9} + \)\(58\!\cdots\!80\)\( T^{10} + \)\(23\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!56\)\( T^{12} + \)\(43\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!40\)\( T^{14} + \)\(29\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!56\)\( T^{16} \)
$3$ \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(42\!\cdots\!20\)\( T^{2} + \)\(59\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!36\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(32\!\cdots\!86\)\( T^{8} + \)\(28\!\cdots\!80\)\( T^{9} + \)\(43\!\cdots\!60\)\( T^{10} + \)\(48\!\cdots\!40\)\( T^{11} + \)\(51\!\cdots\!16\)\( T^{12} + \)\(52\!\cdots\!40\)\( T^{13} + \)\(58\!\cdots\!80\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(32\!\cdots\!61\)\( T^{16} \)
$5$ \( 1 - \)\(38\!\cdots\!00\)\( T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(81\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!00\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!50\)\( T^{8} + \)\(69\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!00\)\( T^{12} + \)\(77\!\cdots\!00\)\( T^{13} + \)\(45\!\cdots\!00\)\( T^{14} - \)\(56\!\cdots\!00\)\( T^{15} + \)\(58\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + \)\(57\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!96\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(32\!\cdots\!06\)\( T^{8} + \)\(66\!\cdots\!00\)\( T^{9} + \)\(70\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(24\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - \)\(46\!\cdots\!96\)\( T + \)\(53\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(68\!\cdots\!68\)\( T^{5} + \)\(32\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!40\)\( T^{7} + \)\(64\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!40\)\( T^{9} + \)\(75\!\cdots\!08\)\( T^{10} - \)\(23\!\cdots\!08\)\( T^{11} + \)\(71\!\cdots\!20\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{13} + \)\(64\!\cdots\!20\)\( T^{14} - \)\(85\!\cdots\!16\)\( T^{15} + \)\(27\!\cdots\!81\)\( T^{16} \)
$13$ \( 1 - \)\(25\!\cdots\!80\)\( T + \)\(17\!\cdots\!40\)\( T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} - \)\(30\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!66\)\( T^{8} - \)\(47\!\cdots\!40\)\( T^{9} + \)\(86\!\cdots\!20\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!36\)\( T^{12} - \)\(13\!\cdots\!20\)\( T^{13} + \)\(19\!\cdots\!60\)\( T^{14} - \)\(90\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \)
$17$ \( 1 + \)\(39\!\cdots\!20\)\( T + \)\(16\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!60\)\( T^{6} + \)\(53\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(97\!\cdots\!80\)\( T^{13} + \)\(74\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} \)
$19$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(87\!\cdots\!52\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!08\)\( T^{4} + \)\(48\!\cdots\!80\)\( T^{5} + \)\(92\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!70\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!68\)\( T^{12} + \)\(85\!\cdots\!60\)\( T^{13} + \)\(73\!\cdots\!12\)\( T^{14} + \)\(25\!\cdots\!20\)\( T^{15} + \)\(17\!\cdots\!41\)\( T^{16} \)
$23$ \( 1 + \)\(13\!\cdots\!80\)\( T + \)\(17\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} + \)\(96\!\cdots\!60\)\( T^{5} + \)\(71\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!80\)\( T^{7} + \)\(28\!\cdots\!46\)\( T^{8} + \)\(15\!\cdots\!40\)\( T^{9} + \)\(83\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(18\!\cdots\!56\)\( T^{12} + \)\(71\!\cdots\!20\)\( T^{13} + \)\(28\!\cdots\!40\)\( T^{14} + \)\(75\!\cdots\!60\)\( T^{15} + \)\(18\!\cdots\!81\)\( T^{16} \)
$29$ \( 1 - \)\(15\!\cdots\!80\)\( T + \)\(17\!\cdots\!32\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!48\)\( T^{4} - \)\(97\!\cdots\!80\)\( T^{5} + \)\(74\!\cdots\!84\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!70\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!44\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(89\!\cdots\!88\)\( T^{12} - \)\(23\!\cdots\!60\)\( T^{13} + \)\(27\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!61\)\( T^{16} \)
$31$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(33\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!20\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} + \)\(62\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!70\)\( T^{8} + \)\(26\!\cdots\!60\)\( T^{9} + \)\(49\!\cdots\!08\)\( T^{10} + \)\(81\!\cdots\!12\)\( T^{11} + \)\(12\!\cdots\!20\)\( T^{12} + \)\(16\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!20\)\( T^{14} + \)\(16\!\cdots\!84\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} \)
$37$ \( 1 - \)\(39\!\cdots\!40\)\( T + \)\(16\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} - \)\(31\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!66\)\( T^{8} - \)\(76\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!20\)\( T^{10} - \)\(19\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!36\)\( T^{12} - \)\(31\!\cdots\!60\)\( T^{13} + \)\(34\!\cdots\!60\)\( T^{14} - \)\(20\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!21\)\( T^{16} \)
$41$ \( 1 - \)\(56\!\cdots\!36\)\( T + \)\(41\!\cdots\!20\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!20\)\( T^{4} - \)\(28\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} - \)\(30\!\cdots\!40\)\( T^{7} + \)\(96\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{9} + \)\(64\!\cdots\!08\)\( T^{10} - \)\(13\!\cdots\!28\)\( T^{11} + \)\(30\!\cdots\!20\)\( T^{12} - \)\(49\!\cdots\!60\)\( T^{13} + \)\(91\!\cdots\!20\)\( T^{14} - \)\(97\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} \)
$43$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(58\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!96\)\( T^{4} + \)\(26\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + \)\(45\!\cdots\!80\)\( T + \)\(42\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!36\)\( T^{4} + \)\(22\!\cdots\!60\)\( T^{5} + \)\(99\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} + \)\(18\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!60\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(26\!\cdots\!16\)\( T^{12} + \)\(36\!\cdots\!20\)\( T^{13} + \)\(82\!\cdots\!80\)\( T^{14} + \)\(68\!\cdots\!40\)\( T^{15} + \)\(11\!\cdots\!61\)\( T^{16} \)
$53$ \( 1 - \)\(13\!\cdots\!40\)\( T + \)\(52\!\cdots\!20\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!36\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!40\)\( T^{7} + \)\(58\!\cdots\!86\)\( T^{8} - \)\(31\!\cdots\!20\)\( T^{9} + \)\(71\!\cdots\!60\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(63\!\cdots\!16\)\( T^{12} - \)\(17\!\cdots\!60\)\( T^{13} + \)\(42\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} \)
$59$ \( 1 - \)\(21\!\cdots\!60\)\( T + \)\(28\!\cdots\!72\)\( T^{2} - \)\(86\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!68\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{5} + \)\(50\!\cdots\!24\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!44\)\( T^{10} - \)\(57\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{13} + \)\(39\!\cdots\!52\)\( T^{14} - \)\(20\!\cdots\!40\)\( T^{15} + \)\(70\!\cdots\!21\)\( T^{16} \)
$61$ \( 1 + \)\(33\!\cdots\!04\)\( T + \)\(13\!\cdots\!20\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!20\)\( T^{4} + \)\(18\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!48\)\( T^{6} + \)\(56\!\cdots\!60\)\( T^{7} + \)\(86\!\cdots\!70\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!92\)\( T^{11} + \)\(16\!\cdots\!20\)\( T^{12} + \)\(13\!\cdots\!40\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} + \)\(57\!\cdots\!84\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} \)
$67$ \( 1 + \)\(61\!\cdots\!20\)\( T + \)\(25\!\cdots\!80\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!56\)\( T^{4} + \)\(57\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} + \)\(41\!\cdots\!26\)\( T^{8} + \)\(64\!\cdots\!60\)\( T^{9} + \)\(90\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!76\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(67\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} \)
$71$ \( 1 + \)\(15\!\cdots\!24\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!20\)\( T^{4} + \)\(37\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!88\)\( T^{6} + \)\(61\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!70\)\( T^{8} + \)\(58\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} + \)\(32\!\cdots\!52\)\( T^{11} + \)\(61\!\cdots\!20\)\( T^{12} + \)\(95\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} + \)\(10\!\cdots\!84\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$73$ \( 1 + \)\(20\!\cdots\!80\)\( T + \)\(67\!\cdots\!60\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!46\)\( T^{8} + \)\(16\!\cdots\!40\)\( T^{9} + \)\(52\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(92\!\cdots\!56\)\( T^{12} + \)\(72\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(47\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!81\)\( T^{16} \)
$79$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(34\!\cdots\!32\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!48\)\( T^{4} - \)\(44\!\cdots\!80\)\( T^{5} + \)\(44\!\cdots\!84\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(93\!\cdots\!44\)\( T^{10} - \)\(42\!\cdots\!20\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} - \)\(74\!\cdots\!60\)\( T^{13} + \)\(31\!\cdots\!72\)\( T^{14} - \)\(61\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!61\)\( T^{16} \)
$83$ \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(94\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!56\)\( T^{4} - \)\(38\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!60\)\( T^{6} - \)\(44\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(20\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!76\)\( T^{12} - \)\(24\!\cdots\!40\)\( T^{13} + \)\(86\!\cdots\!20\)\( T^{14} - \)\(20\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!41\)\( T^{16} \)
$89$ \( 1 + \)\(62\!\cdots\!60\)\( T + \)\(43\!\cdots\!12\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!88\)\( T^{4} + \)\(37\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} + \)\(32\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!44\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(35\!\cdots\!08\)\( T^{12} + \)\(57\!\cdots\!20\)\( T^{13} + \)\(93\!\cdots\!32\)\( T^{14} + \)\(10\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$97$ \( 1 - \)\(64\!\cdots\!20\)\( T + \)\(50\!\cdots\!20\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(94\!\cdots\!36\)\( T^{4} - \)\(29\!\cdots\!40\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!80\)\( T^{7} + \)\(55\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{9} + \)\(20\!\cdots\!60\)\( T^{10} - \)\(28\!\cdots\!20\)\( T^{11} + \)\(42\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{13} + \)\(48\!\cdots\!80\)\( T^{14} - \)\(28\!\cdots\!60\)\( T^{15} + \)\(20\!\cdots\!61\)\( T^{16} \)
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