Properties

Label 1.76.a
Level 1
Weight 76
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{76}(\Gamma_0(1))\).

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

Trace form

\( 6q - 57080822040q^{2} - 785092363818710040q^{3} + 172600466200162028593728q^{4} - 38982947396479621874420940q^{5} + 31673308599187435504995050592q^{6} + 1924474634802918779239478643600q^{7} + 4434903997968330342413437320645120q^{8} + 2119498765962992970568250560046009982q^{9} + O(q^{10}) \) \( 6q - 57080822040q^{2} - 785092363818710040q^{3} + 172600466200162028593728q^{4} - 38982947396479621874420940q^{5} + 31673308599187435504995050592q^{6} + 1924474634802918779239478643600q^{7} + 4434903997968330342413437320645120q^{8} + 2119498765962992970568250560046009982q^{9} + 133267935677460241037537052811576902960q^{10} - 945659951949983071650740113735111214088q^{11} - 115150003643542480663442829411536163866880q^{12} + 533185179266852519941169625089098069764420q^{13} + 8213946953162263647781694565284746353072576q^{14} - 309603328848013864510094045586663272616092880q^{15} + 2681978005634517266622433394712415602648354816q^{16} + 18261354095776132566172834671579513215483393580q^{17} - 439276854274063146050579040790064492160853960440q^{18} + 1061534856238354155312794852768417904375328867080q^{19} + 9296557595621768320156983882258818695571369489280q^{20} - 100259704699697889376866621361897766152209395224128q^{21} + 153315952874260268106194796405555863940172791589920q^{22} + 1514929449877410771078348320690686508954384916853680q^{23} - 7662424782495949976733296347258884300404709513308160q^{24} + 19310380328261093150952553553121815533899364092239850q^{25} + 117730941166816926498584842564210483616872200198362352q^{26} - 1074213478466937258572560754951503958791056763891563120q^{27} + 1465885017268620651444113586751990931877154821040657920q^{28} + 14718767377678125816030788545177391174717388984245534820q^{29} - 25550285934396345286029563779360588150614380222554150080q^{30} - 41742874787360498453628839393549579933009794702042980288q^{31} + 1174660148719991756036751189163540495635749096221862952960q^{32} + 592070005239220743874129160617285870488467768711527333920q^{33} + 3034358950124956876257603978888653679838045546922892395856q^{34} + 27299672493656883988453292135086250689495956464846953371360q^{35} + 179176299070088941546343921251311713064325363113137876471616q^{36} + 98517772547646219221591850851651469983797181732879949140340q^{37} + 1257660981102762853948269222064665311191946222029530979162080q^{38} + 2481484269951087933260675611242651218102907933795356739592944q^{39} + 8807901623729426290853411185831313455271917763097593104051200q^{40} + 5039211762207387369185984529408567568647239506257650214613212q^{41} + 43775096530001421204709465357678460315555651574798436125415680q^{42} + 27938749238581541451173935836639198120068956697568041153196600q^{43} - 86388814623354110240559749283671522466589238091463431115801344q^{44} - 237966843556414434005003864386125719471571617536701728123349180q^{45} - 824145803870667307795487213022331547039596870954611528203613888q^{46} - 1386854970430427381646742231466788117324292253074354560479038880q^{47} - 8212215903718675701941242530692185525776363042145818555572305920q^{48} - 5707868790800975784856935495088511133239544554833878775910738442q^{49} + 3144970672685230615695776810799603542575044196614572896663392600q^{50} + 283284046705467654069692648207344729396412677737329641196945232q^{51} + 41677832028141179840113326797899392222702339479233001780216150400q^{52} + 64283727174659970247345252518647455744441825702701764187054106260q^{53} + 780320815712097175721587535401535151075152637184894982834238943680q^{54} + 436876078674971654403803167180324983329224614451635548418098895120q^{55} + 289950797286325881623798664040853221823087783432246887785270865920q^{56} - 671954656600624358379440578322093895610678807911638904039760657440q^{57} - 1763005756244374137396902189067511762184157619699352574614137403280q^{58} - 2478360370933764187935912640189307752170553390931716617591182395560q^{59} - 31792921597254948192279089096854272160617709162962021109081069309440q^{60} - 25641115321477658815889269525514418267074652297034046321782023152988q^{61} - 29014701474354643819936101890527484086880802356768235703173807694080q^{62} + 42929132343455570369238205341690645702689225620502517839786884793040q^{63} + 47683232974651133862719671893097862706511055838864188064811716968448q^{64} + 122110621788131890149667214965152432737153070969379698276880591977720q^{65} + 930528105441795930909966787493137282089786427774739900252891166297984q^{66} + 955579073553339181030887342220313909838725364297810737998207107813480q^{67} + 1237210897217068882936524414359446185548686871817993659403939290386560q^{68} - 1408360468152512186955736181867470428051018587723566870538767575903936q^{69} - 3476902593928340346610001926714053112321275314411181267396530677050240q^{70} - 255682319306096297182489045146623979225804710540351927309367569173488q^{71} - 21189962510693790174049281425747631548583047301946833368066212728860160q^{72} - 30605188898291905719532895918028188508004452865621142257485594115042020q^{73} - 24065024027894599201472448362529876891121294631891709256708009194181584q^{74} + 19463431192157531038934013606638975862465120463567077259244690008282200q^{75} + 100020936372377387464010564774983213070641689274186088752725827576144640q^{76} + 158930415774536768880543239288689183540801263714134710336228496440507200q^{77} + 135490258190773913577197940793442636786883342442786889886654637067611200q^{78} + 116627678166642324332643069323234092884951450625988452824960767925148320q^{79} + 1239310052714502333188756625709169306278438274940582996524184215206748160q^{80} + 294279639721966061337542444477716876114885567907972876173645426056517686q^{81} - 2557649813228840791149140948716215474705151452500921688788291587877236080q^{82} - 798780284119889426515570523310162377392459257634002033911814028088739960q^{83} - 9175580553861138859894612698264948459030459925276038063627639364295153664q^{84} - 3615500806340871685633297735534398118961343923835923286123685237305018840q^{85} + 7245194180836090971883743617494948296636529538939897523257528311698092832q^{86} - 14159312341342352586604709816323870081891838395129433788110753788803266960q^{87} - 4827001398586167572057494186308233100206033194553368013345656010632181760q^{88} + 53575610216125238704584826863934604186542249469498220298044079954014547260q^{89} + 85648802068325073636273691125068892379540509730396664184966156591547899120q^{90} + 34502419684367451829908407139028781594042460695428817715772307436403470432q^{91} + 187445224353587479072026786339601818260697439565538078689222489106585838080q^{92} - 166703912520796821824619485532876155501181179490548097775502842243408462080q^{93} - 296189325151987924691405038954735773676385503565643130818482899021002578304q^{94} + 194934959385332686251927514630723902507701573195047654345319995220933314800q^{95} - 892786287645978875167763571227022376777771426534553548077964657447900151808q^{96} - 747928220344118582344222376208028579242743608466383778453296378427532508980q^{97} - 1663265696927109513221859402309517858944410070626232817304447619248634722520q^{98} - 187241780708811231174609640862544518464431146740439798697006075456653933736q^{99} + O(q^{100}) \)

Decomposition of \(S_{76}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.76.a.a \(6\) \(35.623\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-57080822040\) \(-7\!\cdots\!40\) \(-3\!\cdots\!40\) \(19\!\cdots\!00\) \(+\) \(q+(-9513470340+\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 57080822040 T + \)\(28\!\cdots\!40\)\( T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!72\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{5} + \)\(27\!\cdots\!20\)\( T^{6} - \)\(70\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!28\)\( T^{8} - \)\(33\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!40\)\( T^{10} + \)\(43\!\cdots\!20\)\( T^{11} + \)\(29\!\cdots\!24\)\( T^{12} \)
$3$ \( 1 + 785092363818710040 T + \)\(10\!\cdots\!30\)\( T^{2} + \)\(88\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!47\)\( T^{4} + \)\(93\!\cdots\!20\)\( T^{5} + \)\(67\!\cdots\!40\)\( T^{6} + \)\(57\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!03\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} + \)\(65\!\cdots\!80\)\( T^{11} + \)\(50\!\cdots\!49\)\( T^{12} \)
$5$ \( 1 + \)\(38\!\cdots\!40\)\( T + \)\(70\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!75\)\( T^{4} + \)\(61\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!75\)\( T^{8} + \)\(20\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!25\)\( T^{12} \)
$7$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!47\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!03\)\( T^{8} + \)\(93\!\cdots\!00\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} - \)\(15\!\cdots\!00\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} \)
$11$ \( 1 + \)\(94\!\cdots\!88\)\( T + \)\(58\!\cdots\!66\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!95\)\( T^{8} + \)\(97\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!66\)\( T^{10} + \)\(31\!\cdots\!88\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \)
$13$ \( 1 - \)\(53\!\cdots\!20\)\( T + \)\(98\!\cdots\!10\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!80\)\( T^{6} - \)\(51\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!03\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!10\)\( T^{10} - \)\(28\!\cdots\!40\)\( T^{11} + \)\(18\!\cdots\!49\)\( T^{12} \)
$17$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(10\!\cdots\!70\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!47\)\( T^{4} - \)\(48\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!60\)\( T^{6} - \)\(92\!\cdots\!20\)\( T^{7} + \)\(15\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!70\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(50\!\cdots\!49\)\( T^{12} \)
$19$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(24\!\cdots\!94\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!80\)\( T^{6} - \)\(22\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!15\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!94\)\( T^{10} - \)\(36\!\cdots\!20\)\( T^{11} + \)\(27\!\cdots\!01\)\( T^{12} \)
$23$ \( 1 - \)\(15\!\cdots\!80\)\( T + \)\(60\!\cdots\!90\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!20\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(29\!\cdots\!03\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!90\)\( T^{10} - \)\(67\!\cdots\!60\)\( T^{11} + \)\(59\!\cdots\!49\)\( T^{12} \)
$29$ \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(30\!\cdots\!94\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!15\)\( T^{8} - \)\(32\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!94\)\( T^{10} - \)\(36\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \)
$31$ \( 1 + \)\(41\!\cdots\!88\)\( T + \)\(16\!\cdots\!66\)\( T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!95\)\( T^{4} + \)\(99\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(70\!\cdots\!08\)\( T^{7} + \)\(83\!\cdots\!95\)\( T^{8} + \)\(26\!\cdots\!80\)\( T^{9} + \)\(41\!\cdots\!66\)\( T^{10} + \)\(76\!\cdots\!88\)\( T^{11} + \)\(12\!\cdots\!01\)\( T^{12} \)
$37$ \( 1 - \)\(98\!\cdots\!40\)\( T + \)\(15\!\cdots\!10\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!47\)\( T^{4} - \)\(96\!\cdots\!20\)\( T^{5} + \)\(74\!\cdots\!80\)\( T^{6} - \)\(39\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!10\)\( T^{10} - \)\(11\!\cdots\!20\)\( T^{11} + \)\(49\!\cdots\!49\)\( T^{12} \)
$41$ \( 1 - \)\(50\!\cdots\!12\)\( T + \)\(29\!\cdots\!66\)\( T^{2} - \)\(66\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(19\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!95\)\( T^{8} - \)\(50\!\cdots\!20\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} - \)\(31\!\cdots\!12\)\( T^{11} + \)\(56\!\cdots\!01\)\( T^{12} \)
$43$ \( 1 - \)\(27\!\cdots\!00\)\( T + \)\(10\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!47\)\( T^{4} - \)\(84\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!00\)\( T^{6} - \)\(27\!\cdots\!00\)\( T^{7} + \)\(55\!\cdots\!03\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!50\)\( T^{10} - \)\(99\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + \)\(13\!\cdots\!80\)\( T + \)\(16\!\cdots\!30\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(65\!\cdots\!40\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(68\!\cdots\!03\)\( T^{8} + \)\(23\!\cdots\!80\)\( T^{9} + \)\(70\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!40\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \)
$53$ \( 1 - \)\(64\!\cdots\!60\)\( T + \)\(76\!\cdots\!30\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!47\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(78\!\cdots\!60\)\( T^{7} + \)\(79\!\cdots\!03\)\( T^{8} - \)\(26\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} - \)\(25\!\cdots\!20\)\( T^{11} + \)\(83\!\cdots\!49\)\( T^{12} \)
$59$ \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(20\!\cdots\!94\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!15\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(37\!\cdots\!94\)\( T^{10} + \)\(29\!\cdots\!40\)\( T^{11} + \)\(76\!\cdots\!01\)\( T^{12} \)
$61$ \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(55\!\cdots\!66\)\( T^{2} + \)\(85\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(99\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!95\)\( T^{8} + \)\(42\!\cdots\!80\)\( T^{9} + \)\(22\!\cdots\!66\)\( T^{10} + \)\(80\!\cdots\!88\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} \)
$67$ \( 1 - \)\(95\!\cdots\!80\)\( T + \)\(72\!\cdots\!70\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(60\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!60\)\( T^{6} - \)\(54\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{9} + \)\(48\!\cdots\!70\)\( T^{10} - \)\(57\!\cdots\!40\)\( T^{11} + \)\(54\!\cdots\!49\)\( T^{12} \)
$71$ \( 1 + \)\(25\!\cdots\!88\)\( T + \)\(24\!\cdots\!66\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(27\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} + \)\(54\!\cdots\!80\)\( T^{9} + \)\(57\!\cdots\!66\)\( T^{10} + \)\(42\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \)
$73$ \( 1 + \)\(30\!\cdots\!20\)\( T + \)\(58\!\cdots\!90\)\( T^{2} + \)\(81\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!47\)\( T^{4} + \)\(83\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!20\)\( T^{6} + \)\(46\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!03\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(58\!\cdots\!90\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!49\)\( T^{12} \)
$79$ \( 1 - \)\(11\!\cdots\!20\)\( T + \)\(92\!\cdots\!94\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(70\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(84\!\cdots\!00\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(47\!\cdots\!80\)\( T^{11} + \)\(85\!\cdots\!01\)\( T^{12} \)
$83$ \( 1 + \)\(79\!\cdots\!60\)\( T + \)\(37\!\cdots\!70\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} + \)\(36\!\cdots\!80\)\( T^{5} + \)\(67\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(46\!\cdots\!03\)\( T^{8} + \)\(15\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!70\)\( T^{10} + \)\(36\!\cdots\!20\)\( T^{11} + \)\(38\!\cdots\!49\)\( T^{12} \)
$89$ \( 1 - \)\(53\!\cdots\!60\)\( T + \)\(17\!\cdots\!94\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!15\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!94\)\( T^{10} - \)\(56\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(63\!\cdots\!30\)\( T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} + \)\(61\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(62\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{9} + \)\(68\!\cdots\!30\)\( T^{10} + \)\(81\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \)
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