Properties

Label 2.74
Level 2
Weight 74
Dimension 7
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2\( 2 \) \)
Weight: \( k \) = \( 74 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{74}(\Gamma_1(2))\).

Total New Old
Modular forms 19 7 12
Cusp forms 17 7 10
Eisenstein series 2 0 2

Trace form

\( 7q + 68719476736q^{2} + 1027124703510204q^{3} + 33056565380087516495872q^{4} - 40558853897019556618457910q^{5} + 41857413074783112692842364928q^{6} + 1319013575200782974071512211448q^{7} + 324518553658426726783156020576256q^{8} + 197377606713669685301132051646756411q^{9} + O(q^{10}) \) \( 7q + 68719476736q^{2} + 1027124703510204q^{3} + 33056565380087516495872q^{4} - 40558853897019556618457910q^{5} + 41857413074783112692842364928q^{6} + 1319013575200782974071512211448q^{7} + 324518553658426726783156020576256q^{8} + 197377606713669685301132051646756411q^{9} + 1708704931234681219841889137212784640q^{10} - 27600640793130220835781688640423160396q^{11} + 4850459273584009196858899998896553984q^{12} + 16382312624706684550801940331533298616514q^{13} + 409136272750615877298547410438906536001536q^{14} - 405584036700872141132474996756634202511640q^{15} + 156105216389714361990750027908538530541862912q^{16} - 1066999187182198043388511037865747532008417282q^{17} + 3255842558478021353341847588179331692256821248q^{18} + 53935121388410443644614146326771216282991503020q^{19} - 191533772226892047038045311404722620279931535360q^{20} + 2849533425198940897609224165597780267210940964064q^{21} + 9150072434788992021693852445701475242905343885312q^{22} + 11487021927231577132753977745688995443452136273384q^{23} + 197666044563985429738489049912218491753452575653888q^{24} + 4442341753390189342611612360612615751804313224567225q^{25} - 939464668009392503125801142449113923737510721093632q^{26} + 20892584344354699149244907986957825120077530692347480q^{27} + 6228865497978237779384994213988670550246569497591808q^{28} - 242659031370254704226112349973038557876808498312734030q^{29} - 1048349114291221124294475848520501250019442670845296640q^{30} + 4754620827269862681198675317057706435180399687073524704q^{31} + 1532495540865888858358347027150309183618739122183602176q^{32} + 79761089777578567682110917335748475775550335479302834768q^{33} + 204488278882446941187474408379912421395940926602866589696q^{34} + 1376295186694272619908786072829995978621818296087679962320q^{35} + 932089394413660384054868395110595432490074489749353005056q^{36} + 6897014531797115945750823550982798156661639192166851483818q^{37} + 9412804191936413646891909754675577992402077739949049774080q^{38} + 31375857019095066098997026182603358399649004449542485849032q^{39} + 8069130896376740533603203935689390719884189644076026429440q^{40} + 207561036577549239977618711632076555016582409617518912741094q^{41} - 104694486954058443037028594421282234244035423158989737689088q^{42} - 669515376121954138996540636248178131700536910793579100133996q^{43} - 130340340987202815896841738532383521291484781898447103983616q^{44} - 11325235544690847000804714518542546119529809765856199775650430q^{45} - 10055187030076398392878405673493410750203033255974000931110912q^{46} - 39353768626244150141187396234568307076650970187101967232623152q^{47} + 22905646300097371733925919643989710940845052141613680164864q^{48} + 129493632285583023439988456210141822988159886108545342585511119q^{49} + 170350141534045943234869791869403118604747023091471721535897600q^{50} + 976530651006378911356405853843811633989428798214510502430152504q^{51} + 77363284050807091967402032577698632221310401846446098684575744q^{52} + 1476036265911713078264026743636826897017112209944569985002104474q^{53} + 225136682955043068034732808337908589915161904277953973950873600q^{54} - 9172316756840955451381907105110761413428399399663556429421224520q^{55} + 1932091421363721765158679284673494912285910202569978892104237056q^{56} - 19246343545762558322604740763123478340316280134768466377718019280q^{57} - 39513537017273788236988247971454006076366280736916740878054195200q^{58} - 74921937563676145846070393075894640398784399890648016828516609660q^{59} - 1915316460903170675719989706391312359075569008799417707727421440q^{60} + 222903490578238099115364191117178893194949750880947971935694990994q^{61} + 195504071699382183303554505973484078352271385303884221638122668032q^{62} + 678865034144871199370209925607341661497036129209431559643530266264q^{63} + 737186041679900306885426193785693026232265667803843778780176842752q^{64} + 6840147896488363686507276668005205679632480056088891731592702691660q^{65} + 3069860275698184453678654113958670789622501761274434325754744406016q^{66} - 7587584248381195618707457069538120550412828037866999031232038314372q^{67} - 5038761198798366803334407150633833722506784674618586571275629494272q^{68} - 21242639944698603837144967984504325973796588470663724540063832232288q^{69} - 35236659370606449042543653930245981331402869469140142926019092807680q^{70} - 54190860046980946062970823072982786833668067514475229703471473178696q^{71} + 15375281771657160870229969994716837558235628710503758158730633412608q^{72} + 302204276459526144519916965475340033188079342798263447893361558739654q^{73} + 460788297382010659258858143952886782081979620906639511666754999812096q^{74} + 1321988182453308808957469062463764651479002141783122360382572395384900q^{75} + 254701409494135202491619664917666445418981268382573756093980129361920q^{76} + 1136290782245844952967529647847166832148473964718146517215659790189216q^{77} - 599357099842967606540357177425328183707393143418058291814117182275584q^{78} - 2063644198299887356015584664968287745082399564774871245601417500935760q^{79} - 904492666301863930243089081419238237195853008895807926928396580290560q^{80} - 16891815155353385788713120786633925888175683542107829174659509500480753q^{81} - 6446672193337464673511795801397809916038289746403453740221072542793728q^{82} + 20637316524023922777695817093166723059825032985062213786640521552119404q^{83} + 13456541138976215781259821921857622832470000144438510839450677536620544q^{84} + 17020911204842079191578418414879669130697869935588821344636303582004020q^{85} + 19124031929912910639273030116541437792145765208839809191835113952903168q^{86} - 11898630724605207728038010903252775519339509401543700937344046348372600q^{87} + 43209995381876983363698417646408761043080555936919912203426521155633152q^{88} + 207336285159197918258375121859361162714964365933301696623144396043125430q^{89} - 317728036129547092571635979865630963286270874163859138802031801957089280q^{90} - 534340988981832565293850366847600554295341775746604779176000263013148016q^{91} + 54245927337147076542352991515708572016052829493901493648502741757067264q^{92} - 2106624457661050548013889439781993705135349795614255107465567210376551552q^{93} - 984126126936657408329473443273688080352037863578757473463124749675659264q^{94} - 4667409163598269693325856603058210361118245346808039972183000921107828600q^{95} + 933451503650382427298696882819844350924165111346864003082343036693250048q^{96} + 5896832155048876985101920414621646427655604788621115351504190027391728558q^{97} - 2124579222479382129816851660207925419012878646227427237621783845675204608q^{98} + 31261167116176375892644199050595144491934483437936698088824988018572848292q^{99} + O(q^{100}) \)

Decomposition of \(S_{74}^{\mathrm{new}}(\Gamma_1(2))\)

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
2.74.a \(\chi_{2}(1, \cdot)\) 2.74.a.a 3 1
2.74.a.b 4

Decomposition of \(S_{74}^{\mathrm{old}}(\Gamma_1(2))\) into lower level spaces

\( S_{74}^{\mathrm{old}}(\Gamma_1(2)) \cong \) \(S_{74}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 68719476736 T )^{3} \))(\( ( 1 - 68719476736 T )^{4} \))
$3$ (\( 1 + 304039201019733972 T + \)\(11\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + \)\(74\!\cdots\!31\)\( T^{4} + \)\(13\!\cdots\!88\)\( T^{5} + \)\(30\!\cdots\!67\)\( T^{6} \))(\( 1 - 305066325723244176 T + \)\(12\!\cdots\!08\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!32\)\( T^{6} - \)\(94\!\cdots\!92\)\( T^{7} + \)\(20\!\cdots\!41\)\( T^{8} \))
$5$ (\( 1 + \)\(32\!\cdots\!50\)\( T + \)\(16\!\cdots\!75\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!75\)\( T^{4} + \)\(36\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \))(\( 1 + \)\(78\!\cdots\!60\)\( T + \)\(41\!\cdots\!00\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} - \)\(14\!\cdots\!50\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{5} + \)\(46\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + \)\(23\!\cdots\!64\)\( T + \)\(36\!\cdots\!53\)\( T^{2} + \)\(31\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + \)\(56\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(36\!\cdots\!12\)\( T + \)\(80\!\cdots\!32\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(43\!\cdots\!16\)\( T^{7} + \)\(58\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + \)\(80\!\cdots\!44\)\( T + \)\(24\!\cdots\!05\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!55\)\( T^{4} + \)\(88\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(52\!\cdots\!48\)\( T + \)\(19\!\cdots\!88\)\( T^{2} + \)\(50\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(52\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!68\)\( T^{6} - \)\(61\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))
$13$ (\( 1 - \)\(15\!\cdots\!38\)\( T + \)\(52\!\cdots\!07\)\( T^{2} - \)\(56\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!71\)\( T^{4} - \)\(64\!\cdots\!42\)\( T^{5} + \)\(89\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!76\)\( T + \)\(19\!\cdots\!28\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!52\)\( T^{6} - \)\(12\!\cdots\!52\)\( T^{7} + \)\(18\!\cdots\!81\)\( T^{8} \))
$17$ (\( 1 + \)\(20\!\cdots\!34\)\( T + \)\(29\!\cdots\!63\)\( T^{2} + \)\(28\!\cdots\!68\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(89\!\cdots\!46\)\( T^{5} + \)\(29\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(95\!\cdots\!52\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!28\)\( T^{6} - \)\(28\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} \))
$19$ (\( 1 + \)\(41\!\cdots\!80\)\( T + \)\(64\!\cdots\!77\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!43\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(95\!\cdots\!00\)\( T + \)\(10\!\cdots\!36\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!16\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!61\)\( T^{8} \))
$23$ (\( 1 - \)\(78\!\cdots\!88\)\( T + \)\(33\!\cdots\!97\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(84\!\cdots\!51\)\( T^{4} - \)\(51\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!87\)\( T^{6} \))(\( 1 + \)\(67\!\cdots\!04\)\( T + \)\(40\!\cdots\!88\)\( T^{2} + \)\(25\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} + \)\(66\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!48\)\( T^{7} + \)\(42\!\cdots\!21\)\( T^{8} \))
$29$ (\( 1 - \)\(16\!\cdots\!10\)\( T + \)\(20\!\cdots\!67\)\( T^{2} + \)\(91\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!63\)\( T^{4} - \)\(53\!\cdots\!10\)\( T^{5} + \)\(18\!\cdots\!69\)\( T^{6} \))(\( 1 + \)\(40\!\cdots\!40\)\( T + \)\(22\!\cdots\!56\)\( T^{2} + \)\(65\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!20\)\( T^{5} + \)\(73\!\cdots\!76\)\( T^{6} + \)\(75\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \))
$31$ (\( 1 - \)\(95\!\cdots\!96\)\( T + \)\(16\!\cdots\!45\)\( T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(52\!\cdots\!76\)\( T^{5} + \)\(40\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(37\!\cdots\!08\)\( T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(64\!\cdots\!16\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(47\!\cdots\!56\)\( T^{5} + \)\(98\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(30\!\cdots\!61\)\( T^{8} \))
$37$ (\( 1 - \)\(95\!\cdots\!66\)\( T + \)\(31\!\cdots\!43\)\( T^{2} + \)\(49\!\cdots\!48\)\( T^{3} + \)\(94\!\cdots\!71\)\( T^{4} - \)\(86\!\cdots\!94\)\( T^{5} + \)\(27\!\cdots\!73\)\( T^{6} \))(\( 1 - \)\(68\!\cdots\!52\)\( T + \)\(27\!\cdots\!52\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} - \)\(22\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!96\)\( T^{7} + \)\(82\!\cdots\!81\)\( T^{8} \))
$41$ (\( 1 - \)\(15\!\cdots\!46\)\( T + \)\(23\!\cdots\!35\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!35\)\( T^{4} - \)\(44\!\cdots\!86\)\( T^{5} + \)\(15\!\cdots\!61\)\( T^{6} \))(\( 1 - \)\(56\!\cdots\!48\)\( T + \)\(97\!\cdots\!48\)\( T^{2} - \)\(10\!\cdots\!36\)\( T^{3} + \)\(58\!\cdots\!70\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{5} + \)\(28\!\cdots\!68\)\( T^{6} - \)\(90\!\cdots\!28\)\( T^{7} + \)\(85\!\cdots\!81\)\( T^{8} \))
$43$ (\( 1 + \)\(47\!\cdots\!92\)\( T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(45\!\cdots\!56\)\( T^{3} + \)\(23\!\cdots\!31\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(53\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!04\)\( T + \)\(65\!\cdots\!28\)\( T^{2} + \)\(91\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} + \)\(93\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 + \)\(12\!\cdots\!64\)\( T + \)\(35\!\cdots\!13\)\( T^{2} + \)\(27\!\cdots\!28\)\( T^{3} + \)\(41\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!56\)\( T^{5} + \)\(15\!\cdots\!83\)\( T^{6} \))(\( 1 + \)\(26\!\cdots\!88\)\( T + \)\(32\!\cdots\!12\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!40\)\( T^{5} + \)\(42\!\cdots\!48\)\( T^{6} + \)\(41\!\cdots\!04\)\( T^{7} + \)\(17\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 - \)\(14\!\cdots\!58\)\( T + \)\(22\!\cdots\!07\)\( T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} - \)\(79\!\cdots\!82\)\( T^{5} + \)\(41\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(45\!\cdots\!16\)\( T + \)\(11\!\cdots\!88\)\( T^{2} - \)\(84\!\cdots\!60\)\( T^{3} + \)\(70\!\cdots\!46\)\( T^{4} - \)\(62\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!41\)\( T^{8} \))
$59$ (\( 1 + \)\(38\!\cdots\!80\)\( T + \)\(60\!\cdots\!37\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(36\!\cdots\!80\)\( T + \)\(48\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 + \)\(13\!\cdots\!14\)\( T + \)\(38\!\cdots\!75\)\( T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(81\!\cdots\!75\)\( T^{4} + \)\(60\!\cdots\!54\)\( T^{5} + \)\(97\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(35\!\cdots\!08\)\( T + \)\(12\!\cdots\!48\)\( T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{5} + \)\(59\!\cdots\!28\)\( T^{6} - \)\(34\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 + \)\(10\!\cdots\!44\)\( T + \)\(72\!\cdots\!73\)\( T^{2} + \)\(37\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} + \)\(40\!\cdots\!36\)\( T^{5} + \)\(81\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(24\!\cdots\!72\)\( T + \)\(33\!\cdots\!92\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(53\!\cdots\!46\)\( T^{4} + \)\(45\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} - \)\(20\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} \))
$71$ (\( 1 + \)\(11\!\cdots\!24\)\( T + \)\(83\!\cdots\!25\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} + \)\(22\!\cdots\!04\)\( T^{5} + \)\(26\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(61\!\cdots\!28\)\( T + \)\(53\!\cdots\!88\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} - \)\(28\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(16\!\cdots\!68\)\( T^{7} + \)\(36\!\cdots\!41\)\( T^{8} \))
$73$ (\( 1 + \)\(15\!\cdots\!82\)\( T + \)\(34\!\cdots\!07\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(36\!\cdots\!31\)\( T^{4} + \)\(17\!\cdots\!98\)\( T^{5} + \)\(11\!\cdots\!37\)\( T^{6} \))(\( 1 - \)\(45\!\cdots\!36\)\( T + \)\(11\!\cdots\!68\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(53\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))
$79$ (\( 1 + \)\(40\!\cdots\!00\)\( T + \)\(12\!\cdots\!17\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!63\)\( T^{4} + \)\(46\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(20\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!26\)\( T^{4} - \)\(56\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(77\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 - \)\(48\!\cdots\!88\)\( T + \)\(83\!\cdots\!37\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(73\!\cdots\!72\)\( T^{5} + \)\(18\!\cdots\!47\)\( T^{6} \))(\( 1 - \)\(15\!\cdots\!16\)\( T + \)\(44\!\cdots\!48\)\( T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{5} + \)\(67\!\cdots\!12\)\( T^{6} - \)\(29\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} \))
$89$ (\( 1 - \)\(45\!\cdots\!70\)\( T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!83\)\( T^{4} - \)\(18\!\cdots\!70\)\( T^{5} + \)\(82\!\cdots\!09\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!40\)\( T + \)\(87\!\cdots\!76\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!66\)\( T^{4} + \)\(29\!\cdots\!20\)\( T^{5} + \)\(35\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \))
$97$ (\( 1 - \)\(76\!\cdots\!26\)\( T + \)\(30\!\cdots\!23\)\( T^{2} - \)\(42\!\cdots\!92\)\( T^{3} + \)\(32\!\cdots\!71\)\( T^{4} - \)\(90\!\cdots\!54\)\( T^{5} + \)\(12\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(58\!\cdots\!32\)\( T + \)\(30\!\cdots\!92\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!68\)\( T^{6} - \)\(73\!\cdots\!56\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \))
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