Properties

Label 2.86.a
Level 2
Weight 86
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 2
Sturm bound 21
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 86 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{86}(\Gamma_0(2))\).

Total New Old
Modular forms 22 8 14
Cusp forms 20 8 12
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(4\)
\(-\)\(4\)

Trace form

\( 8q + 269553027322002295200q^{3} + 154742504910672534362390528q^{4} + 1484419974003616205488098222000q^{5} + 826842279871891938749146151780352q^{6} - 615523597339907391982903341468497600q^{7} + 118828885947059966775185311634015921096424q^{9} + O(q^{10}) \) \( 8q + 269553027322002295200q^{3} + 154742504910672534362390528q^{4} + 1484419974003616205488098222000q^{5} + 826842279871891938749146151780352q^{6} - 615523597339907391982903341468497600q^{7} + 118828885947059966775185311634015921096424q^{9} + 467765378440194361291224013224855522508800q^{10} - 101797481860206574106717351656968700225898784q^{11} + 5213913831757698498672213602352359321842483200q^{12} + 479112641772940819648401363972751216889009628400q^{13} - 4970712848829318407030506490397860087671403053056q^{14} - 271745842789897203677825843886989463236276342644800q^{15} + 2993155353253689176481146537402947624255349848014848q^{16} + 55754425565484547496180681745937601424459350789331600q^{17} - 252682171279403821081304108731828847541902253647462400q^{18} - 2929591165061188546903035525219403377807083997751247840q^{19} + 28712858139594372059529236779806969160727870048305152000q^{20} - 177837756224578674727338565916857153704267407366365634304q^{21} + 1731453221265625162743138887790712836702697145195141529600q^{22} - 8326608631124368674563983949658834032240169517280962804800q^{23} + 15993455694178489043546292624851516862727062593102237663232q^{24} + 50440983628024394799954236936815371007845526315736871515000q^{25} - 1740356260062666646895169756863700990277016303011003197554688q^{26} + 13715447878465367754004449232495137063430285698294322855310400q^{27} - 11905957910500680414116629208459082705745212431581442578841600q^{28} - 80804886823838884071775456652113469944063607328695210326972560q^{29} + 571083860688419042422776990904949923888548001591164854599680000q^{30} + 1633567803274849645661476153182778402652823922406345625782646016q^{31} + 51629154124411494241455192195981789763524293220492860915597116800q^{33} + 76810738067911368171154268692467615587532838104833686639805988864q^{34} - 303907195718456817012846584882539686080918551354269325209569385600q^{35} + 2298484933399084177227191587423576584872237130977009353119429033984q^{36} - 3570318342481673237747139074956703063851617532078736009959140760400q^{37} + 13557340070278076771546516361269540434280107191837609612694624665600q^{38} - 52826197364072519061410300097668400287772548032981050331137250627392q^{39} + 9047898296290546547999898244249449210914774552627163603109989580800q^{40} - 101879485641498700390198644632399823374919799815471748501171952382384q^{41} + 3430430536466684160965519739670099420062166857006963306538602804019200q^{42} - 983017174328970333173953499578367781476596926607126329920567679034400q^{43} - 1969049667080889253984850502657705669675769736731414525891979451039744q^{44} + 129414053322130030478694471180705803462601292852690242215929564439862000q^{45} + 52217657421151487659816109775539445219348550536042728197335901349609472q^{46} + 418464261387472986972541171943537580545821035775417913504888854487318400q^{47} + 100851760839323638759435620035232536926471244341500257660581027459891200q^{48} + 4861201030765380588086678338224369843323136135029448905719677358494048456q^{49} + 3507024174205801426674718518642244780162117239187213725517524731166720000q^{50} + 14458555377434229192130842989239979469835897417393174031425068906319738176q^{51} + 9267386290289323200578772342872056249213880282578624968309635279119974400q^{52} + 79238893661066286548123396135403859078450913300218508816248269289127876400q^{53} + 127479891442271163284555108637214012971880501702375711073997768996547461120q^{54} - 115892409687490903974845422600674579019444744881898202357235531152417400000q^{55} - 96147569677439233318535669969921558315737813875377064640901250850221981696q^{56} - 708291507196858496950336584814158730962641225991642476431323602773416585600q^{57} - 1872002936537433921837667385553562778861363634278890764697018281522678988800q^{58} - 9718564654902213878699228800763515278155682743529550250614492725480982626720q^{59} - 5256329051546314319453026985277683016572662308968704521552881311019748556800q^{60} - 22502293106918750966308876870780925084969607013559207911952914084993042461584q^{61} - 512745364371250743679940949316115524515609735680232267031558635700145356800q^{62} - 33253188453235282602659365361968943040833119870915187924088567977510775768000q^{63} + 57896044618658097711785492504343953926634992332820282019728792003956564819968q^{64} + 208538001796442375993409864803085821949183354617266498590081927679837556887200q^{65} + 731160993111948540733699201721170861720074557308310226128580428775765105967104q^{66} + 618430087497352208039439690042770354046234497149407493289606954052900779447200q^{67} + 1078447433982339860563125081874542097822167715470030508613160573169686911385600q^{68} + 3749053568809890429242703488244876040784090696484290653048341072416446306222848q^{69} - 1712687631740715700813202036984572612382504801817892498537480528754292817920000q^{70} + 2914456738203768985035713583890518823624642690727941357848894790247547420996416q^{71} - 4887584016255318026061913823771859917295703698729015999704548287676928124518400q^{72} - 52943131008350183167528071295913452178374790158418012532048380044880083916135600q^{73} - 5385145831472910827359428561344620582528728363972547468922141298607300433412096q^{74} - 244065785632991520690690211282530753260324493030420362754725144723791313638020000q^{75} - 56666534405717979986979761320503763991405624446638989149707898025031509674557440q^{76} + 7111955725060403646131701118241159104315578946674911063489391126414174064505600q^{77} - 261791324240468500546822182213269410234570008409044716200403595760397197089177600q^{78} + 1025784172367515922042732760261822950646306776701865615471484090898577406684787840q^{79} + 555387448958203245946423939712902268301983657299063771266156608526982192300032000q^{80} + 6626221899858755588629003462743365911014779696849691608947928310864505539575554248q^{81} + 1237202594219701351105933788823014042420042676149637742207414672365206952004812800q^{82} + 7219217659129802650581247135505194878762696845871793297216243109006748312900372000q^{83} - 3439882483235606329833492514826381898240253814840602267894292173353667249060184064q^{84} + 3912830459186382949391201016846469773029483089249926950618811281750507636796962400q^{85} - 2342186222009889804507850275471051931820653834701123909729421408779546889769975808q^{86} - 200105984362001069230238288458032592723516056569427552223412430020167042682139643200q^{87} + 33491176074286972493521577678464668166400076298916295145398472140103501043308953600q^{88} - 85965578199760485237519211339136793968971795955905134183109192927336430636082565680q^{89} - 94533535251611151078408544072096720829273563268460968512105519589964501772507545600q^{90} - 17240791454229674589215417657881373257899663802715180384931132222938185892409821824q^{91} - 161060034623876366108872021610492466260287712176813826455663807597989574797479116800q^{92} + 1290240338494900619547839525109983192448007231583110758481049139812759993872897459200q^{93} + 1433869637842692892430214853576316577021589437801857515062390740922072334616833818624q^{94} + 3159648538409298714208978848652977445274561175020435690323251204202011070319474616000q^{95} + 309358424536879805970631109977718021156076018263400592941652229972136641831016333312q^{96} + 2815820736417570017461296806434083982402251944048464366157428863016933480912055050000q^{97} - 1287533752711804733389579391300849145998644069973215505103549747592962474281376153600q^{98} - 13821486497634205785853128483475818467151466908122344585095762540941651078331897454752q^{99} + O(q^{100}) \)

Decomposition of \(S_{86}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.86.a.a \(4\) \(91.510\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-1\!\cdots\!16\) \(40\!\cdots\!56\) \(68\!\cdots\!00\) \(25\!\cdots\!32\) \(+\) \(q-2^{42}q^{2}+(10193857391958586164+\cdots)q^{3}+\cdots\)
2.86.a.b \(4\) \(91.510\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(17\!\cdots\!16\) \(22\!\cdots\!44\) \(79\!\cdots\!00\) \(-8\!\cdots\!32\) \(-\) \(q+2^{42}q^{2}+(57194399438541987636+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{86}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{86}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{86}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 4398046511104 T )^{4} \))(\( ( 1 - 4398046511104 T )^{4} \))
$3$ (\( 1 - 40775429567834344656 T + \)\(28\!\cdots\!48\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(48\!\cdots\!46\)\( T^{4} + \)\(67\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!92\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(22\!\cdots\!44\)\( T + \)\(82\!\cdots\!48\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} - \)\(69\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!08\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \))
$5$ (\( 1 - \)\(68\!\cdots\!00\)\( T + \)\(94\!\cdots\!00\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!50\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(62\!\cdots\!00\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!25\)\( T^{8} \))(\( 1 - \)\(79\!\cdots\!00\)\( T + \)\(62\!\cdots\!00\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(41\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 - \)\(25\!\cdots\!32\)\( T + \)\(10\!\cdots\!12\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!66\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(49\!\cdots\!88\)\( T^{6} - \)\(81\!\cdots\!76\)\( T^{7} + \)\(21\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(87\!\cdots\!32\)\( T + \)\(60\!\cdots\!12\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!66\)\( T^{4} + \)\(30\!\cdots\!40\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} + \)\(27\!\cdots\!76\)\( T^{7} + \)\(21\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + \)\(24\!\cdots\!92\)\( T + \)\(31\!\cdots\!28\)\( T^{2} + \)\(37\!\cdots\!44\)\( T^{3} + \)\(40\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!28\)\( T^{6} + \)\(88\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(14\!\cdots\!08\)\( T + \)\(87\!\cdots\!28\)\( T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!70\)\( T^{4} - \)\(54\!\cdots\!56\)\( T^{5} + \)\(95\!\cdots\!28\)\( T^{6} - \)\(52\!\cdots\!08\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))
$13$ (\( 1 - \)\(43\!\cdots\!36\)\( T + \)\(18\!\cdots\!08\)\( T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} - \)\(22\!\cdots\!80\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} - \)\(49\!\cdots\!52\)\( T^{7} + \)\(55\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(41\!\cdots\!64\)\( T + \)\(10\!\cdots\!08\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!26\)\( T^{4} - \)\(62\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!92\)\( T^{6} - \)\(47\!\cdots\!48\)\( T^{7} + \)\(55\!\cdots\!01\)\( T^{8} \))
$17$ (\( 1 - \)\(19\!\cdots\!92\)\( T + \)\(67\!\cdots\!52\)\( T^{2} - \)\(93\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!46\)\( T^{4} - \)\(36\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(11\!\cdots\!56\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(36\!\cdots\!08\)\( T + \)\(18\!\cdots\!52\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} - \)\(21\!\cdots\!44\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \))
$19$ (\( 1 + \)\(30\!\cdots\!20\)\( T + \)\(19\!\cdots\!96\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!06\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(48\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!80\)\( T^{7} + \)\(59\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(76\!\cdots\!80\)\( T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(38\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!06\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!96\)\( T^{6} - \)\(92\!\cdots\!20\)\( T^{7} + \)\(59\!\cdots\!01\)\( T^{8} \))
$23$ (\( 1 + \)\(10\!\cdots\!84\)\( T + \)\(19\!\cdots\!68\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!06\)\( T^{4} + \)\(80\!\cdots\!40\)\( T^{5} + \)\(62\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!88\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(17\!\cdots\!84\)\( T + \)\(54\!\cdots\!68\)\( T^{2} - \)\(49\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!06\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!32\)\( T^{6} - \)\(30\!\cdots\!88\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \))
$29$ (\( 1 - \)\(17\!\cdots\!20\)\( T + \)\(54\!\cdots\!96\)\( T^{2} - \)\(93\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!96\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(25\!\cdots\!80\)\( T + \)\(38\!\cdots\!96\)\( T^{2} + \)\(60\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} + \)\(12\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!96\)\( T^{6} + \)\(20\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \))
$31$ (\( 1 - \)\(87\!\cdots\!08\)\( T + \)\(17\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(76\!\cdots\!56\)\( T^{5} + \)\(59\!\cdots\!28\)\( T^{6} - \)\(17\!\cdots\!08\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(75\!\cdots\!08\)\( T + \)\(80\!\cdots\!28\)\( T^{2} + \)\(47\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} + \)\(27\!\cdots\!44\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!08\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))
$37$ (\( 1 + \)\(11\!\cdots\!88\)\( T + \)\(33\!\cdots\!32\)\( T^{2} + \)\(58\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} + \)\(91\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(23\!\cdots\!12\)\( T + \)\(61\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!06\)\( T^{4} + \)\(28\!\cdots\!20\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \))
$41$ (\( 1 + \)\(19\!\cdots\!92\)\( T + \)\(25\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(31\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!28\)\( T^{6} + \)\(34\!\cdots\!92\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(89\!\cdots\!08\)\( T + \)\(20\!\cdots\!28\)\( T^{2} - \)\(93\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(30\!\cdots\!28\)\( T^{6} - \)\(16\!\cdots\!08\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} \))
$43$ (\( 1 + \)\(22\!\cdots\!24\)\( T + \)\(18\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!66\)\( T^{4} + \)\(75\!\cdots\!80\)\( T^{5} + \)\(90\!\cdots\!12\)\( T^{6} + \)\(77\!\cdots\!68\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(75\!\cdots\!76\)\( T + \)\(18\!\cdots\!88\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!66\)\( T^{4} + \)\(32\!\cdots\!20\)\( T^{5} + \)\(88\!\cdots\!12\)\( T^{6} + \)\(25\!\cdots\!32\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - \)\(46\!\cdots\!72\)\( T + \)\(37\!\cdots\!72\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(64\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(67\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!96\)\( T^{7} + \)\(32\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(37\!\cdots\!28\)\( T + \)\(10\!\cdots\!72\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!28\)\( T^{6} - \)\(90\!\cdots\!04\)\( T^{7} + \)\(32\!\cdots\!01\)\( T^{8} \))
$53$ (\( 1 - \)\(27\!\cdots\!56\)\( T + \)\(92\!\cdots\!48\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!46\)\( T^{4} - \)\(75\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!52\)\( T^{6} - \)\(13\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(51\!\cdots\!44\)\( T + \)\(22\!\cdots\!48\)\( T^{2} - \)\(60\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!46\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!52\)\( T^{6} - \)\(25\!\cdots\!08\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))
$59$ (\( 1 + \)\(45\!\cdots\!60\)\( T + \)\(18\!\cdots\!96\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(51\!\cdots\!60\)\( T + \)\(17\!\cdots\!96\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(80\!\cdots\!06\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!96\)\( T^{6} + \)\(19\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \))
$61$ (\( 1 + \)\(20\!\cdots\!92\)\( T + \)\(34\!\cdots\!28\)\( T^{2} + \)\(36\!\cdots\!44\)\( T^{3} + \)\(32\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!28\)\( T^{6} + \)\(37\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(20\!\cdots\!92\)\( T + \)\(15\!\cdots\!28\)\( T^{2} + \)\(30\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!70\)\( T^{4} + \)\(17\!\cdots\!44\)\( T^{5} + \)\(51\!\cdots\!28\)\( T^{6} + \)\(37\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} \))
$67$ (\( 1 + \)\(61\!\cdots\!08\)\( T + \)\(33\!\cdots\!52\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} + \)\(35\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!44\)\( T^{7} + \)\(73\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(67\!\cdots\!08\)\( T + \)\(39\!\cdots\!52\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!46\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!48\)\( T^{6} - \)\(30\!\cdots\!44\)\( T^{7} + \)\(73\!\cdots\!01\)\( T^{8} \))
$71$ (\( 1 - \)\(19\!\cdots\!08\)\( T + \)\(71\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} - \)\(24\!\cdots\!56\)\( T^{5} + \)\(36\!\cdots\!28\)\( T^{6} - \)\(22\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(98\!\cdots\!08\)\( T + \)\(57\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(30\!\cdots\!56\)\( T^{5} + \)\(29\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \))
$73$ (\( 1 + \)\(26\!\cdots\!84\)\( T + \)\(70\!\cdots\!68\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!06\)\( T^{4} + \)\(30\!\cdots\!40\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} + \)\(37\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 + \)\(26\!\cdots\!16\)\( T + \)\(11\!\cdots\!68\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!06\)\( T^{4} + \)\(46\!\cdots\!60\)\( T^{5} + \)\(68\!\cdots\!32\)\( T^{6} + \)\(36\!\cdots\!12\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \))
$79$ (\( 1 - \)\(26\!\cdots\!20\)\( T + \)\(61\!\cdots\!96\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!96\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(75\!\cdots\!20\)\( T + \)\(90\!\cdots\!96\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!06\)\( T^{4} - \)\(89\!\cdots\!60\)\( T^{5} + \)\(35\!\cdots\!96\)\( T^{6} - \)\(59\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \))
$83$ (\( 1 + \)\(25\!\cdots\!04\)\( T + \)\(45\!\cdots\!28\)\( T^{2} + \)\(97\!\cdots\!20\)\( T^{3} + \)\(84\!\cdots\!86\)\( T^{4} + \)\(12\!\cdots\!60\)\( T^{5} + \)\(79\!\cdots\!72\)\( T^{6} + \)\(59\!\cdots\!28\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(98\!\cdots\!04\)\( T + \)\(82\!\cdots\!28\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(55\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!72\)\( T^{6} - \)\(22\!\cdots\!28\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \))
$89$ (\( 1 + \)\(17\!\cdots\!40\)\( T + \)\(18\!\cdots\!96\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!06\)\( T^{4} + \)\(52\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} + \)\(21\!\cdots\!60\)\( T^{7} + \)\(62\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(84\!\cdots\!60\)\( T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(52\!\cdots\!80\)\( T^{5} + \)\(40\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!01\)\( T^{8} \))
$97$ (\( 1 + \)\(78\!\cdots\!28\)\( T + \)\(32\!\cdots\!72\)\( T^{2} + \)\(78\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} + \)\(59\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!28\)\( T^{6} + \)\(33\!\cdots\!04\)\( T^{7} + \)\(31\!\cdots\!01\)\( T^{8} \))(\( 1 - \)\(10\!\cdots\!28\)\( T + \)\(68\!\cdots\!72\)\( T^{2} - \)\(28\!\cdots\!60\)\( T^{3} + \)\(92\!\cdots\!86\)\( T^{4} - \)\(21\!\cdots\!20\)\( T^{5} + \)\(38\!\cdots\!28\)\( T^{6} - \)\(45\!\cdots\!04\)\( T^{7} + \)\(31\!\cdots\!01\)\( T^{8} \))
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