Properties

Label 1.108.a
Level 1
Weight 108
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 108 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{108}(\Gamma_0(1))\).

Total New Old
Modular forms 10 10 0
Cusp forms 9 9 0
Eisenstein series 1 1 0

Trace form

\( 9q + 5465779613435496q^{2} + 15983866788128062714366812q^{3} + 641838178072986908923140040636992q^{4} + 24730262360170403173006735448079708750q^{5} - 120659696701040014765473752948751733206432q^{6} - 935936272579440558945885185297012359442693544q^{7} - 1000549071290096181876510601182828415482105239040q^{8} + 3695400441386851970702053152997489358579707799946813q^{9} + O(q^{10}) \) \( 9q + 5465779613435496q^{2} + 15983866788128062714366812q^{3} + 641838178072986908923140040636992q^{4} + 24730262360170403173006735448079708750q^{5} - 120659696701040014765473752948751733206432q^{6} - 935936272579440558945885185297012359442693544q^{7} - 1000549071290096181876510601182828415482105239040q^{8} + 3695400441386851970702053152997489358579707799946813q^{9} - 1051865624814321749095840304599907252969934693398466000q^{10} + 75371386674400378520973626363253224857093312063724664948q^{11} - 7994600333787230851514661173029738426135745580020653426944q^{12} + 386688410581830829569835770029121092890073212075170355706982q^{13} - 7425120169660057221925189838061614099801999208262592119204416q^{14} + 1114849595057357604173216029035423209082636701646415622289453000q^{15} + 29408614076374287218252370499930154042728180304312580290885259264q^{16} + 557190110555184717584150469172018747011155217501263013200292053826q^{17} + 23686310065847344603267400477778156225688961326793890396260803764872q^{18} + 858726527367010511176780653766124295103204700130810063275302960078860q^{19} + 13293825016749054587175558577837356326662860013460552585283415249200000q^{20} + 98208620574357072907810930779441373553105765539483211558155794396236448q^{21} - 2120482037145318757833662337561674615184814159100624269907546121294686688q^{22} - 10034721240753129475726648205629192527604953514063764213610508652545603448q^{23} - 2941281585435310612767332425157841570788095859273373502643916660565125120q^{24} + 2090668146509277585662418878235593338432069837730236508072952837108480609375q^{25} + 297618678938898468541825094241452097503242676900281081118135240117081112048q^{26} + 12010637132277372007849219431146961646841705965134800444369356096651346807640q^{27} - 772836919356016231875276488741314175452898152873144799220188421863529684675072q^{28} + 3591314969861592690040264647853795847194812597017039069680951706167032656423190q^{29} + 1723952423459255307102435810928767115448833503290363214135969710252703496200000q^{30} + 80089747856260124585510286878769275145759211608657844798932602922226546761434208q^{31} - 1446206593191149472500309116447393072011146060673364310104554905463216959190564864q^{32} + 3945091609919542459300423121384432970721444207912781947382890849249898482566767664q^{33} + 11119656315304296004606294411853238631199272487238804497227126821893639468830096464q^{34} + 49067962462953733482940013097788952961901858200557588372758479947688558453676754000q^{35} - 471823079431745456450508655131148234440317478341437039102283657625130709182178187456q^{36} - 211704323021230989692835910298149365559694037564054725936839839132550744753936919634q^{37} + 8222885444046313840674570479741432975521521666418514540085950022601909450574547779040q^{38} + 12361178896398609619161562511793017908827480821637222996756699411815138902405469354856q^{39} - 46215645594071398383759344185895857310354869228136299740138569713991972194487637120000q^{40} - 152723040798290839871602016171161581257438821965138798374317736485575056512069694567862q^{41} + 1126972287891090202493176150041627427283154361091115578522922093800965746370272835782912q^{42} + 588069515895858848125340008029591292900452115124091837264351724714905019444084944743892q^{43} - 6139643207231705808426527030461811499402203060055049552369302116418471883341886274962176q^{44} - 31911427383192526363701563735501578120849438309590564668521639113629216238459194946116250q^{45} - 18263182784420532146882371818610257236316553965008962373113458516888555657572617999188672q^{46} + 411127412634686500562199358253262979305374320422388983156864942033640143172424484152539536q^{47} + 143930370214090114738516122425367964462847873560413052946256979092891611198246163544522752q^{48} - 4997644842464582200845249613678276624838382931152181846690817178995051025280570789312693263q^{49} - 327953810706856692980141167207241667907351028782006717494410015078645777795786692667625000q^{50} + 42691516908125956684809245526355840553570893389914391645941980034335859681366662250925257208q^{51} + 59364881289634100249284358230824565528876555321811374467351490113119501371198417490401094016q^{52} - 53335374129204409504769785775612138052491162110126476217063770562215392060493506708504818338q^{53} - 1554605781923117836337527850943594545571965959146902578434980467357667330247224121116945547840q^{54} + 1474216898342963983363455770095216024051493885851745466905891181181387017915931645476666255000q^{55} + 7453692229427932770491342947786818133880887558034867118652861854414844393630476278780284170240q^{56} - 9382173343921294589567308072318971089362548602163794938031557584052656917238640814769133635120q^{57} - 14675317362256339836240584983997175224307357531571668080947702647986363061369632064127758276240q^{58} - 17201443369933521380448054479438020840215748691907958949057650623559478137031846309102734186620q^{59} + 119942427235571024011403351926139340364016919293031353278124824892037701815579444998816654144000q^{60} + 996825825169496808828278558344476432289838610202339535134269176536499940712988317319113083438198q^{61} - 1806664642208894952880819938644089164868010113998872083788864908221482430118942433714202980872448q^{62} - 525131716387364530569619107989883435883820178537003544594931993029001820968120281192738395294408q^{63} + 8859800242100397923173345055025991374254635563298611761326132852774863587391298255452246238298112q^{64} + 3354685369129755322181059460881157417967535549386685974710181837734684369198862379954288278640500q^{65} + 57953332747637118130796913061247956114095776689582635716826448321044474448975464944502438440474496q^{66} + 67272594646508937763531278535250984988721264552212406692490519409868542593541008988303512955099676q^{67} - 5271355091203561642375555737902479878900668615857946963901492781607485228195805621489749763440512q^{68} + 1211310643069343900708768635208063407543461033111793008132153244460934861341593025873966799465486816q^{69} + 3012778566653438568685732696962102874663629250451589330912197198332559170935750394136926553795280000q^{70} + 557606230265736297562017362716168472788611890356025926095319973789617215901574196175404049892853528q^{71} + 16960010475120540092713267652235654327415255109531175122804202418279614126703951005828887393908830720q^{72} + 13389828619022747318968894782762272535480348540089259518080823938522165418472525950139881426529375402q^{73} + 72532783100036668116960948230568216484424073462645933664305672446671831241015488437308573521198008624q^{74} + 181746227987304022931719943233189267636401541776890365701313509214346868335764760358972397994607562500q^{75} + 369203366658532984115827909838760339113448992029578983924004973867253851696995490992782273693567066880q^{76} + 388491114360440288787870394325327319503105866790556270795138762844335429572119358188629406545039198432q^{77} + 1684225666426130585491319943814839912966891568955799622030995821490516792520001316121300784238905446464q^{78} + 1767099872706468468805139692411742252367926917059947532799767578463589430448822266501638541217663494640q^{79} + 6734051899360870505834986379536968461908477509814468240268598557190207937484585956747946194493844480000q^{80} + 8587274871353378729934471647154494323432295856816917259751843692725713832863430007711361770205995287889q^{81} + 8604324829509629266993327118230721908083540408244010940656031675054140623869235247377840945490064899472q^{82} + 10573584024255166860785482858544789933175527426032260694531283318949407366175819470225352840720947271372q^{83} + 29398865778509538549966771421066766907861809604721299372151512034272864117018289488710440505099182516224q^{84} + 28217101873426464307527276596170727373321741101882319418731811087750656251683841267824081156479505981500q^{85} - 5660340715902714082316421136856625160042953949488572010019823412763437338500312808862231168781884748512q^{86} - 238801934255165782785740400186129085029676437910074769136670805304976544209174607407124977464878965608280q^{87} - 649669481755409892912473717222108679378769149016402953512733477402495574055782427323588758822348679034880q^{88} - 887044331989597087426634052611019082958734414422732136340380168408138968025139915003605988823387508725030q^{89} - 2358060024132454293950038004387227695091155243819311445899211024200212376406755931436993187026324923882000q^{90} - 2028610692814038362284016228595385975371756633777610476239575578728190535138139517202725484712408010671472q^{91} - 9605953523619230804079170889956757296920712871591046684448269611692869761363895640295324681873778639117824q^{92} - 10277229372847627124269654885108201819576328825280371977906378287406641199658335902623697673880879481373056q^{93} - 13250229074063549042347396359745980974584115253516850536254551366146505318484654226001161612613989202932096q^{94} + 1932498314000384011251898358613786886453409028619258552182745516014729315280931175089689966876445460945000q^{95} + 8188689385662642160607934226103033776391463701323345460407294147322816918802843383187307295905239187980288q^{96} + 18319527847032942156630290559559475585838161401404427251154943474039162663641018432069269659324962153774386q^{97} + 91489091329550984599525415507771543780042234198589939082127267861983786873859650920519076378970080872245928q^{98} + 211118154688143874137117803870035099572707397285369760168480766428388700807018832202269547436760962791157636q^{99} + O(q^{100}) \)

Decomposition of \(S_{108}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.108.a.a \(9\) \(72.504\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(54\!\cdots\!96\) \(15\!\cdots\!12\) \(24\!\cdots\!50\) \(-9\!\cdots\!44\) \(+\) \(q+(607308845937277-\beta _{1})q^{2}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5465779613435496 T + \)\(42\!\cdots\!88\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!16\)\( T^{4} - \)\(47\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} + \)\(78\!\cdots\!80\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} + \)\(19\!\cdots\!44\)\( T^{9} + \)\(40\!\cdots\!88\)\( T^{10} + \)\(20\!\cdots\!20\)\( T^{11} + \)\(73\!\cdots\!56\)\( T^{12} - \)\(32\!\cdots\!36\)\( T^{13} + \)\(11\!\cdots\!88\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(12\!\cdots\!56\)\( T^{16} - \)\(26\!\cdots\!56\)\( T^{17} + \)\(77\!\cdots\!08\)\( T^{18} \)
$3$ \( 1 - \)\(15\!\cdots\!12\)\( T + \)\(33\!\cdots\!07\)\( T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!16\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(47\!\cdots\!52\)\( T^{6} - \)\(20\!\cdots\!80\)\( T^{7} + \)\(69\!\cdots\!46\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{9} + \)\(77\!\cdots\!02\)\( T^{10} - \)\(26\!\cdots\!20\)\( T^{11} + \)\(68\!\cdots\!56\)\( T^{12} - \)\(16\!\cdots\!32\)\( T^{13} + \)\(81\!\cdots\!12\)\( T^{14} - \)\(10\!\cdots\!60\)\( T^{15} + \)\(77\!\cdots\!81\)\( T^{16} - \)\(41\!\cdots\!52\)\( T^{17} + \)\(29\!\cdots\!27\)\( T^{18} \)
$5$ \( 1 - \)\(24\!\cdots\!50\)\( T + \)\(20\!\cdots\!25\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!00\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!50\)\( T^{10} + \)\(38\!\cdots\!00\)\( T^{11} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(93\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!00\)\( T^{14} - \)\(15\!\cdots\!00\)\( T^{15} + \)\(68\!\cdots\!25\)\( T^{16} - \)\(51\!\cdots\!50\)\( T^{17} + \)\(12\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 + \)\(93\!\cdots\!44\)\( T + \)\(14\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!96\)\( T^{4} + \)\(57\!\cdots\!24\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(45\!\cdots\!64\)\( T^{9} + \)\(42\!\cdots\!58\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(91\!\cdots\!96\)\( T^{12} + \)\(28\!\cdots\!24\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(40\!\cdots\!00\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} + \)\(23\!\cdots\!44\)\( T^{17} + \)\(67\!\cdots\!43\)\( T^{18} \)
$11$ \( 1 - \)\(75\!\cdots\!48\)\( T + \)\(13\!\cdots\!63\)\( T^{2} - \)\(79\!\cdots\!96\)\( T^{3} + \)\(90\!\cdots\!60\)\( T^{4} - \)\(49\!\cdots\!68\)\( T^{5} + \)\(42\!\cdots\!04\)\( T^{6} - \)\(20\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!98\)\( T^{8} - \)\(62\!\cdots\!20\)\( T^{9} + \)\(39\!\cdots\!58\)\( T^{10} - \)\(14\!\cdots\!64\)\( T^{11} + \)\(82\!\cdots\!44\)\( T^{12} - \)\(25\!\cdots\!08\)\( T^{13} + \)\(12\!\cdots\!60\)\( T^{14} - \)\(29\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!33\)\( T^{16} - \)\(20\!\cdots\!28\)\( T^{17} + \)\(72\!\cdots\!31\)\( T^{18} \)
$13$ \( 1 - \)\(38\!\cdots\!82\)\( T + \)\(55\!\cdots\!57\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!36\)\( T^{4} - \)\(55\!\cdots\!92\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!62\)\( T^{10} - \)\(25\!\cdots\!40\)\( T^{11} + \)\(14\!\cdots\!16\)\( T^{12} - \)\(32\!\cdots\!32\)\( T^{13} + \)\(15\!\cdots\!52\)\( T^{14} - \)\(28\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!61\)\( T^{16} - \)\(13\!\cdots\!62\)\( T^{17} + \)\(53\!\cdots\!97\)\( T^{18} \)
$17$ \( 1 - \)\(55\!\cdots\!26\)\( T + \)\(20\!\cdots\!53\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(62\!\cdots\!96\)\( T^{9} + \)\(47\!\cdots\!98\)\( T^{10} - \)\(24\!\cdots\!40\)\( T^{11} + \)\(16\!\cdots\!76\)\( T^{12} - \)\(70\!\cdots\!56\)\( T^{13} + \)\(42\!\cdots\!08\)\( T^{14} - \)\(12\!\cdots\!20\)\( T^{15} + \)\(83\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!06\)\( T^{17} + \)\(83\!\cdots\!13\)\( T^{18} \)
$19$ \( 1 - \)\(85\!\cdots\!60\)\( T + \)\(67\!\cdots\!51\)\( T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} - \)\(63\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!96\)\( T^{6} - \)\(70\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!66\)\( T^{8} - \)\(54\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!74\)\( T^{10} - \)\(31\!\cdots\!40\)\( T^{11} + \)\(68\!\cdots\!24\)\( T^{12} - \)\(12\!\cdots\!80\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} - \)\(31\!\cdots\!20\)\( T^{15} + \)\(41\!\cdots\!29\)\( T^{16} - \)\(35\!\cdots\!60\)\( T^{17} + \)\(27\!\cdots\!59\)\( T^{18} \)
$23$ \( 1 + \)\(10\!\cdots\!48\)\( T + \)\(26\!\cdots\!07\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!56\)\( T^{4} + \)\(27\!\cdots\!28\)\( T^{5} + \)\(35\!\cdots\!12\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!26\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!22\)\( T^{10} + \)\(58\!\cdots\!40\)\( T^{11} + \)\(46\!\cdots\!76\)\( T^{12} + \)\(18\!\cdots\!68\)\( T^{13} + \)\(12\!\cdots\!92\)\( T^{14} + \)\(37\!\cdots\!20\)\( T^{15} + \)\(23\!\cdots\!41\)\( T^{16} + \)\(43\!\cdots\!28\)\( T^{17} + \)\(22\!\cdots\!67\)\( T^{18} \)
$29$ \( 1 - \)\(35\!\cdots\!90\)\( T + \)\(18\!\cdots\!81\)\( T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!16\)\( T^{4} - \)\(34\!\cdots\!20\)\( T^{5} + \)\(91\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} - \)\(57\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(24\!\cdots\!44\)\( T^{12} - \)\(27\!\cdots\!20\)\( T^{13} + \)\(39\!\cdots\!84\)\( T^{14} - \)\(34\!\cdots\!80\)\( T^{15} + \)\(40\!\cdots\!89\)\( T^{16} - \)\(23\!\cdots\!90\)\( T^{17} + \)\(19\!\cdots\!89\)\( T^{18} \)
$31$ \( 1 - \)\(80\!\cdots\!08\)\( T + \)\(20\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!60\)\( T^{4} - \)\(95\!\cdots\!68\)\( T^{5} + \)\(84\!\cdots\!84\)\( T^{6} - \)\(39\!\cdots\!64\)\( T^{7} + \)\(30\!\cdots\!78\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{9} + \)\(11\!\cdots\!58\)\( T^{10} - \)\(56\!\cdots\!44\)\( T^{11} + \)\(45\!\cdots\!04\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{13} + \)\(13\!\cdots\!60\)\( T^{14} - \)\(38\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!93\)\( T^{16} - \)\(32\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!91\)\( T^{18} \)
$37$ \( 1 + \)\(21\!\cdots\!34\)\( T + \)\(25\!\cdots\!73\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!76\)\( T^{4} + \)\(97\!\cdots\!04\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} + \)\(99\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!66\)\( T^{8} + \)\(73\!\cdots\!84\)\( T^{9} + \)\(17\!\cdots\!78\)\( T^{10} + \)\(39\!\cdots\!80\)\( T^{11} + \)\(87\!\cdots\!36\)\( T^{12} + \)\(15\!\cdots\!84\)\( T^{13} + \)\(34\!\cdots\!68\)\( T^{14} + \)\(38\!\cdots\!40\)\( T^{15} + \)\(97\!\cdots\!21\)\( T^{16} + \)\(50\!\cdots\!94\)\( T^{17} + \)\(15\!\cdots\!53\)\( T^{18} \)
$41$ \( 1 + \)\(15\!\cdots\!62\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(34\!\cdots\!64\)\( T^{3} + \)\(23\!\cdots\!60\)\( T^{4} + \)\(36\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(85\!\cdots\!18\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(31\!\cdots\!16\)\( T^{11} + \)\(85\!\cdots\!84\)\( T^{12} + \)\(67\!\cdots\!72\)\( T^{13} + \)\(16\!\cdots\!60\)\( T^{14} + \)\(88\!\cdots\!84\)\( T^{15} + \)\(20\!\cdots\!73\)\( T^{16} + \)\(53\!\cdots\!42\)\( T^{17} + \)\(12\!\cdots\!21\)\( T^{18} \)
$43$ \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(33\!\cdots\!07\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!32\)\( T^{5} + \)\(57\!\cdots\!72\)\( T^{6} - \)\(81\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{9} + \)\(27\!\cdots\!42\)\( T^{10} - \)\(29\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!32\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} - \)\(56\!\cdots\!00\)\( T^{15} + \)\(97\!\cdots\!01\)\( T^{16} - \)\(10\!\cdots\!92\)\( T^{17} + \)\(10\!\cdots\!07\)\( T^{18} \)
$47$ \( 1 - \)\(41\!\cdots\!36\)\( T + \)\(65\!\cdots\!83\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(60\!\cdots\!36\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} - \)\(93\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!86\)\( T^{8} - \)\(93\!\cdots\!76\)\( T^{9} + \)\(35\!\cdots\!18\)\( T^{10} - \)\(62\!\cdots\!60\)\( T^{11} + \)\(19\!\cdots\!16\)\( T^{12} - \)\(27\!\cdots\!96\)\( T^{13} + \)\(74\!\cdots\!48\)\( T^{14} - \)\(71\!\cdots\!80\)\( T^{15} + \)\(16\!\cdots\!61\)\( T^{16} - \)\(85\!\cdots\!56\)\( T^{17} + \)\(16\!\cdots\!23\)\( T^{18} \)
$53$ \( 1 + \)\(53\!\cdots\!38\)\( T + \)\(18\!\cdots\!57\)\( T^{2} + \)\(66\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!16\)\( T^{4} + \)\(47\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} + \)\(22\!\cdots\!20\)\( T^{7} + \)\(43\!\cdots\!46\)\( T^{8} + \)\(81\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} + \)\(22\!\cdots\!80\)\( T^{11} + \)\(31\!\cdots\!56\)\( T^{12} + \)\(46\!\cdots\!68\)\( T^{13} + \)\(51\!\cdots\!12\)\( T^{14} + \)\(64\!\cdots\!40\)\( T^{15} + \)\(55\!\cdots\!81\)\( T^{16} + \)\(50\!\cdots\!98\)\( T^{17} + \)\(30\!\cdots\!77\)\( T^{18} \)
$59$ \( 1 + \)\(17\!\cdots\!20\)\( T + \)\(18\!\cdots\!71\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!96\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(92\!\cdots\!56\)\( T^{6} + \)\(96\!\cdots\!80\)\( T^{7} + \)\(37\!\cdots\!46\)\( T^{8} + \)\(33\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} + \)\(88\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!04\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(41\!\cdots\!04\)\( T^{14} + \)\(21\!\cdots\!40\)\( T^{15} + \)\(42\!\cdots\!69\)\( T^{16} + \)\(12\!\cdots\!20\)\( T^{17} + \)\(21\!\cdots\!79\)\( T^{18} \)
$61$ \( 1 - \)\(99\!\cdots\!98\)\( T + \)\(10\!\cdots\!13\)\( T^{2} - \)\(67\!\cdots\!96\)\( T^{3} + \)\(41\!\cdots\!60\)\( T^{4} - \)\(20\!\cdots\!68\)\( T^{5} + \)\(94\!\cdots\!04\)\( T^{6} - \)\(37\!\cdots\!04\)\( T^{7} + \)\(14\!\cdots\!98\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(15\!\cdots\!58\)\( T^{10} - \)\(43\!\cdots\!64\)\( T^{11} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(27\!\cdots\!08\)\( T^{13} + \)\(59\!\cdots\!60\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(17\!\cdots\!33\)\( T^{16} - \)\(17\!\cdots\!78\)\( T^{17} + \)\(18\!\cdots\!81\)\( T^{18} \)
$67$ \( 1 - \)\(67\!\cdots\!76\)\( T + \)\(17\!\cdots\!03\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{5} + \)\(72\!\cdots\!28\)\( T^{6} - \)\(36\!\cdots\!60\)\( T^{7} + \)\(25\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(61\!\cdots\!98\)\( T^{10} - \)\(21\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!76\)\( T^{12} - \)\(29\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!08\)\( T^{14} - \)\(23\!\cdots\!20\)\( T^{15} + \)\(93\!\cdots\!41\)\( T^{16} - \)\(88\!\cdots\!56\)\( T^{17} + \)\(32\!\cdots\!63\)\( T^{18} \)
$71$ \( 1 - \)\(55\!\cdots\!28\)\( T + \)\(30\!\cdots\!23\)\( T^{2} - \)\(98\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!60\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} + \)\(90\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(37\!\cdots\!04\)\( T^{11} + \)\(16\!\cdots\!24\)\( T^{12} - \)\(49\!\cdots\!48\)\( T^{13} + \)\(14\!\cdots\!60\)\( T^{14} - \)\(31\!\cdots\!16\)\( T^{15} + \)\(11\!\cdots\!13\)\( T^{16} - \)\(26\!\cdots\!88\)\( T^{17} + \)\(57\!\cdots\!11\)\( T^{18} \)
$73$ \( 1 - \)\(13\!\cdots\!02\)\( T + \)\(21\!\cdots\!57\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!56\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!12\)\( T^{6} - \)\(52\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!26\)\( T^{8} - \)\(14\!\cdots\!72\)\( T^{9} + \)\(69\!\cdots\!22\)\( T^{10} - \)\(29\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} - \)\(41\!\cdots\!32\)\( T^{13} + \)\(13\!\cdots\!92\)\( T^{14} - \)\(35\!\cdots\!80\)\( T^{15} + \)\(90\!\cdots\!41\)\( T^{16} - \)\(13\!\cdots\!22\)\( T^{17} + \)\(23\!\cdots\!17\)\( T^{18} \)
$79$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(21\!\cdots\!31\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!16\)\( T^{4} - \)\(82\!\cdots\!20\)\( T^{5} + \)\(42\!\cdots\!36\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(77\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!74\)\( T^{10} - \)\(23\!\cdots\!60\)\( T^{11} + \)\(58\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{13} + \)\(23\!\cdots\!84\)\( T^{14} - \)\(35\!\cdots\!80\)\( T^{15} + \)\(45\!\cdots\!89\)\( T^{16} - \)\(41\!\cdots\!40\)\( T^{17} + \)\(25\!\cdots\!39\)\( T^{18} \)
$83$ \( 1 - \)\(10\!\cdots\!72\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(92\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(38\!\cdots\!92\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{9} + \)\(24\!\cdots\!82\)\( T^{10} - \)\(95\!\cdots\!80\)\( T^{11} + \)\(40\!\cdots\!36\)\( T^{12} - \)\(13\!\cdots\!32\)\( T^{13} + \)\(46\!\cdots\!32\)\( T^{14} - \)\(11\!\cdots\!40\)\( T^{15} + \)\(35\!\cdots\!21\)\( T^{16} - \)\(56\!\cdots\!32\)\( T^{17} + \)\(11\!\cdots\!87\)\( T^{18} \)
$89$ \( 1 + \)\(88\!\cdots\!30\)\( T + \)\(55\!\cdots\!61\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!76\)\( T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(59\!\cdots\!06\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(22\!\cdots\!74\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{11} + \)\(56\!\cdots\!64\)\( T^{12} + \)\(74\!\cdots\!40\)\( T^{13} + \)\(86\!\cdots\!24\)\( T^{14} + \)\(84\!\cdots\!60\)\( T^{15} + \)\(68\!\cdots\!49\)\( T^{16} + \)\(42\!\cdots\!30\)\( T^{17} + \)\(18\!\cdots\!69\)\( T^{18} \)
$97$ \( 1 - \)\(18\!\cdots\!86\)\( T + \)\(30\!\cdots\!33\)\( T^{2} - \)\(47\!\cdots\!20\)\( T^{3} + \)\(42\!\cdots\!36\)\( T^{4} - \)\(55\!\cdots\!36\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} - \)\(39\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!86\)\( T^{8} - \)\(18\!\cdots\!76\)\( T^{9} + \)\(76\!\cdots\!18\)\( T^{10} - \)\(57\!\cdots\!60\)\( T^{11} + \)\(20\!\cdots\!16\)\( T^{12} - \)\(12\!\cdots\!96\)\( T^{13} + \)\(35\!\cdots\!48\)\( T^{14} - \)\(15\!\cdots\!80\)\( T^{15} + \)\(38\!\cdots\!61\)\( T^{16} - \)\(86\!\cdots\!06\)\( T^{17} + \)\(18\!\cdots\!73\)\( T^{18} \)
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