Properties

Label 1.114.a
Level 1
Weight 114
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: \( k \) \(=\) \( 114 \)
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 \(M_{114}(\Gamma_0(1))\).

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

Trace form

\( 9q + 4973724357874032q^{2} - 1047822666262699970761104924q^{3} + 57140122287714158706660944854495488q^{4} - 3116148576934475303759287082847998426250q^{5} - 8679197271676415971774420800017032303605312q^{6} - 175592521264923853936268390919613238158926276408q^{7} - 1152754689437068715781677022006928573086863264747520q^{8} + 551049940125998155825143115188713609620516409643418677q^{9} + O(q^{10}) \) \( 9q + 4973724357874032q^{2} - 1047822666262699970761104924q^{3} + 57140122287714158706660944854495488q^{4} - 3116148576934475303759287082847998426250q^{5} - 8679197271676415971774420800017032303605312q^{6} - 175592521264923853936268390919613238158926276408q^{7} - 1152754689437068715781677022006928573086863264747520q^{8} + 551049940125998155825143115188713609620516409643418677q^{9} - 153264347969576737314363653067831776074945972002322140000q^{10} + 124303845165751869554468963179928310347446325358753066991788q^{11} - 4766393536299397973462835582969587848650683059861216359840768q^{12} + 118066950692074908165278563837084569941718111528864100316810046q^{13} + 23370095428037254282484566813709955199818716752021690625741210496q^{14} - 801369990352485680694849420185298838396866837392656002948893145000q^{15} + 109009769902909383382120055717425408918319663897461125806267782201344q^{16} + 1031208472354491758575110721181756882175051278670426497304478795027362q^{17} + 21853761778376511004199260652941739695840479087427332697148033133176496q^{18} - 2054770397164032300993860357285529834117448193071848654506031346813189260q^{19} - 128965280615606341221181395517996603639092921677759187360515198160536640000q^{20} - 2051981423257402292087521454906945230917134727108225442350522950612144271072q^{21} - 31233703450481121590220064319836416371256690633810598891839492722859409642176q^{22} - 167806258064059379045373533912927060845638327737895149615950186261514952182184q^{23} + 837647327683589821169163577629460913691878109446800095361456758053616583557120q^{24} + 36110808046585671891862329281725549351642982282250931138010564349682541171484375q^{25} + 273284118611845700597932385482119882778869175349924218776704957190525454972143648q^{26} - 1293409658858973603349494974301711616276027502548100865754773226934344693652580120q^{27} - 12449742560521692406151296738508394024549837700960492852336353346600639157797779456q^{28} + 16393803312375561825707281499219064392567318107169713827635118635692166495008904910q^{29} + 1444554724925215896891120206778115916774229741452396468041800381952830143736681360000q^{30} + 1314403569804018815618492241812394006579677928452900483535632508652944766037998410528q^{31} - 21448902248775261233416755741924844958908937761160756654268584846741870841621352808448q^{32} - 211847819746714833550323551974005344232426451287015334370441057211901425852772059609168q^{33} + 319367478280300365454292314290661290493902375851140946818660251269274780967076833561056q^{34} + 3439596420817236254046899424410940097643944233598368039111722250253904941775326256910000q^{35} + 2704504833410501582895107271001450900815653199167021310209481973200374905160451231300864q^{36} - 59867320490204404665587120670290466524223984034118562335350833430431530376567774885723498q^{37} - 82453523727277361450274687052865709436318494611960850831592535368825105943678502733442880q^{38} + 871232822588974033828162396761465282346428529310140460376676860192325312944522604565747064q^{39} + 6827166757827354756060064429590450961456578801730020944860911233033409499443258590438400000q^{40} - 34550149215769251780094496181290849916810545798144119670305052821692690196704735942577072902q^{41} - 106065652173490837329551409984862217118049589651335199831107089722564993162529690865908896256q^{42} + 207231754793163377176542763271534728167453514398489730764515211376339059665627233099893879756q^{43} + 2312703333865421082227940066456566870684549523781445620538452769155749838714861346004252527616q^{44} + 256198961065429853532686454584243663537866293764106258783025404071252757628746328749329658750q^{45} - 28138310036672836977617659816100322616155099852864640775866866896982035999177616032492961297792q^{46} - 49338615100567586008533951288135845393852650707485360531368204467394665532910688983139358306128q^{47} + 244064292546444123908052536984208158793401136481260186300744503278739285380247288265778682658816q^{48} + 752433840599750193271015544145910422089938845099283047668758541087980882087499887025758157789313q^{49} - 1756648703190643407679915187540746659445068152182448123783254139837888034842840604352884543750000q^{50} - 4186453060070964308543321243867301742071184523159619262981910395937624269639977914668070815961592q^{51} - 299921527358232778097628046170322710643014254923572622961503877233695051976123243837021461385728q^{52} + 45439534345102239290843163912544095765806982832307502313849525819073728936151813167470772365925926q^{53} + 38680516880674041302480509047884889106404603448054948500904779898349149372610056782201281117946240q^{54} - 432576867358466335617986215621945887405045763707273014611343529132044150112466209509767824829515000q^{55} - 159465105485502254328469891445580976480576866382179762995668936857815143816140668281747609575260160q^{56} + 1990014681911564315630720992224179214491287391034406566984522978256103716439101105615294767535358160q^{57} + 4618158171946557697639499111957754813666889826993148515766803610504911213619043847144587412863790880q^{58} - 7994556888742389963754891914386272086617532494092324481666622701281195647537712229818800059078242980q^{59} - 51547668272403014021235031153713789291688796805654426950141340288411508848106756189101821863952640000q^{60} - 20010787184525995584720289117304568167948599723976550204852777948468547299349907237396329738847967762q^{61} + 292035427959193157957119307134870542698832076716753725989233844815905689373991776544712306185499282944q^{62} + 1084750421819290453616791018398299860321649524662704337157840276934678472457615663242574941575258587176q^{63} - 2173182954777058552426966649041133062639998025794768537871770595713126062249675532560302339710750031872q^{64} - 6050492601584073293182223610830486690452314196108303412288731430157341922477676622337759152417832607500q^{65} - 2811384209515387361103707894564511021023407746835224054361358951726305534503884893008505470528471177984q^{66} + 24573799368134891916564011470928671317077382485972351268274803174871528123414635568434858147580241074212q^{67} + 63690775145054712155234191070401247673413728435364958188409307863633237295553890585548320729107243037184q^{68} - 24003285462463676531332362285165839175525050544235078632804235222118832623175439122841457438544940522656q^{69} - 863169709419813836575212169649584339603968815833753646438175375944308193587969610263821675032671814880000q^{70} + 91658679156386468719245489455910792073607873475232020252166152112986794966382882952820365384103518048008q^{71} + 873882756027094209986251517892994706349418254125301044053004437567652817562627690659257226446504466493440q^{72} + 541694808507779980546116004112644787310665887489575442433845489271276265038276918082847569639076524144666q^{73} - 9378706212502591696314586127924566164346165127102664760442762960199173362938872227498081812178947973840224q^{74} - 29348285812521570443266953567594350343794068770628558899049305240540166160078653563092442717138745889062500q^{75} - 60564820012723729139203913493989896690950347961424832880261930310236667425867853870924430511072293881605120q^{76} + 1068940833620544673782151517761484277047755630607342022951044010736792743275783369575718784146165485947744q^{77} - 286969622560772923408733006352532687901626912195145659167094759389964370273396505496077341678419418050144128q^{78} - 596276255135400928959400589889218117555952939149585650758445593510201386502571903994653015956197756762453040q^{79} - 2420836681564557797101060397107174542916086634827557275495079675925079838120631425602247114230349809336320000q^{80} - 3030606522373897290470630089608148759339991046893156580144516813393469703984963913807719049058860501442725951q^{81} - 6936458118821589036700513874923262497738432667781968228952468701667638701536840125370171023576003636856609696q^{82} - 11004287102773590749349691331049602909161139219297045772286679604418555775574362523281340925221936593079346764q^{83} - 57855325032373861757970130314922242695268575192193159587088758458166711077597077741056893950844377121225826304q^{84} - 64354197775882201370863569029422599518759619010400232504975477030343845397959668657595763753959731762134552500q^{85} - 210819753145942403520859363592361681829074054483701088761378978430433240411156075153802810720648226504181908672q^{86} - 154008309759204716022098762324487951847505839469072421264586325306012373529922474900817730928920322507018969160q^{87} - 609999238537345062842609214560983373867364799260253404096252032888004927138583268244389069934087697492178288640q^{88} - 654568028165810774739876366172819409302687924287909841936841472021672950876376689746932135695962343969713322070q^{89} - 2734380769439054007162335818387627599332566037560181838266985769883367234594498859125630424774118397399333420000q^{90} - 2323627076132845640660102608242767489368371452521120033992540027246745766005131800187294415915449799674836938512q^{91} - 4256037231999230829372236189816770749190987971335840398392358871486563229805945773322404171318956972524277721088q^{92} + 154296866852621529872615733195411660158426537898618117528867810834868770230718469048319703695743778039088292992q^{93} + 1827384388290191849012043193156273334448650299359011463985703887856827570148696897626054924182121847958990279936q^{94} - 4096663912916778389991331111354003035604483111168308740332618943380051360421354423075027168404452962459195325000q^{95} + 8457146062137046129332319587633700387597377410609689871416901579530268721609757152896450994557102813871989587968q^{96} + 36807391546013154433957869141705553075329468371786108764367006729276625672848917612484272062474831101244619876722q^{97} + 225562248663489353389248034966442578767140826293081704735030344217680122252159170139356951685061050587472992483824q^{98} + 321571830004348537474522583988170945954403617464147621640077047560490989900478751999197984675098698758537853855164q^{99} + O(q^{100}) \)

Decomposition of \(S_{114}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.114.a.a \(9\) \(80.863\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(49\!\cdots\!32\) \(-1\!\cdots\!24\) \(-3\!\cdots\!50\) \(-1\!\cdots\!08\) \(+\) \(q+(552636039763781+\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4973724357874032 T + \)\(18\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!36\)\( T^{4} + \)\(77\!\cdots\!08\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} + \)\(14\!\cdots\!40\)\( T^{7} + \)\(43\!\cdots\!36\)\( T^{8} + \)\(12\!\cdots\!68\)\( T^{9} + \)\(45\!\cdots\!12\)\( T^{10} + \)\(15\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!16\)\( T^{12} + \)\(90\!\cdots\!68\)\( T^{13} + \)\(34\!\cdots\!52\)\( T^{14} + \)\(39\!\cdots\!80\)\( T^{15} + \)\(23\!\cdots\!36\)\( T^{16} - \)\(67\!\cdots\!12\)\( T^{17} + \)\(14\!\cdots\!72\)\( T^{18} \)
$3$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(39\!\cdots\!03\)\( T^{2} + \)\(39\!\cdots\!20\)\( T^{3} + \)\(88\!\cdots\!56\)\( T^{4} + \)\(78\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(14\!\cdots\!26\)\( T^{8} + \)\(10\!\cdots\!04\)\( T^{9} + \)\(11\!\cdots\!98\)\( T^{10} + \)\(71\!\cdots\!60\)\( T^{11} + \)\(73\!\cdots\!76\)\( T^{12} + \)\(35\!\cdots\!44\)\( T^{13} + \)\(32\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} + \)\(21\!\cdots\!44\)\( T^{17} + \)\(17\!\cdots\!63\)\( T^{18} \)
$5$ \( 1 + \)\(31\!\cdots\!50\)\( T + \)\(30\!\cdots\!25\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(77\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!50\)\( T^{8} + \)\(22\!\cdots\!00\)\( T^{9} + \)\(86\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(68\!\cdots\!00\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} + \)\(75\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} + \)\(23\!\cdots\!50\)\( T^{17} + \)\(71\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 + \)\(17\!\cdots\!08\)\( T + \)\(10\!\cdots\!07\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!96\)\( T^{4} + \)\(39\!\cdots\!68\)\( T^{5} + \)\(36\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!06\)\( T^{8} + \)\(72\!\cdots\!48\)\( T^{9} + \)\(46\!\cdots\!42\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!96\)\( T^{12} + \)\(38\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!72\)\( T^{14} + \)\(49\!\cdots\!00\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} + \)\(16\!\cdots\!08\)\( T^{17} + \)\(29\!\cdots\!07\)\( T^{18} \)
$11$ \( 1 - \)\(12\!\cdots\!88\)\( T + \)\(17\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(60\!\cdots\!68\)\( T^{5} + \)\(53\!\cdots\!24\)\( T^{6} - \)\(17\!\cdots\!44\)\( T^{7} + \)\(16\!\cdots\!18\)\( T^{8} - \)\(48\!\cdots\!20\)\( T^{9} + \)\(80\!\cdots\!58\)\( T^{10} - \)\(40\!\cdots\!84\)\( T^{11} + \)\(58\!\cdots\!84\)\( T^{12} - \)\(31\!\cdots\!28\)\( T^{13} + \)\(28\!\cdots\!60\)\( T^{14} - \)\(13\!\cdots\!16\)\( T^{15} + \)\(97\!\cdots\!73\)\( T^{16} - \)\(32\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!71\)\( T^{18} \)
$13$ \( 1 - \)\(11\!\cdots\!46\)\( T + \)\(41\!\cdots\!13\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(89\!\cdots\!16\)\( T^{4} - \)\(90\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} - \)\(97\!\cdots\!56\)\( T^{9} + \)\(96\!\cdots\!38\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(53\!\cdots\!56\)\( T^{12} - \)\(28\!\cdots\!36\)\( T^{13} + \)\(21\!\cdots\!88\)\( T^{14} - \)\(88\!\cdots\!40\)\( T^{15} + \)\(55\!\cdots\!81\)\( T^{16} - \)\(11\!\cdots\!06\)\( T^{17} + \)\(75\!\cdots\!33\)\( T^{18} \)
$17$ \( 1 - \)\(10\!\cdots\!62\)\( T + \)\(46\!\cdots\!57\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(17\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!52\)\( T^{6} + \)\(50\!\cdots\!20\)\( T^{7} + \)\(32\!\cdots\!46\)\( T^{8} + \)\(82\!\cdots\!08\)\( T^{9} + \)\(35\!\cdots\!02\)\( T^{10} + \)\(60\!\cdots\!80\)\( T^{11} + \)\(30\!\cdots\!56\)\( T^{12} + \)\(25\!\cdots\!68\)\( T^{13} + \)\(19\!\cdots\!12\)\( T^{14} - \)\(52\!\cdots\!60\)\( T^{15} + \)\(90\!\cdots\!81\)\( T^{16} - \)\(21\!\cdots\!02\)\( T^{17} + \)\(23\!\cdots\!77\)\( T^{18} \)
$19$ \( 1 + \)\(20\!\cdots\!60\)\( T + \)\(24\!\cdots\!31\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!16\)\( T^{4} + \)\(44\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!36\)\( T^{6} + \)\(26\!\cdots\!40\)\( T^{7} + \)\(86\!\cdots\!86\)\( T^{8} + \)\(10\!\cdots\!00\)\( T^{9} + \)\(27\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!44\)\( T^{12} + \)\(44\!\cdots\!80\)\( T^{13} + \)\(87\!\cdots\!84\)\( T^{14} + \)\(44\!\cdots\!20\)\( T^{15} + \)\(77\!\cdots\!89\)\( T^{16} + \)\(20\!\cdots\!60\)\( T^{17} + \)\(31\!\cdots\!39\)\( T^{18} \)
$23$ \( 1 + \)\(16\!\cdots\!84\)\( T + \)\(57\!\cdots\!23\)\( T^{2} + \)\(79\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} + \)\(17\!\cdots\!04\)\( T^{5} + \)\(24\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!66\)\( T^{8} + \)\(21\!\cdots\!84\)\( T^{9} + \)\(20\!\cdots\!78\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!36\)\( T^{12} + \)\(56\!\cdots\!84\)\( T^{13} + \)\(36\!\cdots\!68\)\( T^{14} + \)\(14\!\cdots\!40\)\( T^{15} + \)\(76\!\cdots\!21\)\( T^{16} + \)\(16\!\cdots\!44\)\( T^{17} + \)\(75\!\cdots\!03\)\( T^{18} \)
$29$ \( 1 - \)\(16\!\cdots\!10\)\( T + \)\(70\!\cdots\!01\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!56\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(78\!\cdots\!96\)\( T^{6} - \)\(41\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!66\)\( T^{8} - \)\(85\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!74\)\( T^{10} - \)\(13\!\cdots\!40\)\( T^{11} + \)\(44\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!80\)\( T^{13} + \)\(49\!\cdots\!44\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(40\!\cdots\!29\)\( T^{16} - \)\(16\!\cdots\!10\)\( T^{17} + \)\(18\!\cdots\!09\)\( T^{18} \)
$31$ \( 1 - \)\(13\!\cdots\!28\)\( T + \)\(20\!\cdots\!23\)\( T^{2} - \)\(19\!\cdots\!76\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(14\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(70\!\cdots\!84\)\( T^{7} + \)\(44\!\cdots\!38\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} - \)\(78\!\cdots\!04\)\( T^{11} + \)\(39\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!48\)\( T^{13} + \)\(75\!\cdots\!60\)\( T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(93\!\cdots\!13\)\( T^{16} - \)\(20\!\cdots\!88\)\( T^{17} + \)\(51\!\cdots\!11\)\( T^{18} \)
$37$ \( 1 + \)\(59\!\cdots\!98\)\( T + \)\(57\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!56\)\( T^{4} + \)\(89\!\cdots\!28\)\( T^{5} + \)\(46\!\cdots\!12\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(92\!\cdots\!26\)\( T^{8} + \)\(33\!\cdots\!28\)\( T^{9} + \)\(14\!\cdots\!22\)\( T^{10} + \)\(48\!\cdots\!40\)\( T^{11} + \)\(19\!\cdots\!76\)\( T^{12} + \)\(60\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!20\)\( T^{15} + \)\(16\!\cdots\!41\)\( T^{16} + \)\(27\!\cdots\!78\)\( T^{17} + \)\(72\!\cdots\!17\)\( T^{18} \)
$41$ \( 1 + \)\(34\!\cdots\!02\)\( T + \)\(12\!\cdots\!13\)\( T^{2} + \)\(27\!\cdots\!04\)\( T^{3} + \)\(60\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!04\)\( T^{6} + \)\(28\!\cdots\!96\)\( T^{7} + \)\(43\!\cdots\!98\)\( T^{8} + \)\(57\!\cdots\!80\)\( T^{9} + \)\(76\!\cdots\!58\)\( T^{10} + \)\(87\!\cdots\!36\)\( T^{11} + \)\(10\!\cdots\!44\)\( T^{12} + \)\(99\!\cdots\!92\)\( T^{13} + \)\(10\!\cdots\!60\)\( T^{14} + \)\(79\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!33\)\( T^{16} + \)\(31\!\cdots\!22\)\( T^{17} + \)\(15\!\cdots\!81\)\( T^{18} \)
$43$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(13\!\cdots\!43\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!96\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!28\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!06\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{9} + \)\(77\!\cdots\!58\)\( T^{10} - \)\(30\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!96\)\( T^{12} - \)\(17\!\cdots\!76\)\( T^{13} + \)\(80\!\cdots\!28\)\( T^{14} - \)\(52\!\cdots\!00\)\( T^{15} + \)\(16\!\cdots\!01\)\( T^{16} - \)\(93\!\cdots\!56\)\( T^{17} + \)\(17\!\cdots\!43\)\( T^{18} \)
$47$ \( 1 + \)\(49\!\cdots\!28\)\( T + \)\(76\!\cdots\!07\)\( T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!76\)\( T^{4} + \)\(80\!\cdots\!48\)\( T^{5} + \)\(48\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!66\)\( T^{8} + \)\(14\!\cdots\!88\)\( T^{9} + \)\(55\!\cdots\!82\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!36\)\( T^{12} + \)\(49\!\cdots\!68\)\( T^{13} + \)\(13\!\cdots\!32\)\( T^{14} + \)\(14\!\cdots\!60\)\( T^{15} + \)\(32\!\cdots\!21\)\( T^{16} + \)\(18\!\cdots\!68\)\( T^{17} + \)\(33\!\cdots\!87\)\( T^{18} \)
$53$ \( 1 - \)\(45\!\cdots\!26\)\( T + \)\(58\!\cdots\!53\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!56\)\( T^{4} - \)\(46\!\cdots\!16\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} - \)\(59\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!26\)\( T^{8} - \)\(50\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!98\)\( T^{10} - \)\(29\!\cdots\!40\)\( T^{11} + \)\(77\!\cdots\!76\)\( T^{12} - \)\(10\!\cdots\!56\)\( T^{13} + \)\(24\!\cdots\!08\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(46\!\cdots\!41\)\( T^{16} - \)\(25\!\cdots\!06\)\( T^{17} + \)\(38\!\cdots\!13\)\( T^{18} \)
$59$ \( 1 + \)\(79\!\cdots\!80\)\( T + \)\(66\!\cdots\!11\)\( T^{2} + \)\(55\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!76\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(50\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!20\)\( T^{7} + \)\(84\!\cdots\!06\)\( T^{8} + \)\(50\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} + \)\(56\!\cdots\!20\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(46\!\cdots\!40\)\( T^{13} + \)\(76\!\cdots\!24\)\( T^{14} + \)\(24\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!49\)\( T^{16} + \)\(56\!\cdots\!80\)\( T^{17} + \)\(90\!\cdots\!19\)\( T^{18} \)
$61$ \( 1 + \)\(20\!\cdots\!62\)\( T + \)\(32\!\cdots\!93\)\( T^{2} + \)\(41\!\cdots\!64\)\( T^{3} + \)\(51\!\cdots\!60\)\( T^{4} + \)\(42\!\cdots\!32\)\( T^{5} + \)\(52\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!56\)\( T^{7} + \)\(39\!\cdots\!18\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{11} + \)\(88\!\cdots\!84\)\( T^{12} + \)\(39\!\cdots\!72\)\( T^{13} + \)\(26\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!84\)\( T^{15} + \)\(50\!\cdots\!73\)\( T^{16} + \)\(17\!\cdots\!42\)\( T^{17} + \)\(47\!\cdots\!21\)\( T^{18} \)
$67$ \( 1 - \)\(24\!\cdots\!12\)\( T + \)\(82\!\cdots\!07\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!16\)\( T^{4} - \)\(62\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!52\)\( T^{6} - \)\(19\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!46\)\( T^{8} - \)\(50\!\cdots\!92\)\( T^{9} + \)\(69\!\cdots\!02\)\( T^{10} - \)\(96\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(15\!\cdots\!32\)\( T^{13} + \)\(18\!\cdots\!12\)\( T^{14} - \)\(20\!\cdots\!60\)\( T^{15} + \)\(21\!\cdots\!81\)\( T^{16} - \)\(14\!\cdots\!52\)\( T^{17} + \)\(13\!\cdots\!27\)\( T^{18} \)
$71$ \( 1 - \)\(91\!\cdots\!08\)\( T + \)\(72\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(25\!\cdots\!60\)\( T^{4} - \)\(73\!\cdots\!68\)\( T^{5} + \)\(60\!\cdots\!84\)\( T^{6} - \)\(22\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{9} + \)\(17\!\cdots\!58\)\( T^{10} - \)\(55\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!04\)\( T^{12} - \)\(43\!\cdots\!88\)\( T^{13} + \)\(23\!\cdots\!60\)\( T^{14} - \)\(19\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!93\)\( T^{16} - \)\(31\!\cdots\!48\)\( T^{17} + \)\(53\!\cdots\!91\)\( T^{18} \)
$73$ \( 1 - \)\(54\!\cdots\!66\)\( T + \)\(14\!\cdots\!73\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!76\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(76\!\cdots\!28\)\( T^{6} - \)\(78\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!66\)\( T^{8} - \)\(32\!\cdots\!16\)\( T^{9} + \)\(12\!\cdots\!78\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!36\)\( T^{12} - \)\(21\!\cdots\!16\)\( T^{13} + \)\(75\!\cdots\!68\)\( T^{14} - \)\(29\!\cdots\!60\)\( T^{15} + \)\(10\!\cdots\!21\)\( T^{16} - \)\(15\!\cdots\!06\)\( T^{17} + \)\(99\!\cdots\!53\)\( T^{18} \)
$79$ \( 1 + \)\(59\!\cdots\!40\)\( T + \)\(26\!\cdots\!51\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!56\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(78\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!66\)\( T^{8} + \)\(30\!\cdots\!00\)\( T^{9} + \)\(52\!\cdots\!74\)\( T^{10} + \)\(91\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!24\)\( T^{12} + \)\(23\!\cdots\!20\)\( T^{13} + \)\(30\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!80\)\( T^{15} + \)\(28\!\cdots\!29\)\( T^{16} + \)\(16\!\cdots\!40\)\( T^{17} + \)\(77\!\cdots\!59\)\( T^{18} \)
$83$ \( 1 + \)\(11\!\cdots\!64\)\( T + \)\(81\!\cdots\!83\)\( T^{2} + \)\(42\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!36\)\( T^{4} + \)\(56\!\cdots\!64\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + \)\(34\!\cdots\!60\)\( T^{7} + \)\(70\!\cdots\!86\)\( T^{8} + \)\(15\!\cdots\!24\)\( T^{9} + \)\(50\!\cdots\!18\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(56\!\cdots\!16\)\( T^{12} + \)\(15\!\cdots\!04\)\( T^{13} + \)\(33\!\cdots\!48\)\( T^{14} + \)\(57\!\cdots\!20\)\( T^{15} + \)\(80\!\cdots\!61\)\( T^{16} + \)\(77\!\cdots\!44\)\( T^{17} + \)\(50\!\cdots\!23\)\( T^{18} \)
$89$ \( 1 + \)\(65\!\cdots\!70\)\( T + \)\(29\!\cdots\!21\)\( T^{2} + \)\(93\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!96\)\( T^{4} + \)\(56\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(32\!\cdots\!46\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(61\!\cdots\!74\)\( T^{10} + \)\(73\!\cdots\!80\)\( T^{11} + \)\(78\!\cdots\!04\)\( T^{12} + \)\(75\!\cdots\!60\)\( T^{13} + \)\(63\!\cdots\!04\)\( T^{14} + \)\(45\!\cdots\!40\)\( T^{15} + \)\(26\!\cdots\!69\)\( T^{16} + \)\(11\!\cdots\!70\)\( T^{17} + \)\(33\!\cdots\!29\)\( T^{18} \)
$97$ \( 1 - \)\(36\!\cdots\!22\)\( T + \)\(17\!\cdots\!57\)\( T^{2} - \)\(47\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!76\)\( T^{4} - \)\(28\!\cdots\!52\)\( T^{5} + \)\(64\!\cdots\!92\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} - \)\(41\!\cdots\!12\)\( T^{9} + \)\(77\!\cdots\!82\)\( T^{10} - \)\(12\!\cdots\!80\)\( T^{11} + \)\(21\!\cdots\!36\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{13} + \)\(44\!\cdots\!32\)\( T^{14} - \)\(50\!\cdots\!40\)\( T^{15} + \)\(59\!\cdots\!21\)\( T^{16} - \)\(40\!\cdots\!82\)\( T^{17} + \)\(35\!\cdots\!37\)\( T^{18} \)
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