Properties

Label 3.66.a
Level 3
Weight 66
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 2
Sturm bound 22
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{66}(\Gamma_0(3))\).

Total New Old
Modular forms 23 11 12
Cusp forms 21 11 10
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.

\(3\)Dim.
\(+\)\(5\)
\(-\)\(6\)

Trace form

\( 11q + 3624451998q^{2} + 1853020188851841q^{3} + 182297552527310043284q^{4} - 51169986413727134695494q^{5} + 16301970916583221148237766q^{6} + 7045353328705962403513562128q^{7} + 541385166531412015722258480840q^{8} + 37770522023217637331236339982091q^{9} + O(q^{10}) \) \( 11q + 3624451998q^{2} + 1853020188851841q^{3} + 182297552527310043284q^{4} - 51169986413727134695494q^{5} + 16301970916583221148237766q^{6} + 7045353328705962403513562128q^{7} + 541385166531412015722258480840q^{8} + 37770522023217637331236339982091q^{9} + 619438979330879307901236008580564q^{10} + 102779711445945907142926706248476q^{11} + 712037078029045459308073879334772q^{12} + 2236803413364017652432096558025062154q^{13} - 73111611730347705398280846705393926640q^{14} + 225109105168650616306993446179603443854q^{15} + 8817209428494398737767550388646103253264q^{16} - 27609328203731450230952135670799573996362q^{17} + 12445222182959469819458085478027000833438q^{18} + 1065990310826627973682259321160159414012964q^{19} + 246794839789344647728935680687316114866424q^{20} - 8171282585963302364950263465218503870918320q^{21} - 28668791923198331709444352896329603161079544q^{22} - 182492217093463660457127832766047149369557976q^{23} + 1638080104783332686130517758253203702575980872q^{24} + 6033734467104240936011911951865076088470091061q^{25} + 31812883785035805790949474374544042440107416292q^{26} + 6362685441135942358474828762538534230890216321q^{27} + 556953666945918751121810394028515640552302262816q^{28} + 837406052539935135744172735690636063803312665778q^{29} + 3593435577263205095009874222016368333438517446276q^{30} + 6204942990396592055727778050910347502994230809160q^{31} + 52223294277213143793777144293463538847496097862688q^{32} - 8791769474212247803686864706545373515889094737548q^{33} - 91042367601556570993107311981868509595349326151332q^{34} - 777717848664297055678304731493335836504718134510240q^{35} + 625952156591948913784578682387396529311104090438804q^{36} - 3141739618754499413856358770196284438399889685346302q^{37} - 6036614145839925099522178576851323348170126849961160q^{38} - 582888262261790931525868963652868241702547875866978q^{39} + 55889769356959167054720226735842263678164045891513008q^{40} + 12739729951808557040333109565287405814454862195173790q^{41} + 49759175555747646731387174274294407611543691714547984q^{42} + 488348829152958981907302960450395602350600910421253724q^{43} + 1353835521295621066666731060932648717843803788705961584q^{44} - 175701554433402548164666079535853072383364345154399814q^{45} - 1960542575626780874460909785920341300847951270875860400q^{46} - 7121352319512304228172058661585475634385416202818544032q^{47} - 2167508118662845627875366822767497672417459021495793904q^{48} - 14427664230896531569596301522586539803698431635048906957q^{49} + 26585117828425717813350370495953938065014718041165430434q^{50} + 1872571932086963348975911047525363531370172169946045218q^{51} + 224886220990849588873263317315551932575854906960392734488q^{52} - 73989886727661987770599374209738466246244722776741359206q^{53} + 55975813775150906157815590158898102007884377461149986246q^{54} + 848690418695216601906395846388209087315543992709228756616q^{55} - 3295458519956774021732553139133002042499380606708800323520q^{56} + 101163866345173506390137105609484277569000222432861398220q^{57} - 6915715744259737287549218415974213227802833552600845354780q^{58} - 7639459059287158667102529626062912052811345721535049084804q^{59} + 9323693921270020616909687748877565839309632216825322581816q^{60} + 20784670286902374573417399043999449740619353745336993639658q^{61} + 2914477655747437816778316389121165029970249184821991512096q^{62} + 24191515733021658449945058268692810547135628632437012349968q^{63} + 247086266702509679164429313956979051704551077507033417248832q^{64} + 613646632848692467005291059549724741527365282614998711628q^{65} + 457407027308075804282008685417539963697083291754632659195816q^{66} - 12897879862433232313348670441517039750540926988445462820732q^{67} - 3885160235788747929189818670114307838629323071300103922811544q^{68} + 515562382253181099403631333636444425682719138024450821171960q^{69} - 353452742508352350629180180209243686856254792372732375699360q^{70} + 2545358340513829895032036249420402481032689044213009966226072q^{71} + 1858945486865276880358111291017384807342443001621072587876040q^{72} + 3634977497989942338742619849045326181579380691267081786286574q^{73} + 149583309261260981035291220208421583145787523307449684577780q^{74} + 10696723504727567017466932174990112704208756054655428010059199q^{75} + 44361120066033071202297700264247326844962699790882280597066256q^{76} + 2787248977771439879133771326299195120236233874908828948231616q^{77} - 106265191857555529652912639480974518139485750547126143003068268q^{78} - 108655335736204494196797419786154330065071415921774578973266280q^{79} + 179560660082718156951752908874134790268568122556519286027263584q^{80} + 129692030355124414886729601475537705322460327515034252200066571q^{81} - 168419268813352074152696966965808331803790274705328080277502804q^{82} + 666627046677300801346778718286597552251089160284541426205968804q^{83} - 331674076073398934614361651600449683185481641659738305163284960q^{84} + 786799842934726755368361154177969111250132467557215049805840692q^{85} + 3968876794345165854405432735391216331676546435218946806435253576q^{86} - 867601494999151460698344619816437149071908525696131488359152970q^{87} - 2077235424923546695469227475479972617537370996634929957934965920q^{88} - 4536175169951302186672349499218286573665856269622401182182043266q^{89} + 2126957600986948340851269773340826280278381785499943528917334484q^{90} - 314695945056081124071893638454701863459317027331441499799145760q^{91} - 19275072315136785302212715588188105426215864541962442912799580192q^{92} - 1775852494856519753498069159890217952992787969833659295644550088q^{93} + 50765461892948731039539028852525696414382626889183735875700898912q^{94} + 44116504798438205038497735588871663814704993634734088042403260664q^{95} + 105274360098785049511479634643100239619245690316279881631113940000q^{96} + 161640138220417791770996425132391296550980884122633329592007400918q^{97} - 172788674287647322793549412320988559853203256576665666003433893074q^{98} + 352913032246277614719360267290575261406563762380876948694185756q^{99} + O(q^{100}) \)

Decomposition of \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.66.a.a \(5\) \(80.272\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-2586530964\) \(-9\!\cdots\!05\) \(-8\!\cdots\!94\) \(57\!\cdots\!24\) \(+\) \(q+(-517306193-\beta _{1})q^{2}-3^{32}q^{3}+\cdots\)
3.66.a.b \(6\) \(80.272\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(6210982962\) \(11\!\cdots\!46\) \(35\!\cdots\!00\) \(13\!\cdots\!04\) \(-\) \(q+(1035163827+\beta _{1})q^{2}+3^{32}q^{3}+\cdots\)

Decomposition of \(S_{66}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{66}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2586530964 T + 50100467848900449280 T^{2} + \)\(14\!\cdots\!48\)\( T^{3} + \)\(90\!\cdots\!88\)\( T^{4} + \)\(23\!\cdots\!04\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} + \)\(20\!\cdots\!52\)\( T^{7} + \)\(25\!\cdots\!40\)\( T^{8} + \)\(47\!\cdots\!64\)\( T^{9} + \)\(68\!\cdots\!32\)\( T^{10} \))(\( 1 - 6210982962 T + 84298166589242128224 T^{2} - \)\(60\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!52\)\( T^{4} - \)\(34\!\cdots\!44\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(12\!\cdots\!08\)\( T^{7} + \)\(58\!\cdots\!48\)\( T^{8} - \)\(30\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!24\)\( T^{10} - \)\(42\!\cdots\!84\)\( T^{11} + \)\(25\!\cdots\!24\)\( T^{12} \))
$3$ (\( ( 1 + 1853020188851841 T )^{5} \))(\( ( 1 - 1853020188851841 T )^{6} \))
$5$ (\( 1 + \)\(86\!\cdots\!94\)\( T + \)\(10\!\cdots\!25\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!50\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!50\)\( T^{6} + \)\(49\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!25\)\( T^{8} + \)\(46\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} \))(\( 1 - \)\(35\!\cdots\!00\)\( T + \)\(57\!\cdots\!50\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!75\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(80\!\cdots\!00\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!75\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(31\!\cdots\!50\)\( T^{10} - \)\(51\!\cdots\!00\)\( T^{11} + \)\(39\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 - \)\(57\!\cdots\!24\)\( T + \)\(38\!\cdots\!19\)\( T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(56\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(48\!\cdots\!78\)\( T^{6} - \)\(10\!\cdots\!48\)\( T^{7} + \)\(23\!\cdots\!17\)\( T^{8} - \)\(30\!\cdots\!24\)\( T^{9} + \)\(45\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(13\!\cdots\!04\)\( T + \)\(33\!\cdots\!94\)\( T^{2} - \)\(42\!\cdots\!52\)\( T^{3} + \)\(55\!\cdots\!11\)\( T^{4} - \)\(58\!\cdots\!12\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} - \)\(50\!\cdots\!84\)\( T^{7} + \)\(40\!\cdots\!39\)\( T^{8} - \)\(26\!\cdots\!36\)\( T^{9} + \)\(17\!\cdots\!94\)\( T^{10} - \)\(59\!\cdots\!28\)\( T^{11} + \)\(38\!\cdots\!49\)\( T^{12} \))
$11$ (\( 1 - \)\(24\!\cdots\!52\)\( T + \)\(77\!\cdots\!59\)\( T^{2} - \)\(69\!\cdots\!28\)\( T^{3} + \)\(58\!\cdots\!78\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} + \)\(28\!\cdots\!78\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(90\!\cdots\!09\)\( T^{8} - \)\(14\!\cdots\!52\)\( T^{9} + \)\(28\!\cdots\!51\)\( T^{10} \))(\( 1 + \)\(23\!\cdots\!76\)\( T + \)\(39\!\cdots\!94\)\( T^{2} + \)\(32\!\cdots\!32\)\( T^{3} + \)\(15\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(37\!\cdots\!83\)\( T^{8} + \)\(37\!\cdots\!32\)\( T^{9} + \)\(23\!\cdots\!94\)\( T^{10} + \)\(65\!\cdots\!76\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \))
$13$ (\( 1 - \)\(12\!\cdots\!06\)\( T + \)\(41\!\cdots\!01\)\( T^{2} - \)\(70\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(28\!\cdots\!10\)\( T^{6} - \)\(45\!\cdots\!28\)\( T^{7} + \)\(68\!\cdots\!57\)\( T^{8} - \)\(53\!\cdots\!06\)\( T^{9} + \)\(10\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(96\!\cdots\!48\)\( T + \)\(90\!\cdots\!70\)\( T^{2} - \)\(89\!\cdots\!24\)\( T^{3} + \)\(45\!\cdots\!75\)\( T^{4} - \)\(39\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(10\!\cdots\!04\)\( T^{7} + \)\(29\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!68\)\( T^{9} + \)\(38\!\cdots\!70\)\( T^{10} - \)\(10\!\cdots\!64\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \))
$17$ (\( 1 + \)\(14\!\cdots\!30\)\( T + \)\(34\!\cdots\!05\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!10\)\( T^{4} + \)\(21\!\cdots\!48\)\( T^{5} + \)\(37\!\cdots\!70\)\( T^{6} + \)\(23\!\cdots\!80\)\( T^{7} + \)\(30\!\cdots\!65\)\( T^{8} + \)\(11\!\cdots\!30\)\( T^{9} + \)\(78\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(13\!\cdots\!32\)\( T + \)\(10\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} + \)\(17\!\cdots\!32\)\( T^{5} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(85\!\cdots\!02\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{11} + \)\(75\!\cdots\!49\)\( T^{12} \))
$19$ (\( 1 - \)\(50\!\cdots\!72\)\( T + \)\(70\!\cdots\!83\)\( T^{2} - \)\(25\!\cdots\!16\)\( T^{3} + \)\(19\!\cdots\!06\)\( T^{4} - \)\(50\!\cdots\!24\)\( T^{5} + \)\(25\!\cdots\!94\)\( T^{6} - \)\(44\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!17\)\( T^{8} - \)\(15\!\cdots\!72\)\( T^{9} + \)\(39\!\cdots\!99\)\( T^{10} \))(\( 1 - \)\(56\!\cdots\!92\)\( T + \)\(29\!\cdots\!22\)\( T^{2} - \)\(45\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!43\)\( T^{4} + \)\(28\!\cdots\!72\)\( T^{5} + \)\(96\!\cdots\!68\)\( T^{6} + \)\(37\!\cdots\!28\)\( T^{7} + \)\(22\!\cdots\!43\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(89\!\cdots\!22\)\( T^{10} - \)\(22\!\cdots\!08\)\( T^{11} + \)\(51\!\cdots\!01\)\( T^{12} \))
$23$ (\( 1 + \)\(23\!\cdots\!68\)\( T + \)\(94\!\cdots\!91\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!94\)\( T^{4} + \)\(94\!\cdots\!72\)\( T^{5} + \)\(17\!\cdots\!42\)\( T^{6} + \)\(23\!\cdots\!24\)\( T^{7} + \)\(32\!\cdots\!37\)\( T^{8} + \)\(25\!\cdots\!68\)\( T^{9} + \)\(36\!\cdots\!43\)\( T^{10} \))(\( 1 - \)\(47\!\cdots\!92\)\( T + \)\(10\!\cdots\!86\)\( T^{2} + \)\(21\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!91\)\( T^{4} + \)\(19\!\cdots\!64\)\( T^{5} + \)\(21\!\cdots\!12\)\( T^{6} + \)\(64\!\cdots\!52\)\( T^{7} + \)\(56\!\cdots\!59\)\( T^{8} + \)\(74\!\cdots\!92\)\( T^{9} + \)\(11\!\cdots\!86\)\( T^{10} - \)\(17\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \))
$29$ (\( 1 - \)\(65\!\cdots\!74\)\( T + \)\(56\!\cdots\!77\)\( T^{2} - \)\(26\!\cdots\!56\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} - \)\(43\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!26\)\( T^{6} - \)\(34\!\cdots\!56\)\( T^{7} + \)\(83\!\cdots\!73\)\( T^{8} - \)\(10\!\cdots\!74\)\( T^{9} + \)\(19\!\cdots\!49\)\( T^{10} \))(\( 1 - \)\(18\!\cdots\!04\)\( T + \)\(45\!\cdots\!26\)\( T^{2} - \)\(71\!\cdots\!72\)\( T^{3} + \)\(97\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!92\)\( T^{6} - \)\(14\!\cdots\!36\)\( T^{7} + \)\(12\!\cdots\!27\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(76\!\cdots\!26\)\( T^{10} - \)\(35\!\cdots\!96\)\( T^{11} + \)\(21\!\cdots\!01\)\( T^{12} \))
$31$ (\( 1 - \)\(35\!\cdots\!64\)\( T + \)\(21\!\cdots\!79\)\( T^{2} - \)\(54\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!82\)\( T^{4} - \)\(62\!\cdots\!84\)\( T^{5} + \)\(24\!\cdots\!82\)\( T^{6} - \)\(40\!\cdots\!56\)\( T^{7} + \)\(14\!\cdots\!29\)\( T^{8} - \)\(20\!\cdots\!64\)\( T^{9} + \)\(49\!\cdots\!51\)\( T^{10} \))(\( 1 - \)\(26\!\cdots\!96\)\( T + \)\(33\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(57\!\cdots\!03\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(61\!\cdots\!52\)\( T^{6} - \)\(14\!\cdots\!64\)\( T^{7} + \)\(43\!\cdots\!03\)\( T^{8} - \)\(65\!\cdots\!48\)\( T^{9} + \)\(19\!\cdots\!78\)\( T^{10} - \)\(12\!\cdots\!96\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \))
$37$ (\( 1 + \)\(11\!\cdots\!26\)\( T + \)\(40\!\cdots\!69\)\( T^{2} + \)\(33\!\cdots\!48\)\( T^{3} + \)\(65\!\cdots\!54\)\( T^{4} + \)\(40\!\cdots\!24\)\( T^{5} + \)\(55\!\cdots\!78\)\( T^{6} + \)\(24\!\cdots\!52\)\( T^{7} + \)\(25\!\cdots\!17\)\( T^{8} + \)\(63\!\cdots\!26\)\( T^{9} + \)\(46\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(19\!\cdots\!76\)\( T + \)\(53\!\cdots\!74\)\( T^{2} + \)\(68\!\cdots\!08\)\( T^{3} + \)\(11\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} + \)\(90\!\cdots\!36\)\( T^{7} + \)\(81\!\cdots\!59\)\( T^{8} + \)\(43\!\cdots\!44\)\( T^{9} + \)\(29\!\cdots\!74\)\( T^{10} + \)\(91\!\cdots\!32\)\( T^{11} + \)\(39\!\cdots\!49\)\( T^{12} \))
$41$ (\( 1 + \)\(31\!\cdots\!26\)\( T + \)\(13\!\cdots\!89\)\( T^{2} + \)\(60\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!62\)\( T^{4} + \)\(45\!\cdots\!96\)\( T^{5} + \)\(12\!\cdots\!62\)\( T^{6} + \)\(27\!\cdots\!24\)\( T^{7} + \)\(43\!\cdots\!89\)\( T^{8} + \)\(67\!\cdots\!26\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} \))(\( 1 - \)\(44\!\cdots\!16\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + \)\(57\!\cdots\!83\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!12\)\( T^{6} - \)\(93\!\cdots\!44\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(37\!\cdots\!08\)\( T^{9} + \)\(80\!\cdots\!78\)\( T^{10} - \)\(63\!\cdots\!16\)\( T^{11} + \)\(96\!\cdots\!01\)\( T^{12} \))
$43$ (\( 1 - \)\(33\!\cdots\!84\)\( T + \)\(79\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(22\!\cdots\!32\)\( T^{5} + \)\(28\!\cdots\!74\)\( T^{6} - \)\(29\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} - \)\(16\!\cdots\!84\)\( T^{9} + \)\(75\!\cdots\!43\)\( T^{10} \))(\( 1 - \)\(15\!\cdots\!40\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(62\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{5} + \)\(30\!\cdots\!60\)\( T^{6} - \)\(22\!\cdots\!60\)\( T^{7} + \)\(36\!\cdots\!03\)\( T^{8} - \)\(21\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!70\)\( T^{10} - \)\(11\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 + \)\(47\!\cdots\!52\)\( T + \)\(15\!\cdots\!87\)\( T^{2} + \)\(18\!\cdots\!60\)\( T^{3} - \)\(12\!\cdots\!42\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{5} - \)\(58\!\cdots\!94\)\( T^{6} + \)\(43\!\cdots\!40\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} + \)\(26\!\cdots\!52\)\( T^{9} + \)\(27\!\cdots\!07\)\( T^{10} \))(\( 1 + \)\(23\!\cdots\!80\)\( T + \)\(11\!\cdots\!30\)\( T^{2} + \)\(88\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} - \)\(75\!\cdots\!60\)\( T^{5} - \)\(80\!\cdots\!60\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(53\!\cdots\!03\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(64\!\cdots\!30\)\( T^{10} + \)\(64\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!49\)\( T^{12} \))
$53$ (\( 1 + \)\(17\!\cdots\!38\)\( T + \)\(43\!\cdots\!41\)\( T^{2} + \)\(36\!\cdots\!96\)\( T^{3} + \)\(58\!\cdots\!94\)\( T^{4} + \)\(35\!\cdots\!32\)\( T^{5} + \)\(70\!\cdots\!42\)\( T^{6} + \)\(52\!\cdots\!04\)\( T^{7} + \)\(73\!\cdots\!37\)\( T^{8} + \)\(34\!\cdots\!38\)\( T^{9} + \)\(24\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(96\!\cdots\!32\)\( T + \)\(51\!\cdots\!26\)\( T^{2} - \)\(33\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} - \)\(62\!\cdots\!36\)\( T^{5} + \)\(18\!\cdots\!92\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(57\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!26\)\( T^{10} - \)\(23\!\cdots\!76\)\( T^{11} + \)\(29\!\cdots\!49\)\( T^{12} \))
$59$ (\( 1 + \)\(47\!\cdots\!52\)\( T + \)\(53\!\cdots\!03\)\( T^{2} + \)\(22\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} + \)\(42\!\cdots\!64\)\( T^{5} + \)\(15\!\cdots\!74\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} + \)\(12\!\cdots\!52\)\( T^{9} + \)\(33\!\cdots\!99\)\( T^{10} \))(\( 1 + \)\(71\!\cdots\!52\)\( T + \)\(69\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!24\)\( T^{3} + \)\(21\!\cdots\!63\)\( T^{4} + \)\(86\!\cdots\!68\)\( T^{5} + \)\(35\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!32\)\( T^{7} + \)\(34\!\cdots\!63\)\( T^{8} + \)\(76\!\cdots\!76\)\( T^{9} + \)\(18\!\cdots\!22\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \))
$61$ (\( 1 - \)\(16\!\cdots\!58\)\( T + \)\(46\!\cdots\!77\)\( T^{2} - \)\(64\!\cdots\!08\)\( T^{3} + \)\(99\!\cdots\!22\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!22\)\( T^{6} - \)\(79\!\cdots\!08\)\( T^{7} + \)\(64\!\cdots\!77\)\( T^{8} - \)\(25\!\cdots\!58\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} \))(\( 1 - \)\(40\!\cdots\!00\)\( T + \)\(59\!\cdots\!14\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(21\!\cdots\!00\)\( T^{6} - \)\(37\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{9} + \)\(91\!\cdots\!14\)\( T^{10} - \)\(69\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} \))
$67$ (\( 1 + \)\(15\!\cdots\!40\)\( T + \)\(10\!\cdots\!75\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!10\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!70\)\( T^{6} + \)\(48\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!25\)\( T^{8} + \)\(95\!\cdots\!40\)\( T^{9} + \)\(29\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(14\!\cdots\!08\)\( T + \)\(22\!\cdots\!02\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!55\)\( T^{4} - \)\(24\!\cdots\!68\)\( T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - \)\(12\!\cdots\!76\)\( T^{7} + \)\(57\!\cdots\!95\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} - \)\(43\!\cdots\!56\)\( T^{11} + \)\(14\!\cdots\!49\)\( T^{12} \))
$71$ (\( 1 + \)\(20\!\cdots\!40\)\( T + \)\(40\!\cdots\!95\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} + \)\(19\!\cdots\!48\)\( T^{5} + \)\(29\!\cdots\!10\)\( T^{6} + \)\(37\!\cdots\!80\)\( T^{7} + \)\(39\!\cdots\!45\)\( T^{8} + \)\(43\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!51\)\( T^{10} \))(\( 1 - \)\(45\!\cdots\!12\)\( T + \)\(14\!\cdots\!66\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!95\)\( T^{4} - \)\(75\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(24\!\cdots\!95\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!66\)\( T^{10} - \)\(20\!\cdots\!12\)\( T^{11} + \)\(97\!\cdots\!01\)\( T^{12} \))
$73$ (\( 1 + \)\(16\!\cdots\!78\)\( T + \)\(17\!\cdots\!01\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{3} + \)\(41\!\cdots\!34\)\( T^{4} + \)\(52\!\cdots\!92\)\( T^{5} + \)\(53\!\cdots\!62\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} + \)\(38\!\cdots\!57\)\( T^{8} + \)\(48\!\cdots\!78\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(52\!\cdots\!52\)\( T + \)\(75\!\cdots\!26\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(24\!\cdots\!31\)\( T^{4} - \)\(79\!\cdots\!96\)\( T^{5} + \)\(41\!\cdots\!32\)\( T^{6} - \)\(10\!\cdots\!28\)\( T^{7} + \)\(41\!\cdots\!19\)\( T^{8} - \)\(70\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!26\)\( T^{10} - \)\(20\!\cdots\!36\)\( T^{11} + \)\(49\!\cdots\!49\)\( T^{12} \))
$79$ (\( 1 + \)\(16\!\cdots\!20\)\( T + \)\(16\!\cdots\!95\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!10\)\( T^{4} + \)\(39\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!90\)\( T^{6} + \)\(62\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!05\)\( T^{8} + \)\(39\!\cdots\!20\)\( T^{9} + \)\(53\!\cdots\!99\)\( T^{10} \))(\( 1 - \)\(54\!\cdots\!40\)\( T + \)\(10\!\cdots\!94\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!80\)\( T^{6} - \)\(46\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!00\)\( T^{9} + \)\(26\!\cdots\!94\)\( T^{10} - \)\(29\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!01\)\( T^{12} \))
$83$ (\( 1 + \)\(58\!\cdots\!52\)\( T + \)\(30\!\cdots\!15\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!78\)\( T^{4} + \)\(86\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!54\)\( T^{6} + \)\(33\!\cdots\!64\)\( T^{7} + \)\(50\!\cdots\!05\)\( T^{8} + \)\(53\!\cdots\!52\)\( T^{9} + \)\(50\!\cdots\!43\)\( T^{10} \))(\( 1 - \)\(12\!\cdots\!56\)\( T + \)\(88\!\cdots\!26\)\( T^{2} - \)\(43\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!67\)\( T^{4} - \)\(50\!\cdots\!92\)\( T^{5} + \)\(13\!\cdots\!36\)\( T^{6} - \)\(27\!\cdots\!56\)\( T^{7} + \)\(49\!\cdots\!83\)\( T^{8} - \)\(71\!\cdots\!04\)\( T^{9} + \)\(80\!\cdots\!26\)\( T^{10} - \)\(63\!\cdots\!08\)\( T^{11} + \)\(27\!\cdots\!49\)\( T^{12} \))
$89$ (\( 1 + \)\(10\!\cdots\!78\)\( T + \)\(69\!\cdots\!53\)\( T^{2} + \)\(29\!\cdots\!24\)\( T^{3} + \)\(98\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!96\)\( T^{5} + \)\(50\!\cdots\!34\)\( T^{6} + \)\(78\!\cdots\!24\)\( T^{7} + \)\(93\!\cdots\!97\)\( T^{8} + \)\(75\!\cdots\!78\)\( T^{9} + \)\(35\!\cdots\!49\)\( T^{10} \))(\( 1 - \)\(64\!\cdots\!12\)\( T + \)\(35\!\cdots\!62\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!63\)\( T^{4} - \)\(96\!\cdots\!48\)\( T^{5} + \)\(24\!\cdots\!48\)\( T^{6} - \)\(49\!\cdots\!52\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{9} + \)\(24\!\cdots\!62\)\( T^{10} - \)\(22\!\cdots\!88\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} \))
$97$ (\( 1 - \)\(32\!\cdots\!90\)\( T + \)\(40\!\cdots\!65\)\( T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!10\)\( T^{4} - \)\(56\!\cdots\!72\)\( T^{5} + \)\(86\!\cdots\!70\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(11\!\cdots\!90\)\( T^{9} + \)\(50\!\cdots\!57\)\( T^{10} \))(\( 1 - \)\(12\!\cdots\!28\)\( T + \)\(12\!\cdots\!02\)\( T^{2} - \)\(86\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!55\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} + \)\(95\!\cdots\!44\)\( T^{6} - \)\(32\!\cdots\!76\)\( T^{7} + \)\(95\!\cdots\!95\)\( T^{8} - \)\(22\!\cdots\!60\)\( T^{9} + \)\(46\!\cdots\!02\)\( T^{10} - \)\(64\!\cdots\!96\)\( T^{11} + \)\(69\!\cdots\!49\)\( T^{12} \))
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