Properties

Label 3.68.a
Level 3
Weight 68
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: \( k \) \(=\) \( 68 \)
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 \(M_{68}(\Gamma_0(3))\).

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

Trace form

\( 11q - 2519867922q^{2} - 5559060566555523q^{3} + 648863271028424237828q^{4} - 258877034356887158595510q^{5} - 166719440940008987767396842q^{6} - 25170544830943953984840073400q^{7} - 272131678300220551429703131656q^{8} + 339934698208958735981127059838819q^{9} + O(q^{10}) \) \( 11q - 2519867922q^{2} - 5559060566555523q^{3} + 648863271028424237828q^{4} - 258877034356887158595510q^{5} - 166719440940008987767396842q^{6} - 25170544830943953984840073400q^{7} - 272131678300220551429703131656q^{8} + 339934698208958735981127059838819q^{9} + 3944503060971364857819105595064340q^{10} - 105965122956205668536725315302485508q^{11} - 1519910454404886892260883671569531148q^{12} + 17688231681293316260485269239561435314q^{13} - 509052232063568882320625545175592215904q^{14} + 565362485856152742456010877219260184070q^{15} + 9051750287916473810357202192230395555856q^{16} + 133330372581699107679265749161315484721254q^{17} - 77871867417409633801864479590364953496738q^{18} - 13040337294134068839144308626047658096086140q^{19} + 54988444793559595210483530880212176591018520q^{20} + 172135267509365279268261983136215925195673368q^{21} + 120710868797619144040433867652136117884326568q^{22} + 9142561048971637867936642534483948715140109400q^{23} - 3452116095070017493293273116834216225129209320q^{24} - 3525197194209734426041285094375035310391380275q^{25} - 162164976223756373440878305175814074287341335468q^{26} - 171792506910670443678820376588540424234035840667q^{27} - 16470476443489988777201058332130514171432326685760q^{28} - 36068866713106934949248938395316032943799436878430q^{29} - 59281145959231791899475816274132523592940538934780q^{30} - 391344643562339061135153076936220718864516397338848q^{31} - 1309276262268849000007498144078796927814752824588320q^{32} - 603003712613085326026857624125346753194104203232188q^{33} - 2975199360215619634690864358314727419736073767735844q^{34} - 7041403046932475611835220834746467171579832310892560q^{35} + 20051921837811381058238213369581331416844907145695012q^{36} + 57457781141133716968544853029073980942699924483548810q^{37} + 273037151357962998118282341854253997527779527581805848q^{38} + 109808599888079516207125830862371874490637173076997566q^{39} + 2309189307551644134551254353320311274742520413944706000q^{40} + 1852110738442783689943806195728130088971160790190417342q^{41} + 2144183177282803035650585949224212948799203281659316000q^{42} - 5842390887399762902622392068867887060140829125764457828q^{43} - 27802139929471024081892619163418804501918719198434972624q^{44} - 8000116958848970759181885343837070515430820581001554790q^{45} - 32533822466852430654278393234959022930962228902571285968q^{46} + 166293839036999576543683605282444285864233467705417114416q^{47} + 170139889630769603273174811671741371920238910623709116368q^{48} + 1865751593557117794616934081056318812677892680675199888003q^{49} + 5844382073945934593527634801934073077329367207245869430850q^{50} + 989166912692864415929371203386129579041170621013462305258q^{51} + 1536330093619119335871679744550051752177973517745593769624q^{52} - 10429495096003229922240678897188791382107968992707750409670q^{53} - 5152156621955297728754562146488371152633209889764339055418q^{54} - 102501666968205811018888793496313267870999516242000937641720q^{55} - 101154672572301793839435580611322944610368645578131920078720q^{56} + 2105162285151674574986741439004694306249894792277514019772q^{57} - 111828208813929573333107624143959448952772952903437033305244q^{58} + 252548716210294020542691561699088065150545546669888857190700q^{59} - 207770293800652726185877807640387431801445466282119118663240q^{60} + 283226209820039173747657633776576366763291132337429930598562q^{61} + 4345369142610658299154937509670683896200438106802162736418480q^{62} - 777849232805636298599349732238404466045108848899608739028600q^{63} - 7805665869157902053979358662227642813645425063714381097394112q^{64} + 4668993440690419113623096319982795473401340147525472043836380q^{65} - 14961542669388798090697332477106716023880850945355261262111864q^{66} - 2314620678481268001577772268984034016485979448727100838620876q^{67} + 82207762428552901624538678301063190865575489579926465215589384q^{68} - 85250988177390652014779462683100595279881550847203713046433944q^{69} + 13820413890568809915317980103585447540823507652955970335048640q^{70} + 460255303663715904178931222121588166038502364846056549043580552q^{71} - 8409727266916628909244217486962594421824708790404839172414024q^{72} - 85634903877460114460252272808356746822977012686117492852288610q^{73} + 648240813670132428423102389928522637915588558605373850627059236q^{74} - 1409195139958983593947936144311662716008553693165679047539986325q^{75} - 3685284052815532112813248435988602099077453598087926416877276080q^{76} - 2363081644264197898187225621014231762470303992266913069431162080q^{77} - 2543946338943767941330061219206744635222705148916504284959601596q^{78} - 8210167959169806282217136193078952258558533690604555908685023920q^{79} + 13713408283803517531152705441659450587161163619037612475920851040q^{80} + 10505054458765077605825097719518554131119286528717774428205392251q^{81} + 58009044350153153777868562651983519383145258281207610106292318220q^{82} + 33897634062421237699262472155041487846453807911818092490558857796q^{83} + 72012313035210128293108725879623418568493662626850311920735665344q^{84} + 151061067040563944112379649672099615844324876815086467189157546740q^{85} - 178308106695330420524024162029259782148942117035404630163843226328q^{86} + 9327939448445406358913546981457920122926811793405100859823815854q^{87} - 180651392112991925413707155774687023539343348195260038433751726176q^{88} - 977947895723229556311910972672067880892102883647816641125745851730q^{89} + 121897587055964988480437596521608988337667114106980468738394055860q^{90} - 188894031686552534321286247567459651513753623660488773544185244368q^{91} + 1059740178651157065151081742576265174952585384890864916734347625120q^{92} + 315448592759304557292891630024516015816867597668942713621087346880q^{93} + 6398414495724709076267663835507596270667355291388470176380579829696q^{94} + 5557626028701544557060724829649646421715158064426888035730090833400q^{95} - 77195520702726622676767180289558417457325378221314440139190478752q^{96} - 8103947413238212751440300605332526614257313153995383812116932102026q^{97} + 8958086496385928836543368609427172213659751349617003933922570832926q^{98} - 3274656553890270846448478945004511325727974807035915828261505757732q^{99} + O(q^{100}) \)

Decomposition of \(S_{68}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.68.a.a \(5\) \(85.287\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-16255223088\) \(27\!\cdots\!15\) \(-7\!\cdots\!10\) \(28\!\cdots\!08\) \(-\) \(q+(-3251044618-\beta _{1})q^{2}+3^{33}q^{3}+\cdots\)
3.68.a.b \(6\) \(85.287\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(13735355166\) \(-3\!\cdots\!38\) \(-1\!\cdots\!00\) \(-2\!\cdots\!08\) \(+\) \(q+(2289225861-\beta _{1})q^{2}-3^{33}q^{3}+\cdots\)

Decomposition of \(S_{68}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{68}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 16255223088 T + \)\(40\!\cdots\!04\)\( T^{2} + \)\(45\!\cdots\!56\)\( T^{3} + \)\(68\!\cdots\!92\)\( T^{4} + \)\(70\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(98\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!08\)\( T^{8} + \)\(77\!\cdots\!28\)\( T^{9} + \)\(69\!\cdots\!68\)\( T^{10} \))(\( 1 - 13735355166 T + \)\(30\!\cdots\!36\)\( T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!72\)\( T^{4} - \)\(35\!\cdots\!52\)\( T^{5} + \)\(66\!\cdots\!36\)\( T^{6} - \)\(51\!\cdots\!56\)\( T^{7} + \)\(12\!\cdots\!48\)\( T^{8} - \)\(87\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!16\)\( T^{10} - \)\(96\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!04\)\( T^{12} \))
$3$ (\( ( 1 - 5559060566555523 T )^{5} \))(\( ( 1 + 5559060566555523 T )^{6} \))
$5$ (\( 1 + \)\(78\!\cdots\!10\)\( T + \)\(23\!\cdots\!25\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!50\)\( T^{4} + \)\(30\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!50\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!25\)\( T^{8} + \)\(16\!\cdots\!50\)\( T^{9} + \)\(14\!\cdots\!25\)\( T^{10} \))(\( 1 + \)\(18\!\cdots\!00\)\( T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(61\!\cdots\!75\)\( T^{8} + \)\(87\!\cdots\!00\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} + \)\(25\!\cdots\!00\)\( T^{11} + \)\(96\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 - \)\(28\!\cdots\!08\)\( T + \)\(61\!\cdots\!23\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{3} - \)\(57\!\cdots\!58\)\( T^{4} + \)\(17\!\cdots\!92\)\( T^{5} - \)\(23\!\cdots\!94\)\( T^{6} + \)\(19\!\cdots\!72\)\( T^{7} + \)\(44\!\cdots\!61\)\( T^{8} - \)\(88\!\cdots\!08\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} \))(\( 1 + \)\(28\!\cdots\!08\)\( T + \)\(11\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(52\!\cdots\!11\)\( T^{4} + \)\(65\!\cdots\!84\)\( T^{5} + \)\(28\!\cdots\!12\)\( T^{6} + \)\(27\!\cdots\!12\)\( T^{7} + \)\(92\!\cdots\!39\)\( T^{8} + \)\(97\!\cdots\!52\)\( T^{9} + \)\(35\!\cdots\!26\)\( T^{10} + \)\(35\!\cdots\!44\)\( T^{11} + \)\(53\!\cdots\!49\)\( T^{12} \))
$11$ (\( 1 + \)\(10\!\cdots\!32\)\( T + \)\(21\!\cdots\!95\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!78\)\( T^{6} + \)\(67\!\cdots\!20\)\( T^{7} + \)\(45\!\cdots\!45\)\( T^{8} + \)\(13\!\cdots\!92\)\( T^{9} + \)\(73\!\cdots\!51\)\( T^{10} \))(\( 1 - \)\(12\!\cdots\!24\)\( T + \)\(26\!\cdots\!34\)\( T^{2} + \)\(77\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!63\)\( T^{4} + \)\(14\!\cdots\!60\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} + \)\(88\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!83\)\( T^{8} + \)\(16\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!54\)\( T^{10} - \)\(92\!\cdots\!24\)\( T^{11} + \)\(43\!\cdots\!21\)\( T^{12} \))
$13$ (\( 1 - \)\(18\!\cdots\!78\)\( T + \)\(15\!\cdots\!65\)\( T^{2} - \)\(21\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!12\)\( T^{5} + \)\(47\!\cdots\!06\)\( T^{6} - \)\(40\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} - \)\(64\!\cdots\!38\)\( T^{9} + \)\(14\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(10\!\cdots\!64\)\( T + \)\(60\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!15\)\( T^{4} - \)\(86\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} - \)\(37\!\cdots\!32\)\( T^{7} + \)\(50\!\cdots\!35\)\( T^{8} - \)\(92\!\cdots\!56\)\( T^{9} + \)\(20\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!48\)\( T^{11} + \)\(63\!\cdots\!69\)\( T^{12} \))
$17$ (\( 1 - \)\(15\!\cdots\!50\)\( T + \)\(96\!\cdots\!05\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!10\)\( T^{4} - \)\(47\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!30\)\( T^{6} - \)\(95\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!85\)\( T^{8} - \)\(89\!\cdots\!50\)\( T^{9} + \)\(15\!\cdots\!93\)\( T^{10} \))(\( 1 + \)\(22\!\cdots\!96\)\( T + \)\(75\!\cdots\!78\)\( T^{2} - \)\(32\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(69\!\cdots\!56\)\( T^{6} - \)\(81\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!95\)\( T^{8} - \)\(67\!\cdots\!80\)\( T^{9} + \)\(43\!\cdots\!98\)\( T^{10} + \)\(35\!\cdots\!28\)\( T^{11} + \)\(43\!\cdots\!89\)\( T^{12} \))
$19$ (\( 1 + \)\(63\!\cdots\!88\)\( T + \)\(13\!\cdots\!71\)\( T^{2} + \)\(84\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!02\)\( T^{4} + \)\(50\!\cdots\!68\)\( T^{5} + \)\(50\!\cdots\!78\)\( T^{6} + \)\(19\!\cdots\!48\)\( T^{7} + \)\(14\!\cdots\!49\)\( T^{8} + \)\(32\!\cdots\!08\)\( T^{9} + \)\(24\!\cdots\!99\)\( T^{10} \))(\( 1 + \)\(67\!\cdots\!52\)\( T + \)\(82\!\cdots\!42\)\( T^{2} + \)\(39\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(13\!\cdots\!72\)\( T^{5} - \)\(70\!\cdots\!92\)\( T^{6} - \)\(64\!\cdots\!08\)\( T^{7} + \)\(26\!\cdots\!63\)\( T^{8} + \)\(42\!\cdots\!16\)\( T^{9} + \)\(42\!\cdots\!22\)\( T^{10} + \)\(16\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!61\)\( T^{12} \))
$23$ (\( 1 + \)\(30\!\cdots\!64\)\( T + \)\(39\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!64\)\( T^{3} + \)\(87\!\cdots\!02\)\( T^{4} + \)\(26\!\cdots\!64\)\( T^{5} + \)\(15\!\cdots\!94\)\( T^{6} + \)\(35\!\cdots\!76\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} + \)\(27\!\cdots\!84\)\( T^{9} + \)\(15\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(12\!\cdots\!64\)\( T + \)\(91\!\cdots\!54\)\( T^{2} - \)\(39\!\cdots\!52\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} + \)\(43\!\cdots\!48\)\( T^{5} - \)\(92\!\cdots\!12\)\( T^{6} + \)\(75\!\cdots\!56\)\( T^{7} + \)\(30\!\cdots\!79\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{9} + \)\(80\!\cdots\!74\)\( T^{10} - \)\(18\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!29\)\( T^{12} \))
$29$ (\( 1 + \)\(17\!\cdots\!66\)\( T + \)\(34\!\cdots\!49\)\( T^{2} + \)\(38\!\cdots\!96\)\( T^{3} + \)\(52\!\cdots\!18\)\( T^{4} + \)\(48\!\cdots\!36\)\( T^{5} + \)\(50\!\cdots\!62\)\( T^{6} + \)\(35\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} + \)\(14\!\cdots\!26\)\( T^{9} + \)\(80\!\cdots\!49\)\( T^{10} \))(\( 1 + \)\(18\!\cdots\!64\)\( T + \)\(49\!\cdots\!66\)\( T^{2} + \)\(61\!\cdots\!32\)\( T^{3} + \)\(95\!\cdots\!67\)\( T^{4} + \)\(90\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + \)\(86\!\cdots\!96\)\( T^{7} + \)\(87\!\cdots\!27\)\( T^{8} + \)\(53\!\cdots\!28\)\( T^{9} + \)\(41\!\cdots\!26\)\( T^{10} + \)\(15\!\cdots\!36\)\( T^{11} + \)\(76\!\cdots\!41\)\( T^{12} \))
$31$ (\( 1 + \)\(16\!\cdots\!44\)\( T + \)\(29\!\cdots\!67\)\( T^{2} + \)\(35\!\cdots\!28\)\( T^{3} + \)\(42\!\cdots\!06\)\( T^{4} + \)\(37\!\cdots\!68\)\( T^{5} + \)\(35\!\cdots\!66\)\( T^{6} + \)\(24\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!77\)\( T^{8} + \)\(80\!\cdots\!04\)\( T^{9} + \)\(40\!\cdots\!51\)\( T^{10} \))(\( 1 + \)\(22\!\cdots\!04\)\( T + \)\(65\!\cdots\!38\)\( T^{2} + \)\(94\!\cdots\!52\)\( T^{3} + \)\(15\!\cdots\!83\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!56\)\( T^{7} + \)\(10\!\cdots\!43\)\( T^{8} + \)\(54\!\cdots\!12\)\( T^{9} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(90\!\cdots\!04\)\( T^{11} + \)\(33\!\cdots\!61\)\( T^{12} \))
$37$ (\( 1 - \)\(30\!\cdots\!58\)\( T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(11\!\cdots\!92\)\( T^{3} + \)\(82\!\cdots\!02\)\( T^{4} - \)\(19\!\cdots\!48\)\( T^{5} + \)\(97\!\cdots\!66\)\( T^{6} - \)\(16\!\cdots\!88\)\( T^{7} + \)\(67\!\cdots\!41\)\( T^{8} - \)\(56\!\cdots\!18\)\( T^{9} + \)\(22\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(27\!\cdots\!52\)\( T + \)\(27\!\cdots\!26\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!91\)\( T^{4} + \)\(79\!\cdots\!04\)\( T^{5} + \)\(17\!\cdots\!52\)\( T^{6} + \)\(93\!\cdots\!32\)\( T^{7} + \)\(39\!\cdots\!99\)\( T^{8} - \)\(16\!\cdots\!68\)\( T^{9} + \)\(52\!\cdots\!46\)\( T^{10} - \)\(61\!\cdots\!36\)\( T^{11} + \)\(26\!\cdots\!69\)\( T^{12} \))
$41$ (\( 1 + \)\(80\!\cdots\!54\)\( T + \)\(27\!\cdots\!77\)\( T^{2} + \)\(49\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!06\)\( T^{4} - \)\(56\!\cdots\!52\)\( T^{5} + \)\(34\!\cdots\!86\)\( T^{6} + \)\(64\!\cdots\!08\)\( T^{7} + \)\(41\!\cdots\!57\)\( T^{8} + \)\(13\!\cdots\!34\)\( T^{9} + \)\(19\!\cdots\!01\)\( T^{10} \))(\( 1 - \)\(26\!\cdots\!96\)\( T + \)\(71\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!43\)\( T^{4} - \)\(26\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!72\)\( T^{6} - \)\(30\!\cdots\!04\)\( T^{7} + \)\(26\!\cdots\!23\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!18\)\( T^{10} - \)\(50\!\cdots\!96\)\( T^{11} + \)\(21\!\cdots\!81\)\( T^{12} \))
$43$ (\( 1 + \)\(47\!\cdots\!28\)\( T + \)\(68\!\cdots\!47\)\( T^{2} + \)\(21\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!78\)\( T^{4} + \)\(94\!\cdots\!44\)\( T^{5} + \)\(84\!\cdots\!46\)\( T^{6} + \)\(16\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!21\)\( T^{8} + \)\(27\!\cdots\!28\)\( T^{9} + \)\(16\!\cdots\!07\)\( T^{10} \))(\( 1 + \)\(11\!\cdots\!00\)\( T + \)\(98\!\cdots\!10\)\( T^{2} + \)\(79\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!47\)\( T^{4} + \)\(36\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(99\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!03\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(57\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{11} + \)\(45\!\cdots\!49\)\( T^{12} \))
$47$ (\( 1 - \)\(10\!\cdots\!56\)\( T + \)\(34\!\cdots\!03\)\( T^{2} - \)\(33\!\cdots\!40\)\( T^{3} + \)\(62\!\cdots\!58\)\( T^{4} - \)\(48\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!54\)\( T^{6} - \)\(39\!\cdots\!60\)\( T^{7} + \)\(42\!\cdots\!41\)\( T^{8} - \)\(14\!\cdots\!16\)\( T^{9} + \)\(14\!\cdots\!43\)\( T^{10} \))(\( 1 - \)\(58\!\cdots\!60\)\( T + \)\(21\!\cdots\!70\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!20\)\( T^{5} + \)\(37\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{7} + \)\(38\!\cdots\!83\)\( T^{8} - \)\(30\!\cdots\!60\)\( T^{9} + \)\(29\!\cdots\!70\)\( T^{10} - \)\(82\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!09\)\( T^{12} \))
$53$ (\( 1 - \)\(32\!\cdots\!66\)\( T + \)\(94\!\cdots\!17\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(31\!\cdots\!62\)\( T^{4} + \)\(93\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!94\)\( T^{6} + \)\(32\!\cdots\!16\)\( T^{7} + \)\(35\!\cdots\!01\)\( T^{8} - \)\(41\!\cdots\!26\)\( T^{9} + \)\(42\!\cdots\!57\)\( T^{10} \))(\( 1 + \)\(13\!\cdots\!36\)\( T + \)\(21\!\cdots\!74\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(76\!\cdots\!08\)\( T^{6} + \)\(38\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!59\)\( T^{8} + \)\(72\!\cdots\!04\)\( T^{9} + \)\(27\!\cdots\!14\)\( T^{10} + \)\(58\!\cdots\!52\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \))
$59$ (\( 1 + \)\(17\!\cdots\!12\)\( T + \)\(94\!\cdots\!51\)\( T^{2} + \)\(85\!\cdots\!32\)\( T^{3} + \)\(66\!\cdots\!02\)\( T^{4} + \)\(77\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!38\)\( T^{6} + \)\(16\!\cdots\!52\)\( T^{7} + \)\(82\!\cdots\!09\)\( T^{8} + \)\(69\!\cdots\!52\)\( T^{9} + \)\(17\!\cdots\!99\)\( T^{10} \))(\( 1 - \)\(43\!\cdots\!12\)\( T + \)\(22\!\cdots\!62\)\( T^{2} - \)\(61\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!63\)\( T^{4} - \)\(44\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!92\)\( T^{7} + \)\(40\!\cdots\!43\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{9} + \)\(87\!\cdots\!02\)\( T^{10} - \)\(74\!\cdots\!88\)\( T^{11} + \)\(76\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 - \)\(58\!\cdots\!02\)\( T + \)\(19\!\cdots\!17\)\( T^{2} - \)\(90\!\cdots\!92\)\( T^{3} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(54\!\cdots\!72\)\( T^{5} + \)\(63\!\cdots\!42\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(13\!\cdots\!37\)\( T^{8} - \)\(17\!\cdots\!62\)\( T^{9} + \)\(12\!\cdots\!01\)\( T^{10} \))(\( 1 + \)\(30\!\cdots\!40\)\( T + \)\(11\!\cdots\!74\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(11\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!94\)\( T^{10} + \)\(36\!\cdots\!40\)\( T^{11} + \)\(50\!\cdots\!21\)\( T^{12} \))
$67$ (\( 1 - \)\(13\!\cdots\!20\)\( T + \)\(25\!\cdots\!35\)\( T^{2} + \)\(41\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!10\)\( T^{4} - \)\(65\!\cdots\!96\)\( T^{5} + \)\(17\!\cdots\!30\)\( T^{6} + \)\(20\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!45\)\( T^{8} - \)\(32\!\cdots\!20\)\( T^{9} + \)\(54\!\cdots\!43\)\( T^{10} \))(\( 1 + \)\(36\!\cdots\!96\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(32\!\cdots\!55\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(84\!\cdots\!56\)\( T^{6} - \)\(62\!\cdots\!52\)\( T^{7} + \)\(16\!\cdots\!95\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(20\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!28\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \))
$71$ (\( 1 - \)\(20\!\cdots\!80\)\( T + \)\(27\!\cdots\!55\)\( T^{2} - \)\(44\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!30\)\( T^{4} - \)\(53\!\cdots\!52\)\( T^{5} + \)\(59\!\cdots\!30\)\( T^{6} - \)\(51\!\cdots\!80\)\( T^{7} + \)\(34\!\cdots\!05\)\( T^{8} - \)\(28\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} \))(\( 1 - \)\(25\!\cdots\!72\)\( T + \)\(73\!\cdots\!06\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!95\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{5} + \)\(29\!\cdots\!44\)\( T^{6} - \)\(25\!\cdots\!72\)\( T^{7} + \)\(23\!\cdots\!95\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!66\)\( T^{10} - \)\(37\!\cdots\!72\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} \))
$73$ (\( 1 + \)\(25\!\cdots\!34\)\( T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(33\!\cdots\!04\)\( T^{3} + \)\(38\!\cdots\!22\)\( T^{4} - \)\(21\!\cdots\!36\)\( T^{5} + \)\(26\!\cdots\!34\)\( T^{6} + \)\(16\!\cdots\!36\)\( T^{7} + \)\(56\!\cdots\!81\)\( T^{8} + \)\(59\!\cdots\!54\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} \))(\( 1 - \)\(16\!\cdots\!24\)\( T + \)\(21\!\cdots\!14\)\( T^{2} - \)\(26\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!11\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!28\)\( T^{6} - \)\(20\!\cdots\!24\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} - \)\(89\!\cdots\!16\)\( T^{9} + \)\(50\!\cdots\!34\)\( T^{10} - \)\(27\!\cdots\!68\)\( T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \))
$79$ (\( 1 + \)\(34\!\cdots\!80\)\( T + \)\(46\!\cdots\!95\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(93\!\cdots\!10\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!90\)\( T^{6} + \)\(19\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!05\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} + \)\(50\!\cdots\!99\)\( T^{10} \))(\( 1 + \)\(47\!\cdots\!40\)\( T + \)\(66\!\cdots\!54\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!15\)\( T^{4} + \)\(55\!\cdots\!00\)\( T^{5} + \)\(35\!\cdots\!80\)\( T^{6} + \)\(77\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!15\)\( T^{8} + \)\(61\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!94\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} + \)\(70\!\cdots\!41\)\( T^{12} \))
$83$ (\( 1 + \)\(27\!\cdots\!36\)\( T + \)\(10\!\cdots\!31\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{3} + \)\(54\!\cdots\!22\)\( T^{4} + \)\(70\!\cdots\!76\)\( T^{5} + \)\(20\!\cdots\!94\)\( T^{6} + \)\(14\!\cdots\!12\)\( T^{7} + \)\(59\!\cdots\!73\)\( T^{8} + \)\(56\!\cdots\!76\)\( T^{9} + \)\(77\!\cdots\!07\)\( T^{10} \))(\( 1 - \)\(36\!\cdots\!32\)\( T + \)\(21\!\cdots\!74\)\( T^{2} - \)\(63\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} - \)\(45\!\cdots\!64\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} - \)\(17\!\cdots\!28\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} - \)\(34\!\cdots\!48\)\( T^{9} + \)\(43\!\cdots\!34\)\( T^{10} - \)\(28\!\cdots\!24\)\( T^{11} + \)\(29\!\cdots\!89\)\( T^{12} \))
$89$ (\( 1 + \)\(95\!\cdots\!38\)\( T + \)\(53\!\cdots\!81\)\( T^{2} + \)\(20\!\cdots\!08\)\( T^{3} + \)\(60\!\cdots\!42\)\( T^{4} + \)\(13\!\cdots\!48\)\( T^{5} + \)\(24\!\cdots\!18\)\( T^{6} + \)\(34\!\cdots\!28\)\( T^{7} + \)\(36\!\cdots\!09\)\( T^{8} + \)\(26\!\cdots\!78\)\( T^{9} + \)\(11\!\cdots\!49\)\( T^{10} \))(\( 1 + \)\(24\!\cdots\!92\)\( T + \)\(88\!\cdots\!02\)\( T^{2} - \)\(17\!\cdots\!96\)\( T^{3} + \)\(51\!\cdots\!03\)\( T^{4} + \)\(14\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!88\)\( T^{6} + \)\(60\!\cdots\!52\)\( T^{7} + \)\(85\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!62\)\( T^{10} + \)\(27\!\cdots\!08\)\( T^{11} + \)\(45\!\cdots\!21\)\( T^{12} \))
$97$ (\( 1 + \)\(11\!\cdots\!90\)\( T + \)\(91\!\cdots\!45\)\( T^{2} + \)\(52\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!10\)\( T^{4} + \)\(99\!\cdots\!64\)\( T^{5} + \)\(32\!\cdots\!30\)\( T^{6} + \)\(89\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!65\)\( T^{8} + \)\(34\!\cdots\!90\)\( T^{9} + \)\(37\!\cdots\!93\)\( T^{10} \))(\( 1 - \)\(38\!\cdots\!64\)\( T + \)\(36\!\cdots\!18\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!55\)\( T^{4} - \)\(40\!\cdots\!24\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(52\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!95\)\( T^{8} - \)\(43\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!98\)\( T^{10} - \)\(14\!\cdots\!52\)\( T^{11} + \)\(48\!\cdots\!09\)\( T^{12} \))
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