Properties

Label 1.122.a
Level 1
Weight 122
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 122 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{122}(\Gamma_0(1))\).

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

Trace form

\( 9q - 2345005562254519728q^{2} - 45298662406895255983997747004q^{3} + 8462901481545657250924227818768095488q^{4} - 1892628936004587162449482404208025942059050q^{5} - 169155345437237254853213475765636376857034060992q^{6} + 2195742607988216453868262401462060909846066161833992q^{7} - 6498505205013969366907252193209867920059782603760087040q^{8} + 7945503363883566160952357352391639075761121301070127960917q^{9} + O(q^{10}) \) \( 9q - 2345005562254519728q^{2} - 45298662406895255983997747004q^{3} + 8462901481545657250924227818768095488q^{4} - 1892628936004587162449482404208025942059050q^{5} - 169155345437237254853213475765636376857034060992q^{6} + 2195742607988216453868262401462060909846066161833992q^{7} - 6498505205013969366907252193209867920059782603760087040q^{8} + 7945503363883566160952357352391639075761121301070127960917q^{9} - 1722648625287605771733277740799503204814238573365164973293600q^{10} - 946247437041242628794010189459909585820203760454440287613552692q^{11} + 205846961050281878415699597946796712976603670706873231298938012672q^{12} + 20596127771705642483605985955370331334854065669132980339746788779486q^{13} - 6541972694294346334011779110963479954883365352702523229035305330483584q^{14} + 94761650294240619608372286408346994746207880434327141095820600145938200q^{15} + 15125444512952621416063439206177933863772762787209602627890080570611400704q^{16} + 291951493650412549850834598805724750617640895870041885121831690800331035682q^{17} - 2554000175718076948380008722939707666960273565314048023897922364658130426864q^{18} + 30244684445956612332670117612595525326891015746649482369862884848315284461780q^{19} + 4562951142663369182841621952582280345228980283169468426672463108421372089382400q^{20} + 35146001071237823021693437259813836720201207139254364736561723878685716530560288q^{21} - 1908843328157955617823816866863990801015534541375661383005747627202736697921398336q^{22} - 66547277273309692585260941575189421012052896396341670760157778423156238686357058024q^{23} - 752736756299124610400627977416074789098074869027579302337966622137134625951047925760q^{24} - 6063954929776831553687695824368630787209822932539681432479202799269060736255468175625q^{25} - 68842531925347949534946832128658250301897561117051749502543801676148842557574912147872q^{26} - 599094982712902977137778651999579486037062432725587105690436473582035154107389795716440q^{27} + 14581457392452701650088858542793969027985828891624009507886840217572255356341773965625344q^{28} + 9039340472411915035494616975186209221844392206687458522714064361334880069082093346905070q^{29} + 224260392412850366562482895689598093094206490242486949280006808962977358350861560270998400q^{30} - 1034214344782654758430975917538809236085530199413694042134266190861671432181122576926086112q^{31} - 32504525000929540761398714812123921617256100597234986820033520241386493923631009832383807488q^{32} + 189949463814086324300823839088808170132383689196162349649439549175970470599919775108948927152q^{33} + 939230388337351560267419962761669652735221862277825209065568487608538630630957599920217170336q^{34} + 1343144021786280211948320998755385153610555228046229864187689603416027546078320739943060612400q^{35} - 21747907752251375975427518389059436655823261956750397535228718126776457105689705407608776563456q^{36} - 88515341604947210488499310221844243003334043790663530401335727647981800120006279439706998268938q^{37} + 126431143256035943119117270678428356969446546517968603954366283120911078870917992997220500555840q^{38} + 217837571218652472832776380975246155068206021757529611026730729062256423563336349872452629634104q^{39} + 11699272454320878743488987607870545498317393987884112075950101149279048493373249852479799479296000q^{40} - 36266094262194860707738000735265088509116494488735889525314316314221341311667592777599187406913222q^{41} - 24787352510041822155108090768801766655788422577097156255869676682213614031202393800364112862967296q^{42} - 232370696794750496427815530982465444104722554898177200564202440384179721532324359363059913691265044q^{43} + 7755327422511594524608603055387018294701513657236683851986386901343418513803274064133364624004320256q^{44} + 3247131420778261129864754583862668667744851710389803065112927478198766013287651133195234793708696350q^{45} - 32066987947547067490416916557196164509339083485720964163476500441257367761070295505033101035894375552q^{46} - 132970274906173452814801214841116212222140439777467383504750807710754558312919347426848207706456695248q^{47} + 1792993240520940040285669530334719652675802655317197830920369780797223163120464508596735702234386661376q^{48} + 3971615803493637002463922107037659765667323348806045910695714957649057905572825978452768033947465237313q^{49} + 7697223557528977921454596784007095645216064886088735712319363716221302227247960859603436865144030830000q^{50} - 12851187227364644420645646585565612980157473692407810028414511867222315540665892049707079694243262493752q^{51} + 41491246868006007350968584983945986180424656334954442495103497331692372835472248671859508511281974820352q^{52} + 10921805138133777254090784536986505203689729704788186882887817145907729071674310284399320645165776125446q^{53} + 1339620561670595781535503143383501960027220441835890399944886982813214108542142442500405331111067486871680q^{54} - 1312854241531753321574015681587452449577731960880104498235108054018717321131599152359037095554839302748600q^{55} - 14013489107783484379024464711764338924776495535033134010566539603721944518516633929838564532681379280158720q^{56} - 31019370720452892632999266840260901115398140535819179494909299925851052488218537305068762474072132809464880q^{57} + 103566512036259767730142842802003772713045020405103287191675134464175461494809359502286741327442998854097760q^{58} - 212654227372156556243821221093344570156200984495183110384877226881440240904587127248726963633402442388397060q^{59} - 422775085570641982781661820780199899688061805673956240019260238835680896536027096304091002230119430523545600q^{60} + 1192163308009290988314718399559245385155131325092676542388200473202899841355909578157895873548972217626937358q^{61} + 1941577256261942079284748865266177577315768005250991995768914948259042107075822164327427075406958896697915904q^{62} + 451398588142208697776529182978564850520004037586377967283679997710918662905240621028803130635870486918043496q^{63} + 58844915592066748224918413663360071934978670807229624180238002804297923325840628846335879209980431060401389568q^{64} - 41307677239725606429137681882540847206257959638235776751518537901187469657040797622415542482923664688418764300q^{65} - 222086041679950312339781078110357023370750120852496118683208049646663973764702391499379128192551147975013941504q^{66} - 136956255883591433626896738276452320541908670428347212980145870863327851773988143589420054081089733226508975868q^{67} - 141094811799092295974464545753546540483126567546619661037683966526498908594653379053568196980211372687791832576q^{68} + 1325703171238765045361423824776857588261240408135555321965899246235283904667578177535332268855352594559670964064q^{69} + 251934589370384384494289262861906043864096277276788575857738394556055301564200175196895771626792095349002028800q^{70} - 21585422985838182545762327030440661631324036114582384867758341692032917773329287240302446531581657134820891213752q^{71} - 5181834258416339630015764026055527646506354416168007949107924953264135981412995982436721696529253639785101291520q^{72} + 91235460780032725960652165272631372912505283323159668796891496309162099636399114489032757775807008811684035374426q^{73} + 310562981122696891225427716699672479798427405954878722327169646078571868354993002411909262537623251852286369642976q^{74} + 43065854506087315183339457241248482679331791698024061594158993211332750535581795467446263657604087829943816477500q^{75} - 1628928259080846394930799553567522291541981414122456953774556713740025206052658836539555750192733076653498088483840q^{76} + 593398979324677723779464865172133117018002924612325496238799373091215617849415550444705675974821710181802828350304q^{77} - 6873235601534959334329744558974767766222222958184019979663697300957230011165665645661869140437348638585389730736768q^{78} + 12451904914912678516471799458709519750612136226261564829575149335945412565470277464754582893898189772426218896816720q^{79} - 20416950898306426146345946546741629794844904073553249082791509645594723097676639519064500446972597103490830240972800q^{80} - 40856231575215745746855879250733371261216334279125203008573307569275978049909264916389709598337529820618984078592511q^{81} + 32147344949534769988675834932358838024355394065520493232144096322964430678203949090265848270504974541383740855029024q^{82} - 92550876115215159531985566894207205907616571001151579100140340424113698287990359504129121116969620392280333900687084q^{83} + 149786622283872863814961679878718479671325736759801797089969582016734283953793144464030325178406284102124769637769216q^{84} - 457768087179221548903179340844196170221647881541175735561813412251720733334690809762523483946456540131705087221018100q^{85} - 2597491864434159334587628292949145911098969463321219405661773952780499099399232684938429948876042525744324616394370112q^{86} - 6860132173224607170538777320017488364980901889195600241683572931313063475309515779498812905462751134050092231747965320q^{87} - 20006125934411614149040201757441776841868885955363484760412796241321136345672966303804893860326010500704122960860692480q^{88} - 6037951059942933692255237205803554193360594969996068937834548367886153095083618336696729724512143193617265723073840790q^{89} - 63101356532716899775688972110257991639639781786779198684055149646555511091876460642986977143928184313177571126402448800q^{90} - 123641023753197596743476339553493022736411481279426005005225341366808933988856782685240985478245146307872825619974888592q^{91} - 364595199967728361741013600374859006514622011131757842452690960184887037990775378757973866446814260984048497269830739968q^{92} - 660210659270707805852573362768494443983824981820503945313596568722380303376948547337519220648409914639203450549706054528q^{93} - 1077366499492799036766200119308272598652452209354508770841202268675762751723923474906601231339334467643845894468443138304q^{94} - 1952811193153518003415269978917451688485843585663036733937657231131838392410846728976735534304153554615423725772929253000q^{95} - 7973910491229034017269638806130805343748566540122585079441423327960922891988630774830062893111316529474837792857136300032q^{96} - 10257092292553957046050140760321382189656293894451103156433186039799204062322028715705878067672306955149219750865890726798q^{97} - 27691496134014026270502117819887007317432773944983420293676841151578333911041338924275151210232199410900266664836266600496q^{98} - 44066255828766064795825662460182521807738019967004038392602001105306909448583819181101964434724044331056342593475150607396q^{99} + O(q^{100}) \)

Decomposition of \(S_{122}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.122.a.a \(9\) \(92.717\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-2\!\cdots\!28\) \(-4\!\cdots\!04\) \(-1\!\cdots\!50\) \(21\!\cdots\!92\) \(+\) \(q+(-260556173583835525+\beta _{1}+\cdots)q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2345005562254519728 T + \)\(10\!\cdots\!32\)\( T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!76\)\( T^{4} + \)\(95\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!92\)\( T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(57\!\cdots\!16\)\( T^{8} + \)\(96\!\cdots\!88\)\( T^{9} + \)\(15\!\cdots\!32\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(36\!\cdots\!36\)\( T^{12} + \)\(47\!\cdots\!68\)\( T^{13} + \)\(72\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!60\)\( T^{15} + \)\(98\!\cdots\!96\)\( T^{16} + \)\(58\!\cdots\!68\)\( T^{17} + \)\(66\!\cdots\!12\)\( T^{18} \)
$3$ \( 1 + \)\(45\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!16\)\( T^{4} + \)\(13\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!46\)\( T^{8} + \)\(65\!\cdots\!44\)\( T^{9} + \)\(81\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!20\)\( T^{11} + \)\(34\!\cdots\!56\)\( T^{12} + \)\(11\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!60\)\( T^{15} + \)\(28\!\cdots\!81\)\( T^{16} + \)\(32\!\cdots\!44\)\( T^{17} + \)\(38\!\cdots\!83\)\( T^{18} \)
$5$ \( 1 + \)\(18\!\cdots\!50\)\( T + \)\(21\!\cdots\!25\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!00\)\( T^{4} + \)\(52\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(75\!\cdots\!50\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(28\!\cdots\!50\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{11} + \)\(81\!\cdots\!00\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!25\)\( T^{16} + \)\(75\!\cdots\!50\)\( T^{17} + \)\(15\!\cdots\!25\)\( T^{18} \)
$7$ \( 1 - \)\(21\!\cdots\!92\)\( T + \)\(85\!\cdots\!07\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!96\)\( T^{4} - \)\(38\!\cdots\!32\)\( T^{5} + \)\(68\!\cdots\!72\)\( T^{6} - \)\(69\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{9} + \)\(21\!\cdots\!42\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} + \)\(40\!\cdots\!96\)\( T^{12} - \)\(41\!\cdots\!32\)\( T^{13} + \)\(59\!\cdots\!72\)\( T^{14} - \)\(48\!\cdots\!00\)\( T^{15} + \)\(53\!\cdots\!01\)\( T^{16} - \)\(24\!\cdots\!92\)\( T^{17} + \)\(20\!\cdots\!07\)\( T^{18} \)
$11$ \( 1 + \)\(94\!\cdots\!92\)\( T + \)\(52\!\cdots\!83\)\( T^{2} + \)\(54\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!60\)\( T^{4} + \)\(14\!\cdots\!32\)\( T^{5} + \)\(30\!\cdots\!84\)\( T^{6} + \)\(25\!\cdots\!36\)\( T^{7} + \)\(41\!\cdots\!78\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{9} + \)\(42\!\cdots\!58\)\( T^{10} + \)\(26\!\cdots\!56\)\( T^{11} + \)\(31\!\cdots\!04\)\( T^{12} + \)\(15\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!60\)\( T^{14} + \)\(61\!\cdots\!84\)\( T^{15} + \)\(60\!\cdots\!93\)\( T^{16} + \)\(11\!\cdots\!52\)\( T^{17} + \)\(11\!\cdots\!91\)\( T^{18} \)
$13$ \( 1 - \)\(20\!\cdots\!86\)\( T + \)\(30\!\cdots\!33\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(48\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} - \)\(98\!\cdots\!76\)\( T^{9} + \)\(22\!\cdots\!18\)\( T^{10} - \)\(50\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!16\)\( T^{12} - \)\(17\!\cdots\!96\)\( T^{13} + \)\(40\!\cdots\!48\)\( T^{14} - \)\(39\!\cdots\!80\)\( T^{15} + \)\(99\!\cdots\!61\)\( T^{16} - \)\(40\!\cdots\!06\)\( T^{17} + \)\(12\!\cdots\!73\)\( T^{18} \)
$17$ \( 1 - \)\(29\!\cdots\!82\)\( T + \)\(40\!\cdots\!57\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(83\!\cdots\!36\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(30\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!32\)\( T^{9} + \)\(91\!\cdots\!62\)\( T^{10} - \)\(17\!\cdots\!40\)\( T^{11} + \)\(52\!\cdots\!16\)\( T^{12} - \)\(84\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!52\)\( T^{14} - \)\(25\!\cdots\!20\)\( T^{15} + \)\(62\!\cdots\!61\)\( T^{16} - \)\(34\!\cdots\!62\)\( T^{17} + \)\(90\!\cdots\!97\)\( T^{18} \)
$19$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(22\!\cdots\!71\)\( T^{2} - \)\(28\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!96\)\( T^{4} - \)\(46\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!56\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!46\)\( T^{8} - \)\(26\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!04\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{13} + \)\(12\!\cdots\!04\)\( T^{14} - \)\(66\!\cdots\!60\)\( T^{15} + \)\(29\!\cdots\!69\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{17} + \)\(36\!\cdots\!79\)\( T^{18} \)
$23$ \( 1 + \)\(66\!\cdots\!24\)\( T + \)\(47\!\cdots\!03\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!56\)\( T^{4} + \)\(23\!\cdots\!84\)\( T^{5} + \)\(67\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!26\)\( T^{8} + \)\(93\!\cdots\!04\)\( T^{9} + \)\(23\!\cdots\!98\)\( T^{10} + \)\(56\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!76\)\( T^{12} + \)\(28\!\cdots\!44\)\( T^{13} + \)\(55\!\cdots\!08\)\( T^{14} + \)\(81\!\cdots\!80\)\( T^{15} + \)\(11\!\cdots\!41\)\( T^{16} + \)\(94\!\cdots\!44\)\( T^{17} + \)\(83\!\cdots\!63\)\( T^{18} \)
$29$ \( 1 - \)\(90\!\cdots\!70\)\( T + \)\(40\!\cdots\!61\)\( T^{2} - \)\(58\!\cdots\!40\)\( T^{3} + \)\(82\!\cdots\!76\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(27\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} - \)\(21\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{13} + \)\(46\!\cdots\!24\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(18\!\cdots\!49\)\( T^{16} - \)\(36\!\cdots\!70\)\( T^{17} + \)\(35\!\cdots\!69\)\( T^{18} \)
$31$ \( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!60\)\( T^{4} + \)\(11\!\cdots\!32\)\( T^{5} + \)\(51\!\cdots\!24\)\( T^{6} + \)\(55\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!18\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(52\!\cdots\!58\)\( T^{10} + \)\(44\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!84\)\( T^{12} + \)\(75\!\cdots\!72\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} + \)\(82\!\cdots\!84\)\( T^{15} + \)\(22\!\cdots\!73\)\( T^{16} + \)\(44\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!71\)\( T^{18} \)
$37$ \( 1 + \)\(88\!\cdots\!38\)\( T + \)\(28\!\cdots\!57\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(36\!\cdots\!16\)\( T^{4} + \)\(16\!\cdots\!88\)\( T^{5} + \)\(32\!\cdots\!52\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{7} + \)\(23\!\cdots\!46\)\( T^{8} + \)\(88\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!02\)\( T^{10} + \)\(41\!\cdots\!80\)\( T^{11} + \)\(58\!\cdots\!56\)\( T^{12} + \)\(16\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!12\)\( T^{14} + \)\(55\!\cdots\!40\)\( T^{15} + \)\(53\!\cdots\!81\)\( T^{16} + \)\(92\!\cdots\!98\)\( T^{17} + \)\(59\!\cdots\!77\)\( T^{18} \)
$41$ \( 1 + \)\(36\!\cdots\!22\)\( T + \)\(60\!\cdots\!73\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!60\)\( T^{4} + \)\(81\!\cdots\!32\)\( T^{5} + \)\(56\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!38\)\( T^{8} + \)\(30\!\cdots\!80\)\( T^{9} + \)\(14\!\cdots\!58\)\( T^{10} + \)\(36\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!24\)\( T^{12} + \)\(31\!\cdots\!52\)\( T^{13} + \)\(12\!\cdots\!60\)\( T^{14} + \)\(18\!\cdots\!84\)\( T^{15} + \)\(64\!\cdots\!13\)\( T^{16} + \)\(54\!\cdots\!62\)\( T^{17} + \)\(20\!\cdots\!61\)\( T^{18} \)
$43$ \( 1 + \)\(23\!\cdots\!44\)\( T + \)\(29\!\cdots\!43\)\( T^{2} + \)\(59\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!96\)\( T^{4} + \)\(73\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!28\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!06\)\( T^{8} + \)\(30\!\cdots\!64\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{11} + \)\(33\!\cdots\!96\)\( T^{12} + \)\(29\!\cdots\!24\)\( T^{13} + \)\(75\!\cdots\!28\)\( T^{14} + \)\(46\!\cdots\!00\)\( T^{15} + \)\(10\!\cdots\!01\)\( T^{16} + \)\(36\!\cdots\!44\)\( T^{17} + \)\(70\!\cdots\!43\)\( T^{18} \)
$47$ \( 1 + \)\(13\!\cdots\!48\)\( T + \)\(10\!\cdots\!07\)\( T^{2} + \)\(80\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!56\)\( T^{4} + \)\(98\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!12\)\( T^{6} - \)\(41\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!26\)\( T^{8} - \)\(16\!\cdots\!72\)\( T^{9} + \)\(51\!\cdots\!22\)\( T^{10} - \)\(18\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(19\!\cdots\!68\)\( T^{13} + \)\(19\!\cdots\!92\)\( T^{14} + \)\(70\!\cdots\!20\)\( T^{15} + \)\(19\!\cdots\!41\)\( T^{16} + \)\(51\!\cdots\!28\)\( T^{17} + \)\(82\!\cdots\!67\)\( T^{18} \)
$53$ \( 1 - \)\(10\!\cdots\!46\)\( T + \)\(22\!\cdots\!13\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!16\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(34\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!46\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{9} + \)\(38\!\cdots\!38\)\( T^{10} - \)\(64\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!56\)\( T^{12} - \)\(13\!\cdots\!36\)\( T^{13} + \)\(37\!\cdots\!88\)\( T^{14} - \)\(19\!\cdots\!40\)\( T^{15} + \)\(65\!\cdots\!81\)\( T^{16} - \)\(13\!\cdots\!06\)\( T^{17} + \)\(54\!\cdots\!33\)\( T^{18} \)
$59$ \( 1 + \)\(21\!\cdots\!60\)\( T + \)\(10\!\cdots\!31\)\( T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!16\)\( T^{4} + \)\(70\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} + \)\(38\!\cdots\!00\)\( T^{9} + \)\(69\!\cdots\!74\)\( T^{10} + \)\(65\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!44\)\( T^{12} + \)\(86\!\cdots\!80\)\( T^{13} + \)\(11\!\cdots\!84\)\( T^{14} + \)\(75\!\cdots\!20\)\( T^{15} + \)\(84\!\cdots\!89\)\( T^{16} + \)\(32\!\cdots\!60\)\( T^{17} + \)\(28\!\cdots\!39\)\( T^{18} \)
$61$ \( 1 - \)\(11\!\cdots\!58\)\( T + \)\(45\!\cdots\!33\)\( T^{2} - \)\(66\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} - \)\(23\!\cdots\!64\)\( T^{7} + \)\(28\!\cdots\!78\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!58\)\( T^{10} - \)\(26\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!04\)\( T^{12} - \)\(20\!\cdots\!88\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} - \)\(94\!\cdots\!16\)\( T^{15} + \)\(68\!\cdots\!93\)\( T^{16} - \)\(18\!\cdots\!98\)\( T^{17} + \)\(16\!\cdots\!41\)\( T^{18} \)
$67$ \( 1 + \)\(13\!\cdots\!68\)\( T + \)\(32\!\cdots\!07\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!36\)\( T^{4} - \)\(36\!\cdots\!92\)\( T^{5} + \)\(75\!\cdots\!32\)\( T^{6} - \)\(52\!\cdots\!60\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!32\)\( T^{9} + \)\(80\!\cdots\!62\)\( T^{10} - \)\(43\!\cdots\!40\)\( T^{11} + \)\(55\!\cdots\!16\)\( T^{12} - \)\(24\!\cdots\!32\)\( T^{13} + \)\(32\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!80\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} + \)\(59\!\cdots\!88\)\( T^{17} + \)\(39\!\cdots\!47\)\( T^{18} \)
$71$ \( 1 + \)\(21\!\cdots\!52\)\( T + \)\(81\!\cdots\!63\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(30\!\cdots\!60\)\( T^{4} + \)\(44\!\cdots\!32\)\( T^{5} + \)\(69\!\cdots\!04\)\( T^{6} + \)\(82\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!98\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!58\)\( T^{10} + \)\(83\!\cdots\!36\)\( T^{11} + \)\(70\!\cdots\!44\)\( T^{12} + \)\(45\!\cdots\!92\)\( T^{13} + \)\(31\!\cdots\!60\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(84\!\cdots\!33\)\( T^{16} + \)\(22\!\cdots\!72\)\( T^{17} + \)\(10\!\cdots\!31\)\( T^{18} \)
$73$ \( 1 - \)\(91\!\cdots\!26\)\( T + \)\(16\!\cdots\!53\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(93\!\cdots\!16\)\( T^{5} + \)\(76\!\cdots\!28\)\( T^{6} - \)\(45\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!96\)\( T^{9} + \)\(88\!\cdots\!98\)\( T^{10} - \)\(37\!\cdots\!40\)\( T^{11} + \)\(18\!\cdots\!76\)\( T^{12} - \)\(66\!\cdots\!56\)\( T^{13} + \)\(28\!\cdots\!08\)\( T^{14} - \)\(75\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!41\)\( T^{16} - \)\(45\!\cdots\!06\)\( T^{17} + \)\(14\!\cdots\!13\)\( T^{18} \)
$79$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(26\!\cdots\!11\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!76\)\( T^{6} - \)\(10\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} - \)\(51\!\cdots\!00\)\( T^{9} + \)\(44\!\cdots\!74\)\( T^{10} - \)\(18\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!64\)\( T^{12} - \)\(49\!\cdots\!60\)\( T^{13} + \)\(33\!\cdots\!24\)\( T^{14} - \)\(10\!\cdots\!40\)\( T^{15} + \)\(52\!\cdots\!49\)\( T^{16} - \)\(99\!\cdots\!20\)\( T^{17} + \)\(32\!\cdots\!19\)\( T^{18} \)
$83$ \( 1 + \)\(92\!\cdots\!84\)\( T + \)\(79\!\cdots\!23\)\( T^{2} + \)\(51\!\cdots\!60\)\( T^{3} + \)\(32\!\cdots\!76\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(87\!\cdots\!28\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!66\)\( T^{8} + \)\(53\!\cdots\!84\)\( T^{9} + \)\(28\!\cdots\!78\)\( T^{10} + \)\(81\!\cdots\!80\)\( T^{11} + \)\(37\!\cdots\!36\)\( T^{12} + \)\(10\!\cdots\!84\)\( T^{13} + \)\(35\!\cdots\!68\)\( T^{14} + \)\(91\!\cdots\!40\)\( T^{15} + \)\(22\!\cdots\!21\)\( T^{16} + \)\(43\!\cdots\!44\)\( T^{17} + \)\(75\!\cdots\!03\)\( T^{18} \)
$89$ \( 1 + \)\(60\!\cdots\!90\)\( T + \)\(32\!\cdots\!01\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!56\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!96\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(84\!\cdots\!66\)\( T^{8} - \)\(27\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!24\)\( T^{12} + \)\(48\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!44\)\( T^{14} + \)\(23\!\cdots\!80\)\( T^{15} + \)\(44\!\cdots\!29\)\( T^{16} + \)\(61\!\cdots\!90\)\( T^{17} + \)\(76\!\cdots\!09\)\( T^{18} \)
$97$ \( 1 + \)\(10\!\cdots\!98\)\( T + \)\(63\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!56\)\( T^{4} + \)\(29\!\cdots\!28\)\( T^{5} + \)\(74\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!26\)\( T^{8} + \)\(51\!\cdots\!28\)\( T^{9} + \)\(77\!\cdots\!22\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!76\)\( T^{12} + \)\(11\!\cdots\!68\)\( T^{13} + \)\(10\!\cdots\!92\)\( T^{14} + \)\(70\!\cdots\!20\)\( T^{15} + \)\(39\!\cdots\!41\)\( T^{16} + \)\(16\!\cdots\!78\)\( T^{17} + \)\(39\!\cdots\!17\)\( T^{18} \)
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