Properties

Label 2.82.a
Level 2
Weight 82
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 2
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 82 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{82}(\Gamma_0(2))\).

Total New Old
Modular forms 21 7 14
Cusp forms 19 7 12
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.

\(2\)Dim.
\(+\)\(3\)
\(-\)\(4\)

Trace form

\( 7q + 1099511627776q^{2} + 5492623293753256284q^{3} + 8462480737302404222943232q^{4} + 3930103990742699543615489130q^{5} - 21678079061476108854899603668992q^{6} + 17840124989511368450810570225392568q^{7} + 1329227995784915872903807060280344576q^{8} + 895258546468549738568590790199579496731q^{9} + O(q^{10}) \) \( 7q + 1099511627776q^{2} + 5492623293753256284q^{3} + 8462480737302404222943232q^{4} + 3930103990742699543615489130q^{5} - 21678079061476108854899603668992q^{6} + 17840124989511368450810570225392568q^{7} + 1329227995784915872903807060280344576q^{8} + 895258546468549738568590790199579496731q^{9} + 50463440699837666565174903814070395207680q^{10} + 4574205974746866887301110535567689797851924q^{11} + 6640174117235059459063078612734728925609984q^{12} + 364156433571445284139262691120632802372144674q^{13} + 20830971491787782972327917197359525676676284416q^{14} - 288696232916218955954253101780543418545348888280q^{15} + 10230511461316320427425793829013981137591527800832q^{16} + 30797813462159183585571851099038617318785664498558q^{17} + 1243365571050882738535442992934696278613147598716928q^{18} + 10552219769880013876700985537471230307761596684415180q^{19} + 4751204188179343036375915600360129120490090271866880q^{20} - 534204395395054602762771158030159951278109283209553696q^{21} + 2213095756532053785240759362683971572702330104213143552q^{22} - 10513722214019385471431306716226633418248837618951393496q^{23} - 26207189497065736089112950693849540407165504893162094592q^{24} + 1014189466564068690368694254083309919300775340818247344025q^{25} + 846912971011690213566605075357303729752095555778833809408q^{26} - 41047056863228843015433175713593563710927499432759533085800q^{27} + 21567387724972458813165328982714265401134825135065654099968q^{28} + 111326085425145818041443197251224378309458208297450456511890q^{29} - 885918882743471050999671162877544194731846979137678323220480q^{30} + 569235729037298103337199475119942454463628025305311407367904q^{31} + 1606938044258990275541962092341162602522202993782792835301376q^{32} - 61496338392290018712116602303159393277177362460967281584608432q^{33} + 3718529020543483372117866554887703376895140544748422447497216q^{34} + 686063792178605252855004712321340223103017825503448653988751440q^{35} + 1082301172056493072005884950823187296448920008772088480877510656q^{36} + 1905873955549544706440129014763384940073631587560832679296783178q^{37} + 1057413629544062188578721654616793655668978257175191520605634560q^{38} + 55341897122822588861691194763490464172205587304131594611109087752q^{39} + 61006556408625487129528265106360708571198708550074892262498631680q^{40} + 557515048551694176466040728107302945533850171681227845833793371814q^{41} + 555163309707595815805961702714236316200515286126475176926995546112q^{42} + 4816630411633048565398768161252558622962223074904367325846509636084q^{43} + 5529875707106989812609201886273374147571497497532759933118656282624q^{44} + 59280547321194031305904576843492222218648695345165516645244087774690q^{45} + 54763300416573635501069024199006675365149081237125318433109912322048q^{46} + 231600947615420523849173663516598760086361299122810432076865700336208q^{47} + 8027477937062241009643024641139128166341580904865027845844772061184q^{48} + 488884314080089494567910985806986572998097715062788175105646760287759q^{49} + 497356219041719558041226153518799564426131705030743706023953327718400q^{50} - 1837331169353117595512341888565294661065129266300924338435898332834696q^{51} + 440238114923299753371707677529614312190829786646038517437070913306624q^{52} - 21528641216322884743608350078749181085164032178710898763780234892918086q^{53} - 43755715708510428229418018240591061767395048738154899626547797043445760q^{54} - 89816572570227989101559828878826497696785313927220351593090382298274440q^{55} + 25183099284078520120341186590139951248547525837939632864247704207753216q^{56} - 325340069796986008297665253801465753346468478251645125296159579689898960q^{57} + 75286071641180057909408657391764386614076857088308740887661460911554560q^{58} + 662728117869435253821040267452447193179435605293938374690618618632935780q^{59} - 349012329997895887088397976680482613704259662419279599200034977250017280q^{60} + 5048975939077047323128803569069390709396201365602898856556981048321479794q^{61} + 9116117924911410298568489050005822874983045648944636378308346515042598912q^{62} + 35173215480755526937203547670059135126186051786127848366226864546062716504q^{63} + 12367929453448690307083082505200429610792387278129332706851243409048338432q^{64} - 46970532294923265265023129333973302067057379269613739355602364732812844980q^{65} - 85577854048600869442930812288143040550866591698175368757526131512174444544q^{66} - 184322092783437411769758942053951188704733055756695883309417066466667239652q^{67} + 37232271882079251195560462656419419341118428617840476746502613144025694208q^{68} - 1054123015060350205925859853673196722512269353610573887874785135858940609888q^{69} - 996287423890495461170521495492361754626623104274349693454298030309071912960q^{70} - 132669540113241156337773373156585489641837182225722090893045827548208576136q^{71} + 1503136742063299860150000575820714377378196845590270384949419035128997347328q^{72} + 7696425638326647416396579150109019018025911343704474407389216781457420473734q^{73} + 4212064098262262325478369202498743646631535719551325155846923504531753926656q^{74} + 18977872456506974957673383259339807360208907159415673322444899002577185656100q^{75} + 12756850934055889436101651315354615554023393824186865510409514056674894151680q^{76} + 79234255678237437380795543050254678320307999826725923179432696830135390143136q^{77} - 19194055024117071479861721778884928595303790396409596723840106893964354781184q^{78} - 212618578002421286534586833449108430493711472521556413568669436503059830318800q^{79} + 5743853417351171108173897102142362498933090902160685763748905880182985850880q^{80} + 79959423980560659225591806521539208152890161349457691318726618762317783721167q^{81} - 81302500671107676733587921195547926095566395205746092567558139477877051097088q^{82} - 1790164754511083345892373552571735287262201429345682698674912994270538116665716q^{83} - 645813486544703820860776922234667779771591725210663437335867182260916894826496q^{84} + 1564580034919142525696449604971603614529377387637357241796475674956634730426740q^{85} + 3294852804201453056359711287948686818251054091134890322093338475113528057397248q^{86} + 19247758453282545358950677651684018818005971882317880274189014839018404283268040q^{87} + 2675468601351170940457271556225519303767276818916460328318668092982006908977152q^{88} + 18806926689964364108023787783339140105636748008665375255555474899453722318915830q^{89} + 23999109123248994341661836060153247826889034621429882658172676529941076865187840q^{90} + 24928419343167225592864366889445673788456676771135660868978914924860750956057104q^{91} - 12710310244783919270419517984655261594177748110585961976304586108996071557431296q^{92} - 64439148604228135905283217235236807481608325977029514498260568870448376088280192q^{93} - 104782899901035220448955601374807674550698541121210045778060945597865908716437504q^{94} - 52275812910445671050882146644756147065705406087550783931933553231946215194105400q^{95} - 31682548042536096350318607947502050425635048441430143704736754878624462318600192q^{96} - 688354148445678317962501387361884373903301824763889876908335398557166488547019602q^{97} + 351029176762994484562468308831559055488776837904037672844652650080466774672801792q^{98} + 1787417826570478154481491653017470632071111476329584129222745030630914564983314692q^{99} + O(q^{100}) \)

Decomposition of \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.82.a.a \(3\) \(83.100\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-3\!\cdots\!28\) \(12\!\cdots\!88\) \(-2\!\cdots\!50\) \(-5\!\cdots\!24\) \(+\) \(q-2^{40}q^{2}+(4201453557463404996+\cdots)q^{3}+\cdots\)
2.82.a.b \(4\) \(83.100\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(43\!\cdots\!04\) \(-7\!\cdots\!04\) \(24\!\cdots\!80\) \(18\!\cdots\!92\) \(-\) \(q+2^{40}q^{2}+(-1777934344659239676+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{82}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{82}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{82}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 1099511627776 T )^{3} \))(\( ( 1 - 1099511627776 T )^{4} \))
$3$ (\( 1 - 12604360672390214988 T + \)\(80\!\cdots\!57\)\( T^{2} - \)\(73\!\cdots\!64\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} - \)\(24\!\cdots\!92\)\( T^{5} + \)\(87\!\cdots\!27\)\( T^{6} \))(\( 1 + 7111737378636958704 T + \)\(40\!\cdots\!68\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(67\!\cdots\!20\)\( T^{5} + \)\(79\!\cdots\!12\)\( T^{6} + \)\(62\!\cdots\!08\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \))
$5$ (\( 1 + \)\(20\!\cdots\!50\)\( T + \)\(70\!\cdots\!75\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!75\)\( T^{4} + \)\(35\!\cdots\!50\)\( T^{5} + \)\(70\!\cdots\!25\)\( T^{6} \))(\( 1 - \)\(24\!\cdots\!80\)\( T + \)\(77\!\cdots\!00\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!50\)\( T^{4} - \)\(74\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(29\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + \)\(55\!\cdots\!24\)\( T + \)\(38\!\cdots\!13\)\( T^{2} + \)\(13\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!91\)\( T^{4} + \)\(44\!\cdots\!76\)\( T^{5} + \)\(22\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!92\)\( T + \)\(53\!\cdots\!52\)\( T^{2} - \)\(90\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!46\)\( T^{4} - \)\(25\!\cdots\!00\)\( T^{5} + \)\(43\!\cdots\!48\)\( T^{6} - \)\(42\!\cdots\!56\)\( T^{7} + \)\(64\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 - \)\(12\!\cdots\!36\)\( T + \)\(66\!\cdots\!65\)\( T^{2} - \)\(57\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(65\!\cdots\!56\)\( T^{5} + \)\(11\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(32\!\cdots\!88\)\( T + \)\(83\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - \)\(34\!\cdots\!56\)\( T^{5} + \)\(42\!\cdots\!08\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(25\!\cdots\!41\)\( T^{8} \))
$13$ (\( 1 + \)\(20\!\cdots\!42\)\( T + \)\(32\!\cdots\!27\)\( T^{2} + \)\(53\!\cdots\!36\)\( T^{3} + \)\(54\!\cdots\!51\)\( T^{4} + \)\(58\!\cdots\!98\)\( T^{5} + \)\(48\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(56\!\cdots\!16\)\( T + \)\(45\!\cdots\!48\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!66\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} - \)\(27\!\cdots\!52\)\( T^{7} + \)\(82\!\cdots\!61\)\( T^{8} \))
$17$ (\( 1 - \)\(13\!\cdots\!46\)\( T + \)\(40\!\cdots\!23\)\( T^{2} - \)\(34\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!91\)\( T^{4} - \)\(29\!\cdots\!94\)\( T^{5} + \)\(99\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(17\!\cdots\!12\)\( T + \)\(83\!\cdots\!72\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!26\)\( T^{4} - \)\(79\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} - \)\(17\!\cdots\!56\)\( T^{7} + \)\(46\!\cdots\!21\)\( T^{8} \))
$19$ (\( 1 - \)\(47\!\cdots\!60\)\( T + \)\(54\!\cdots\!57\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!83\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(54\!\cdots\!59\)\( T^{6} \))(\( 1 - \)\(57\!\cdots\!20\)\( T + \)\(11\!\cdots\!76\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(57\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} - \)\(31\!\cdots\!80\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} \))
$23$ (\( 1 + \)\(30\!\cdots\!72\)\( T + \)\(82\!\cdots\!97\)\( T^{2} + \)\(12\!\cdots\!36\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(79\!\cdots\!67\)\( T^{6} \))(\( 1 - \)\(19\!\cdots\!76\)\( T + \)\(73\!\cdots\!08\)\( T^{2} - \)\(96\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(29\!\cdots\!32\)\( T^{6} - \)\(15\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \))
$29$ (\( 1 - \)\(21\!\cdots\!90\)\( T + \)\(58\!\cdots\!87\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!89\)\( T^{6} \))(\( 1 - \)\(89\!\cdots\!00\)\( T - \)\(91\!\cdots\!84\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} - \)\(49\!\cdots\!00\)\( T^{5} - \)\(73\!\cdots\!44\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(65\!\cdots\!81\)\( T^{8} \))
$31$ (\( 1 + \)\(38\!\cdots\!04\)\( T + \)\(72\!\cdots\!65\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!15\)\( T^{4} + \)\(15\!\cdots\!44\)\( T^{5} + \)\(25\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(44\!\cdots\!08\)\( T + \)\(25\!\cdots\!48\)\( T^{2} - \)\(72\!\cdots\!76\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!28\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \))
$37$ (\( 1 + \)\(96\!\cdots\!14\)\( T + \)\(20\!\cdots\!43\)\( T^{2} + \)\(28\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!66\)\( T^{5} + \)\(11\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(28\!\cdots\!92\)\( T + \)\(21\!\cdots\!72\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!06\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(33\!\cdots\!76\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \))
$41$ (\( 1 - \)\(31\!\cdots\!26\)\( T + \)\(11\!\cdots\!15\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!15\)\( T^{4} - \)\(58\!\cdots\!06\)\( T^{5} + \)\(80\!\cdots\!21\)\( T^{6} \))(\( 1 - \)\(24\!\cdots\!88\)\( T + \)\(12\!\cdots\!68\)\( T^{2} - \)\(23\!\cdots\!16\)\( T^{3} + \)\(68\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!08\)\( T^{6} - \)\(19\!\cdots\!48\)\( T^{7} + \)\(34\!\cdots\!61\)\( T^{8} \))
$43$ (\( 1 - \)\(90\!\cdots\!68\)\( T + \)\(29\!\cdots\!37\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(59\!\cdots\!91\)\( T^{4} - \)\(38\!\cdots\!32\)\( T^{5} + \)\(85\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(39\!\cdots\!16\)\( T + \)\(11\!\cdots\!68\)\( T^{2} - \)\(21\!\cdots\!20\)\( T^{3} + \)\(35\!\cdots\!06\)\( T^{4} - \)\(44\!\cdots\!60\)\( T^{5} + \)\(47\!\cdots\!32\)\( T^{6} - \)\(33\!\cdots\!12\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - \)\(16\!\cdots\!56\)\( T + \)\(14\!\cdots\!53\)\( T^{2} - \)\(91\!\cdots\!72\)\( T^{3} + \)\(40\!\cdots\!91\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!23\)\( T^{6} \))(\( 1 - \)\(68\!\cdots\!52\)\( T + \)\(12\!\cdots\!52\)\( T^{2} - \)\(57\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!06\)\( T^{4} - \)\(15\!\cdots\!40\)\( T^{5} + \)\(95\!\cdots\!68\)\( T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + \)\(57\!\cdots\!81\)\( T^{8} \))
$53$ (\( 1 + \)\(90\!\cdots\!62\)\( T + \)\(13\!\cdots\!07\)\( T^{2} + \)\(73\!\cdots\!36\)\( T^{3} + \)\(63\!\cdots\!71\)\( T^{4} + \)\(19\!\cdots\!58\)\( T^{5} + \)\(99\!\cdots\!77\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!24\)\( T + \)\(12\!\cdots\!28\)\( T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!46\)\( T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!48\)\( T^{7} + \)\(46\!\cdots\!81\)\( T^{8} \))
$59$ (\( 1 - \)\(17\!\cdots\!80\)\( T + \)\(34\!\cdots\!77\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!43\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(48\!\cdots\!00\)\( T + \)\(63\!\cdots\!36\)\( T^{2} - \)\(75\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(48\!\cdots\!16\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(57\!\cdots\!61\)\( T^{8} \))
$61$ (\( 1 - \)\(36\!\cdots\!86\)\( T + \)\(15\!\cdots\!15\)\( T^{2} - \)\(31\!\cdots\!20\)\( T^{3} + \)\(64\!\cdots\!15\)\( T^{4} - \)\(61\!\cdots\!06\)\( T^{5} + \)\(68\!\cdots\!81\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!08\)\( T + \)\(14\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!96\)\( T^{3} + \)\(88\!\cdots\!70\)\( T^{4} - \)\(67\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!28\)\( T^{6} - \)\(93\!\cdots\!48\)\( T^{7} + \)\(27\!\cdots\!41\)\( T^{8} \))
$67$ (\( 1 + \)\(45\!\cdots\!04\)\( T + \)\(23\!\cdots\!73\)\( T^{2} + \)\(73\!\cdots\!68\)\( T^{3} + \)\(19\!\cdots\!91\)\( T^{4} + \)\(30\!\cdots\!56\)\( T^{5} + \)\(54\!\cdots\!63\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!48\)\( T + \)\(28\!\cdots\!32\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!48\)\( T^{6} + \)\(75\!\cdots\!24\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \))
$71$ (\( 1 + \)\(66\!\cdots\!84\)\( T + \)\(19\!\cdots\!65\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!15\)\( T^{4} + \)\(53\!\cdots\!44\)\( T^{5} + \)\(71\!\cdots\!11\)\( T^{6} \))(\( 1 - \)\(53\!\cdots\!48\)\( T + \)\(13\!\cdots\!48\)\( T^{2} + \)\(47\!\cdots\!64\)\( T^{3} + \)\(91\!\cdots\!70\)\( T^{4} + \)\(42\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} - \)\(38\!\cdots\!28\)\( T^{7} + \)\(64\!\cdots\!81\)\( T^{8} \))
$73$ (\( 1 - \)\(28\!\cdots\!78\)\( T + \)\(14\!\cdots\!47\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!31\)\( T^{4} - \)\(20\!\cdots\!62\)\( T^{5} + \)\(61\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(48\!\cdots\!56\)\( T + \)\(35\!\cdots\!68\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(29\!\cdots\!52\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} \))
$79$ (\( 1 + \)\(16\!\cdots\!60\)\( T + \)\(23\!\cdots\!37\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!23\)\( T^{4} + \)\(42\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!40\)\( T + \)\(13\!\cdots\!16\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!46\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{5} + \)\(35\!\cdots\!56\)\( T^{6} + \)\(67\!\cdots\!60\)\( T^{7} + \)\(67\!\cdots\!81\)\( T^{8} \))
$83$ (\( 1 + \)\(15\!\cdots\!52\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(98\!\cdots\!36\)\( T^{3} + \)\(44\!\cdots\!11\)\( T^{4} + \)\(12\!\cdots\!28\)\( T^{5} + \)\(21\!\cdots\!87\)\( T^{6} \))(\( 1 + \)\(23\!\cdots\!64\)\( T + \)\(57\!\cdots\!68\)\( T^{2} + \)\(21\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!46\)\( T^{4} + \)\(59\!\cdots\!60\)\( T^{5} + \)\(44\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!68\)\( T^{7} + \)\(60\!\cdots\!21\)\( T^{8} \))
$89$ (\( 1 + \)\(86\!\cdots\!30\)\( T + \)\(15\!\cdots\!67\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!63\)\( T^{4} + \)\(54\!\cdots\!30\)\( T^{5} + \)\(50\!\cdots\!69\)\( T^{6} \))(\( 1 - \)\(27\!\cdots\!60\)\( T + \)\(48\!\cdots\!56\)\( T^{2} - \)\(63\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!26\)\( T^{4} - \)\(50\!\cdots\!80\)\( T^{5} + \)\(30\!\cdots\!76\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(40\!\cdots\!41\)\( T^{8} \))
$97$ (\( 1 + \)\(53\!\cdots\!94\)\( T + \)\(31\!\cdots\!03\)\( T^{2} + \)\(91\!\cdots\!28\)\( T^{3} + \)\(26\!\cdots\!91\)\( T^{4} + \)\(38\!\cdots\!46\)\( T^{5} + \)\(61\!\cdots\!73\)\( T^{6} \))(\( 1 + \)\(15\!\cdots\!08\)\( T + \)\(16\!\cdots\!12\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} + \)\(41\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} + \)\(94\!\cdots\!84\)\( T^{7} + \)\(51\!\cdots\!81\)\( T^{8} \))
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