Properties

Label 2.70.a
Level 2
Weight 70
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 2
Sturm bound 17
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 70 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(17\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{70}(\Gamma_0(2))\).

Total New Old
Modular forms 18 6 12
Cusp forms 16 6 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.

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

Trace form

\( 6q - 8116452590112840q^{3} + 1770887431076116955136q^{4} + 1426588595791134963780420q^{5} - 671420440364298705385291776q^{6} - 116335089770189171139669400080q^{7} + 3672920218335960101345358967935918q^{9} + O(q^{10}) \) \( 6q - 8116452590112840q^{3} + 1770887431076116955136q^{4} + 1426588595791134963780420q^{5} - 671420440364298705385291776q^{6} - 116335089770189171139669400080q^{7} + 3672920218335960101345358967935918q^{9} - 44694630231320425476916845509345280q^{10} - 779703160576606393457922593129696088q^{11} - 2395553979459337147563188339909591040q^{12} - 831414022042322482715213091975463416780q^{13} + 5187989122644174036444494786171668267008q^{14} - 83876421624676348116128509439264136521520q^{15} + 522673715590561479879743397015195972796416q^{16} + 5160064732780890948584931977708591074956780q^{17} - 50697077687405732317771321256854337114603520q^{18} - 96720209617985555632965644120595112049929320q^{19} + 421054635600507998121124846727602535682539520q^{20} - 5356902555740701562658844711762649290184017728q^{21} - 4103326903385290096170352248311455812597841920q^{22} - 164654097474915092093554651606397612889938184240q^{23} - 198168336468121353005497984406663765284676960256q^{24} + 343982741924912434789527401140003707518523322650q^{25} + 14552789099046640493456196197291973556637924327424q^{26} + 25155714211205606396679577781710935305545752195120q^{27} - 34336058044523292412054751297808939365960232468480q^{28} + 879870068410485588157677639008614091720978713108500q^{29} + 2089559958453038048649059237850084103950492688711680q^{30} + 4259122099432113349107322108553060230752435308348352q^{31} + 157161258678103110326529414464761195100570778160580640q^{33} + 82026454491425915107367651033873315448326937271861248q^{34} + 753074726090774807188344090582855473677565317663232160q^{35} + 1084054708332749830377861331018406650241808588821495808q^{36} - 636652623424990691325515517408361047564150411099860220q^{37} - 4694383139791008039490962897184690760329752642028830720q^{38} - 13806223698879060730771286205897459990258316450773135984q^{39} - 13191526485539997195796958078789210054068859624415559680q^{40} - 191791730417736693124327119028078040838686789920324526148q^{41} - 115288865168439582157129676803355606163543693357786071040q^{42} + 222668712187261917949796662763484689033402121671249407080q^{43} - 230127754505905934216980450047712202169013964493268451328q^{44} + 1444491732428428584802788675791560402891567009271397550260q^{45} + 4064794469256100882471594230981403067905798236134581469184q^{46} + 5673164716981881150724988345312093468667704328738953212320q^{47} - 707042738781485767498569190104835977946705816581297930240q^{48} + 81779954923206179988738353737029391480925141079386516082582q^{49} + 15803287811891509606962882056150927105541093457607943782400q^{50} - 364949844856183689345233119727781411316590317854997353309968q^{51} - 245390106942531756417387065429345522593765447114642088263680q^{52} - 406633452694943811616842001378244776481572609515416706589980q^{53} - 187081645242416988903852889609217234002259509532516753080320q^{54} - 1797835857210966679887516197020868096626144910701285979606160q^{55} + 1531224121641696536968409057133376930156538993853687134158848q^{56} + 13069469519897085114311216085701708849550808156645490220061920q^{57} + 11566430142000287533300838404276112452935295721295430035701760q^{58} + 21062512847330641124912693100530044494719262798742105351677000q^{59} - 24755950136463393690601035927805564854337808301930386156421120q^{60} + 36768371103239994533332637416006377098791999567704009754382292q^{61} - 54787865268360341812294744812639839924943372124667947662704640q^{62} - 707868338349989750548312160179021462363272069140858941917723600q^{63} + 154266052248863066452028360864751609842131487403148112188932096q^{64} + 534639497417675283882459284729437964669070991626449700979359480q^{65} - 804653824448036580229158423509835184994691233445041152985137152q^{66} + 1326406006182582073344209684677207064451845945433292508661493560q^{67} + 1522982296470136978932443696694277310898766191527739206066503680q^{68} + 54671920854115024388750475056074487604689642845713397550408256q^{69} + 6511044838169186374318590658317774461885349190688181022422466560q^{70} + 22643056367225613143680307144193598140645134565899992273795800432q^{71} - 14963136278152710924458383250847915524950600804499402637044613120q^{72} - 51400594488780097570389583961177998792066383288162297751593550980q^{73} + 3231382658457599610356059568544195417507605233599948910585970688q^{74} - 117330088959572754695533825430631949248493367475945787032781913400q^{75} - 28546767257256329977701138444131901697568859590650380507068497920q^{76} + 18865701184985719352943873329473756021543976106295264965175625280q^{77} - 84434303671106602307621214564075717528150991192925208435574702080q^{78} + 423606095180569714094469495338132714239009705331866372337932019040q^{79} + 124273393663545691315913383727251273909359673458360136630397829120q^{80} + 2453385764702239197044157434265290116889709504359774603461835525526q^{81} + 1904538086325223092870887464177718056013280734274522892582980157440q^{82} - 3045002523975126463520834645594936033966589059048724437948217249000q^{83} - 1581078567576789400656307626855893643174462849849635981764400775168q^{84} - 4505628189732933902709310487664764137455293320208169786611270954040q^{85} - 8314720733504325890579903539129951078801554050414733609465299337216q^{86} - 23529204394619984991514359911328795316718042007179274751658725662960q^{87} - 1211088339800249055235029660691648649889908254615877312839176683520q^{88} + 14003143505188362047371849302974443533510315953330907991413349358940q^{89} + 43350853141951540418947977453233347125365421436483368187326533468160q^{90} + 167289772138632447206259431978594475451098908472584520197750886039072q^{91} - 48597311948918157155190015265274649931047232378083201252601563709440q^{92} - 80750279621043168353472691938113082057424307441354070102345924291840q^{93} + 154725469199997555945931617366067327100611408901075241868816010969088q^{94} - 96537237912260086181775912050035879596382861758350557369478083196400q^{95} - 58488969381443167766078366700419660571031503202374588798009801179136q^{96} - 558843682779118891894438882485091548438973244124374571584133645726900q^{97} - 725284418654469657387428488893703662968788638578887592355322480558080q^{98} - 1307626618097632835783297863284678546131166202128409820640336954275064q^{99} + O(q^{100}) \)

Decomposition of \(S_{70}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.70.a.a \(3\) \(60.303\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-51539607552\) \(15\!\cdots\!12\) \(20\!\cdots\!70\) \(-2\!\cdots\!96\) \(+\) \(q-2^{34}q^{2}+(5160893455497204+\cdots)q^{3}+\cdots\)
2.70.a.b \(3\) \(60.303\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(51539607552\) \(-2\!\cdots\!52\) \(-5\!\cdots\!50\) \(92\!\cdots\!16\) \(-\) \(q+2^{34}q^{2}+(-7866377652201484+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{70}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{70}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 17179869184 T )^{3} \))(\( ( 1 - 17179869184 T )^{3} \))
$3$ (\( 1 - 15482680366491612 T - \)\(28\!\cdots\!03\)\( T^{2} + \)\(39\!\cdots\!44\)\( T^{3} - \)\(23\!\cdots\!49\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{5} + \)\(58\!\cdots\!87\)\( T^{6} \))(\( 1 + 23599132956604452 T + \)\(13\!\cdots\!17\)\( T^{2} + \)\(18\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!11\)\( T^{4} + \)\(16\!\cdots\!28\)\( T^{5} + \)\(58\!\cdots\!87\)\( T^{6} \))
$5$ (\( 1 - \)\(20\!\cdots\!70\)\( T + \)\(47\!\cdots\!75\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(79\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!50\)\( T^{5} + \)\(48\!\cdots\!25\)\( T^{6} \))(\( 1 + \)\(58\!\cdots\!50\)\( T + \)\(23\!\cdots\!75\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!75\)\( T^{4} + \)\(16\!\cdots\!50\)\( T^{5} + \)\(48\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + \)\(20\!\cdots\!96\)\( T + \)\(21\!\cdots\!93\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(44\!\cdots\!51\)\( T^{4} + \)\(87\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!43\)\( T^{6} \))(\( 1 - \)\(92\!\cdots\!16\)\( T + \)\(25\!\cdots\!73\)\( T^{2} - \)\(31\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(39\!\cdots\!84\)\( T^{5} + \)\(86\!\cdots\!43\)\( T^{6} \))
$11$ (\( 1 + \)\(27\!\cdots\!04\)\( T + \)\(11\!\cdots\!45\)\( T^{2} + \)\(64\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!24\)\( T^{5} + \)\(37\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!84\)\( T + \)\(17\!\cdots\!25\)\( T^{2} + \)\(63\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} + \)\(26\!\cdots\!04\)\( T^{5} + \)\(37\!\cdots\!71\)\( T^{6} \))
$13$ (\( 1 + \)\(83\!\cdots\!58\)\( T + \)\(43\!\cdots\!07\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(31\!\cdots\!11\)\( T^{4} + \)\(44\!\cdots\!82\)\( T^{5} + \)\(38\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(78\!\cdots\!78\)\( T + \)\(38\!\cdots\!47\)\( T^{2} + \)\(70\!\cdots\!36\)\( T^{3} + \)\(28\!\cdots\!31\)\( T^{4} - \)\(41\!\cdots\!62\)\( T^{5} + \)\(38\!\cdots\!17\)\( T^{6} \))
$17$ (\( 1 - \)\(19\!\cdots\!54\)\( T + \)\(12\!\cdots\!63\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(95\!\cdots\!11\)\( T^{4} - \)\(12\!\cdots\!86\)\( T^{5} + \)\(50\!\cdots\!73\)\( T^{6} \))(\( 1 - \)\(49\!\cdots\!26\)\( T + \)\(28\!\cdots\!83\)\( T^{2} - \)\(80\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} - \)\(31\!\cdots\!34\)\( T^{5} + \)\(50\!\cdots\!73\)\( T^{6} \))
$19$ (\( 1 - \)\(88\!\cdots\!80\)\( T + \)\(20\!\cdots\!37\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!23\)\( T^{4} - \)\(25\!\cdots\!80\)\( T^{5} + \)\(50\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(18\!\cdots\!00\)\( T + \)\(40\!\cdots\!37\)\( T^{2} + \)\(41\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!23\)\( T^{4} + \)\(54\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!39\)\( T^{6} \))
$23$ (\( 1 + \)\(20\!\cdots\!08\)\( T + \)\(34\!\cdots\!77\)\( T^{2} + \)\(33\!\cdots\!64\)\( T^{3} + \)\(31\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!52\)\( T^{5} + \)\(75\!\cdots\!47\)\( T^{6} \))(\( 1 - \)\(35\!\cdots\!68\)\( T + \)\(19\!\cdots\!97\)\( T^{2} - \)\(96\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!11\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{5} + \)\(75\!\cdots\!47\)\( T^{6} \))
$29$ (\( 1 - \)\(10\!\cdots\!30\)\( T + \)\(24\!\cdots\!07\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!83\)\( T^{4} - \)\(66\!\cdots\!30\)\( T^{5} + \)\(52\!\cdots\!09\)\( T^{6} \))(\( 1 - \)\(77\!\cdots\!70\)\( T + \)\(41\!\cdots\!07\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!83\)\( T^{4} - \)\(50\!\cdots\!70\)\( T^{5} + \)\(52\!\cdots\!09\)\( T^{6} \))
$31$ (\( 1 - \)\(37\!\cdots\!56\)\( T + \)\(10\!\cdots\!25\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(87\!\cdots\!75\)\( T^{4} - \)\(23\!\cdots\!96\)\( T^{5} + \)\(51\!\cdots\!11\)\( T^{6} \))(\( 1 - \)\(53\!\cdots\!96\)\( T + \)\(11\!\cdots\!85\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(88\!\cdots\!35\)\( T^{4} - \)\(34\!\cdots\!36\)\( T^{5} + \)\(51\!\cdots\!11\)\( T^{6} \))
$37$ (\( 1 + \)\(41\!\cdots\!26\)\( T + \)\(42\!\cdots\!23\)\( T^{2} + \)\(11\!\cdots\!92\)\( T^{3} + \)\(67\!\cdots\!71\)\( T^{4} + \)\(10\!\cdots\!54\)\( T^{5} + \)\(41\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(22\!\cdots\!94\)\( T + \)\(38\!\cdots\!43\)\( T^{2} + \)\(37\!\cdots\!68\)\( T^{3} + \)\(62\!\cdots\!11\)\( T^{4} + \)\(57\!\cdots\!26\)\( T^{5} + \)\(41\!\cdots\!33\)\( T^{6} \))
$41$ (\( 1 + \)\(15\!\cdots\!54\)\( T + \)\(12\!\cdots\!55\)\( T^{2} + \)\(68\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!55\)\( T^{4} + \)\(55\!\cdots\!34\)\( T^{5} + \)\(70\!\cdots\!81\)\( T^{6} \))(\( 1 + \)\(40\!\cdots\!94\)\( T + \)\(14\!\cdots\!95\)\( T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!95\)\( T^{4} + \)\(14\!\cdots\!74\)\( T^{5} + \)\(70\!\cdots\!81\)\( T^{6} \))
$43$ (\( 1 - \)\(35\!\cdots\!52\)\( T + \)\(18\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!76\)\( T^{3} + \)\(93\!\cdots\!71\)\( T^{4} - \)\(92\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!72\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(55\!\cdots\!51\)\( T^{4} + \)\(34\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + \)\(16\!\cdots\!56\)\( T + \)\(64\!\cdots\!13\)\( T^{2} + \)\(74\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!71\)\( T^{4} + \)\(93\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!63\)\( T^{6} \))(\( 1 - \)\(73\!\cdots\!76\)\( T + \)\(83\!\cdots\!93\)\( T^{2} - \)\(34\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!63\)\( T^{6} \))
$53$ (\( 1 + \)\(37\!\cdots\!98\)\( T + \)\(16\!\cdots\!67\)\( T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(15\!\cdots\!11\)\( T^{4} + \)\(33\!\cdots\!22\)\( T^{5} + \)\(84\!\cdots\!37\)\( T^{6} \))(\( 1 + \)\(36\!\cdots\!82\)\( T + \)\(21\!\cdots\!07\)\( T^{2} + \)\(57\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(32\!\cdots\!98\)\( T^{5} + \)\(84\!\cdots\!37\)\( T^{6} \))
$59$ (\( 1 + \)\(17\!\cdots\!40\)\( T + \)\(32\!\cdots\!17\)\( T^{2} + \)\(56\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!63\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(38\!\cdots\!40\)\( T + \)\(89\!\cdots\!17\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} - \)\(92\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!19\)\( T^{6} \))
$61$ (\( 1 + \)\(53\!\cdots\!94\)\( T + \)\(35\!\cdots\!35\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!35\)\( T^{4} + \)\(12\!\cdots\!14\)\( T^{5} + \)\(36\!\cdots\!21\)\( T^{6} \))(\( 1 - \)\(42\!\cdots\!86\)\( T + \)\(43\!\cdots\!55\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(67\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!66\)\( T^{5} + \)\(36\!\cdots\!21\)\( T^{6} \))
$67$ (\( 1 + \)\(16\!\cdots\!76\)\( T + \)\(36\!\cdots\!33\)\( T^{2} + \)\(33\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(99\!\cdots\!23\)\( T^{6} \))(\( 1 - \)\(29\!\cdots\!36\)\( T + \)\(53\!\cdots\!73\)\( T^{2} - \)\(61\!\cdots\!12\)\( T^{3} + \)\(53\!\cdots\!31\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(99\!\cdots\!23\)\( T^{6} \))
$71$ (\( 1 + \)\(40\!\cdots\!24\)\( T + \)\(13\!\cdots\!85\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!64\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(26\!\cdots\!56\)\( T + \)\(39\!\cdots\!05\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!55\)\( T^{4} - \)\(79\!\cdots\!16\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))
$73$ (\( 1 + \)\(45\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!11\)\( T^{4} + \)\(62\!\cdots\!42\)\( T^{5} + \)\(51\!\cdots\!97\)\( T^{6} \))(\( 1 + \)\(57\!\cdots\!62\)\( T + \)\(80\!\cdots\!87\)\( T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(29\!\cdots\!31\)\( T^{4} + \)\(78\!\cdots\!78\)\( T^{5} + \)\(51\!\cdots\!97\)\( T^{6} \))
$79$ (\( 1 - \)\(56\!\cdots\!60\)\( T + \)\(30\!\cdots\!57\)\( T^{2} - \)\(89\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!83\)\( T^{4} - \)\(42\!\cdots\!60\)\( T^{5} + \)\(64\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(14\!\cdots\!20\)\( T + \)\(43\!\cdots\!57\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!83\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!59\)\( T^{6} \))
$83$ (\( 1 - \)\(64\!\cdots\!92\)\( T + \)\(16\!\cdots\!97\)\( T^{2} - \)\(12\!\cdots\!96\)\( T^{3} + \)\(42\!\cdots\!91\)\( T^{4} - \)\(44\!\cdots\!28\)\( T^{5} + \)\(17\!\cdots\!27\)\( T^{6} \))(\( 1 + \)\(36\!\cdots\!92\)\( T + \)\(94\!\cdots\!97\)\( T^{2} + \)\(17\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!91\)\( T^{4} + \)\(25\!\cdots\!28\)\( T^{5} + \)\(17\!\cdots\!27\)\( T^{6} \))
$89$ (\( 1 + \)\(33\!\cdots\!30\)\( T + \)\(59\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!43\)\( T^{4} + \)\(34\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!29\)\( T^{6} \))(\( 1 - \)\(17\!\cdots\!70\)\( T + \)\(96\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!43\)\( T^{4} - \)\(18\!\cdots\!70\)\( T^{5} + \)\(33\!\cdots\!29\)\( T^{6} \))
$97$ (\( 1 + \)\(45\!\cdots\!86\)\( T + \)\(37\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(45\!\cdots\!11\)\( T^{4} + \)\(68\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!13\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!14\)\( T + \)\(26\!\cdots\!83\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(32\!\cdots\!11\)\( T^{4} + \)\(15\!\cdots\!46\)\( T^{5} + \)\(18\!\cdots\!13\)\( T^{6} \))
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