Properties

Label 2.68.a
Level 2
Weight 68
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: \( k \) \(=\) \( 68 \)
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 \(M_{68}(\Gamma_0(2))\).

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

\( 6q - 7608049647932760q^{3} + 442721857769029238784q^{4} - 194764873271514971106060q^{5} + 24423269365861391032909824q^{6} - 25472880664845880773818039280q^{7} + 307421164442124171189667730270142q^{9} + O(q^{10}) \) \( 6q - 7608049647932760q^{3} + 442721857769029238784q^{4} - 194764873271514971106060q^{5} + 24423269365861391032909824q^{6} - 25472880664845880773818039280q^{7} + 307421164442124171189667730270142q^{9} + 2875535035976801610919437643284480q^{10} - 194528729227668394789524730397199048q^{11} - 561374979021966724619321603669360640q^{12} + 13426181020022526731021866395504546180q^{13} - 587441586945603476299082757570526445568q^{14} + 4122770279988267930475492594373079791280q^{15} + 32667107224410092492483962313449748299776q^{16} - 406131085963533133004709126325015926289620q^{17} + 652067067677224115958746776055852008734720q^{18} + 1713662164079298007934240347198380555420360q^{19} - 14371111087152442574205268855595512721571840q^{20} - 66346141854663534386120591270533033555664448q^{21} + 52770494517693515321135359377812025224724480q^{22} - 3769508334255602091762314982720872690295230160q^{23} + 1802119197741262615289659556219528907205902336q^{24} - 21948781054012039871502958890283570152890280150q^{25} + 44325291234558192633941056131304299851063230464q^{26} - 3047393276893726176772625798493952476190359997680q^{27} - 1879566841778225497051236475051308430804301905920q^{28} - 18975796776142919104046133207944978990412805385820q^{29} - 84109230747069657039591082214861768901104228106240q^{30} - 538210453228271688025554058616824428270408588170688q^{31} - 2230273135249039517687973903091673636121524287350240q^{33} - 4350397242280047349947151489716990554976165757452288q^{34} - 4248845690187810468893397442798655752873264304508960q^{35} + 22683678173222574361782206025361576031538389490597888q^{36} + 35559936457709464556758247952999374283062296203830580q^{37} + 107169455664530255724674464343510303460228559137669120q^{38} + 916431352383498072696586216280529700475114071206037104q^{39} + 212177035534596989653467784593161439673963097326878720q^{40} - 407329955201306049997844781648386616640068104499856548q^{41} + 2130980994385218201027756734150661109996106241603010560q^{42} - 9340848207868916101624464685354916304734010894906028680q^{43} - 14353686732186968013500031777505950907282157641528246272q^{44} - 119834981267368952169697584837637412860198078177994487420q^{45} + 11574478123089108993384700269341140506077446932554317824q^{46} + 178661178225511677548368815198287111456437152688024045920q^{47} - 41422162269609154150886786250882000265307172367195176960q^{48} + 722819010335989231697641789359655125654982642213921583478q^{49} + 592730057594650750000825972540473615868571008642528051200q^{50} + 5531238256724358302502846555847751927655956603214237953232q^{51} + 990677300654608830965708765452284494890944548069462507520q^{52} - 12241492696715674108926681175491568936325413743323667227820q^{53} - 10130271848908178375798834760765535743509814643026894520320q^{54} - 23004965892657671573434211243380648737175644041642955089520q^{55} - 43345538450557380908604633137015345836892657567550041751552q^{56} - 166694765891788067318477166628066058095660030743552928987680q^{57} + 111599735655643602542412280887578869388802258819321000099840q^{58} + 441265192596176373107574341794337060718049626078708679691160q^{59} + 304206752918557801082388868232467456837135473424958666833920q^{60} + 1190999677431039748060049041408186173577768157943715984330212q^{61} + 177579177194603693817537403470957123090920383246954250895360q^{62} + 1781707618725880102267065796065958417932331622894448119960400q^{63} + 2410407066388485413312943138511743903783304490674189252952064q^{64} - 17014249594153100176941848930235219643875560772265066518598920q^{65} - 18323907210228963216244380862396726996820812018726146033057792q^{66} - 14130250287061935214434024229279303628947271905621898845646040q^{67} - 29967184812588117134987994434872120424430063232587187020103680q^{68} + 59397393705678114949513905408973237584363242832892113003065664q^{69} + 151359529314070680116471083768031130830953932273582601855303680q^{70} + 41340774020983520169015276829401277717319910799136988136562832q^{71} + 48114057265343996319061730166355386093510241153476506765230080q^{72} + 618709200056416488517131921856719346264203807424896740476909980q^{73} - 196714776259890201586090221611088065667543298215726786489090048q^{74} - 460131762587231510466574375997947411790386156041963769297981800q^{75} + 126445949478280303150648712608458038903943943396831095989207040q^{76} - 7651452259806639980201822914203088779370146448853331630607715520q^{77} - 2267492623252981251020624280255026653491793912853595600055173120q^{78} + 7720715696945865590463871323925920624391457706165908903663901600q^{79} - 1060400833118203806395193753656810262457014821830576708528373760q^{80} + 7365026793095143817274580003278450604511726994241946863537222326q^{81} + 15012798411264507710787646918011484815898800989896769888153763840q^{82} + 31489915361822334560188711693604949352841211701312605165190339400q^{83} - 4895481196284031170058973833771616839202181915446482556128591872q^{84} + 78142426591912234333681756244948230271920702940004457876989641640q^{85} - 32986497567414733106619261411311304087985073091878715612721053696q^{86} - 351985235467347889798399589356480953343808396604484464579292665360q^{87} + 3893775228043940948171347826488845188894320286161772231755038720q^{88} - 123426521949058543473593445867186345816512087857189099156785955140q^{89} - 76464107345529467755356214071532252283658907962764212180761968640q^{90} - 564824914214332057553970702582901165038636076565440326180832890528q^{91} - 278140622102913165933437666614113146387926835473920359028479754240q^{92} + 3331276272901403046526870062417073206614851388308318555780388970240q^{93} + 913387847886008990512847111276821560789069894876286047013790810112q^{94} + 4337582135928558136377686930964417676836263985858732036555055052400q^{95} + 132972926524207390900451772141758277608232221976366066300987899904q^{96} - 6824871461188772314037540224803732928542096436791132468788037640500q^{97} - 9653718711210471788616546940897111465443860582012248308386491269120q^{98} + 4165067909549884750024286102826729628775548538024955012922814797464q^{99} + O(q^{100}) \)

Decomposition of \(S_{68}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.68.a.a \(3\) \(56.858\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-25769803776\) \(-5\!\cdots\!16\) \(-2\!\cdots\!50\) \(21\!\cdots\!12\) \(+\) \(q-2^{33}q^{2}+(-1741882068537572+\cdots)q^{3}+\cdots\)
2.68.a.b \(3\) \(56.858\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(25769803776\) \(-2\!\cdots\!44\) \(69\!\cdots\!90\) \(-4\!\cdots\!92\) \(-\) \(q+2^{33}q^{2}+(-794134480773348+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{68}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 8589934592 T )^{3} \))(\( ( 1 - 8589934592 T )^{3} \))
$3$ (\( 1 + 5225646205612716 T + \)\(94\!\cdots\!13\)\( T^{2} + \)\(59\!\cdots\!32\)\( T^{3} + \)\(87\!\cdots\!31\)\( T^{4} + \)\(44\!\cdots\!04\)\( T^{5} + \)\(79\!\cdots\!03\)\( T^{6} \))(\( 1 + 2382403442320044 T + \)\(46\!\cdots\!73\)\( T^{2} + \)\(73\!\cdots\!48\)\( T^{3} + \)\(42\!\cdots\!51\)\( T^{4} + \)\(20\!\cdots\!36\)\( T^{5} + \)\(79\!\cdots\!03\)\( T^{6} \))
$5$ (\( 1 + \)\(26\!\cdots\!50\)\( T + \)\(15\!\cdots\!75\)\( T^{2} + \)\(37\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} + \)\(12\!\cdots\!50\)\( T^{5} + \)\(31\!\cdots\!25\)\( T^{6} \))(\( 1 - \)\(69\!\cdots\!90\)\( T + \)\(92\!\cdots\!75\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!75\)\( T^{4} - \)\(32\!\cdots\!50\)\( T^{5} + \)\(31\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - \)\(21\!\cdots\!12\)\( T + \)\(39\!\cdots\!77\)\( T^{2} + \)\(94\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} - \)\(37\!\cdots\!88\)\( T^{5} + \)\(73\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(46\!\cdots\!92\)\( T + \)\(18\!\cdots\!17\)\( T^{2} + \)\(39\!\cdots\!56\)\( T^{3} + \)\(76\!\cdots\!31\)\( T^{4} + \)\(82\!\cdots\!08\)\( T^{5} + \)\(73\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + \)\(10\!\cdots\!44\)\( T + \)\(12\!\cdots\!25\)\( T^{2} + \)\(62\!\cdots\!40\)\( T^{3} + \)\(73\!\cdots\!75\)\( T^{4} + \)\(35\!\cdots\!04\)\( T^{5} + \)\(20\!\cdots\!11\)\( T^{6} \))(\( 1 + \)\(94\!\cdots\!04\)\( T + \)\(66\!\cdots\!85\)\( T^{2} + \)\(65\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!35\)\( T^{4} + \)\(33\!\cdots\!64\)\( T^{5} + \)\(20\!\cdots\!11\)\( T^{6} \))
$13$ (\( 1 - \)\(41\!\cdots\!94\)\( T + \)\(74\!\cdots\!63\)\( T^{2} - \)\(75\!\cdots\!88\)\( T^{3} + \)\(32\!\cdots\!71\)\( T^{4} - \)\(76\!\cdots\!66\)\( T^{5} + \)\(79\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(92\!\cdots\!86\)\( T - \)\(54\!\cdots\!17\)\( T^{2} + \)\(23\!\cdots\!48\)\( T^{3} - \)\(23\!\cdots\!89\)\( T^{4} - \)\(17\!\cdots\!54\)\( T^{5} + \)\(79\!\cdots\!13\)\( T^{6} \))
$17$ (\( 1 - \)\(50\!\cdots\!22\)\( T + \)\(59\!\cdots\!47\)\( T^{2} - \)\(28\!\cdots\!36\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} - \)\(38\!\cdots\!38\)\( T^{5} + \)\(20\!\cdots\!17\)\( T^{6} \))(\( 1 + \)\(45\!\cdots\!42\)\( T + \)\(14\!\cdots\!07\)\( T^{2} + \)\(28\!\cdots\!76\)\( T^{3} + \)\(40\!\cdots\!11\)\( T^{4} + \)\(34\!\cdots\!18\)\( T^{5} + \)\(20\!\cdots\!17\)\( T^{6} \))
$19$ (\( 1 + \)\(53\!\cdots\!00\)\( T + \)\(86\!\cdots\!17\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!63\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(70\!\cdots\!60\)\( T + \)\(55\!\cdots\!17\)\( T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!63\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!19\)\( T^{6} \))
$23$ (\( 1 + \)\(25\!\cdots\!16\)\( T + \)\(22\!\cdots\!93\)\( T^{2} + \)\(12\!\cdots\!52\)\( T^{3} + \)\(38\!\cdots\!71\)\( T^{4} + \)\(75\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!23\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!44\)\( T + \)\(43\!\cdots\!53\)\( T^{2} + \)\(47\!\cdots\!28\)\( T^{3} + \)\(74\!\cdots\!91\)\( T^{4} + \)\(35\!\cdots\!96\)\( T^{5} + \)\(50\!\cdots\!23\)\( T^{6} \))
$29$ (\( 1 + \)\(15\!\cdots\!70\)\( T + \)\(36\!\cdots\!27\)\( T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!43\)\( T^{4} + \)\(14\!\cdots\!70\)\( T^{5} + \)\(87\!\cdots\!29\)\( T^{6} \))(\( 1 + \)\(29\!\cdots\!50\)\( T + \)\(63\!\cdots\!27\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!43\)\( T^{4} + \)\(27\!\cdots\!50\)\( T^{5} + \)\(87\!\cdots\!29\)\( T^{6} \))
$31$ (\( 1 + \)\(27\!\cdots\!84\)\( T + \)\(46\!\cdots\!85\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!35\)\( T^{4} + \)\(19\!\cdots\!64\)\( T^{5} + \)\(58\!\cdots\!31\)\( T^{6} \))(\( 1 + \)\(25\!\cdots\!04\)\( T + \)\(42\!\cdots\!05\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} + \)\(18\!\cdots\!84\)\( T^{5} + \)\(58\!\cdots\!31\)\( T^{6} \))
$37$ (\( 1 - \)\(29\!\cdots\!62\)\( T + \)\(28\!\cdots\!47\)\( T^{2} - \)\(57\!\cdots\!56\)\( T^{3} + \)\(33\!\cdots\!51\)\( T^{4} - \)\(40\!\cdots\!18\)\( T^{5} + \)\(16\!\cdots\!37\)\( T^{6} \))(\( 1 - \)\(63\!\cdots\!18\)\( T + \)\(11\!\cdots\!07\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!31\)\( T^{4} - \)\(87\!\cdots\!02\)\( T^{5} + \)\(16\!\cdots\!37\)\( T^{6} \))
$41$ (\( 1 + \)\(10\!\cdots\!34\)\( T + \)\(25\!\cdots\!95\)\( T^{2} + \)\(16\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!74\)\( T^{5} + \)\(14\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(67\!\cdots\!86\)\( T + \)\(13\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!75\)\( T^{4} - \)\(86\!\cdots\!46\)\( T^{5} + \)\(14\!\cdots\!41\)\( T^{6} \))
$43$ (\( 1 + \)\(27\!\cdots\!96\)\( T + \)\(24\!\cdots\!93\)\( T^{2} - \)\(21\!\cdots\!88\)\( T^{3} + \)\(67\!\cdots\!51\)\( T^{4} + \)\(21\!\cdots\!04\)\( T^{5} + \)\(21\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(65\!\cdots\!84\)\( T + \)\(77\!\cdots\!73\)\( T^{2} + \)\(31\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} + \)\(50\!\cdots\!16\)\( T^{5} + \)\(21\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 - \)\(36\!\cdots\!92\)\( T + \)\(20\!\cdots\!77\)\( T^{2} - \)\(90\!\cdots\!36\)\( T^{3} + \)\(21\!\cdots\!51\)\( T^{4} - \)\(41\!\cdots\!48\)\( T^{5} + \)\(12\!\cdots\!47\)\( T^{6} \))(\( 1 - \)\(14\!\cdots\!28\)\( T + \)\(20\!\cdots\!17\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!71\)\( T^{4} - \)\(16\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!47\)\( T^{6} \))
$53$ (\( 1 + \)\(43\!\cdots\!06\)\( T + \)\(79\!\cdots\!23\)\( T^{2} + \)\(19\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!51\)\( T^{4} + \)\(49\!\cdots\!14\)\( T^{5} + \)\(37\!\cdots\!53\)\( T^{6} \))(\( 1 + \)\(78\!\cdots\!14\)\( T + \)\(54\!\cdots\!43\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(18\!\cdots\!91\)\( T^{4} + \)\(88\!\cdots\!66\)\( T^{5} + \)\(37\!\cdots\!53\)\( T^{6} \))
$59$ (\( 1 - \)\(49\!\cdots\!60\)\( T + \)\(15\!\cdots\!57\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(70\!\cdots\!83\)\( T^{4} - \)\(97\!\cdots\!60\)\( T^{5} + \)\(87\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(55\!\cdots\!00\)\( T + \)\(83\!\cdots\!57\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!83\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(87\!\cdots\!59\)\( T^{6} \))
$61$ (\( 1 - \)\(27\!\cdots\!26\)\( T + \)\(36\!\cdots\!55\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} - \)\(47\!\cdots\!66\)\( T^{5} + \)\(71\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(15\!\cdots\!14\)\( T + \)\(18\!\cdots\!95\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!95\)\( T^{4} + \)\(27\!\cdots\!74\)\( T^{5} + \)\(71\!\cdots\!61\)\( T^{6} \))
$67$ (\( 1 - \)\(12\!\cdots\!92\)\( T + \)\(51\!\cdots\!57\)\( T^{2} - \)\(47\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!11\)\( T^{4} - \)\(60\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \))(\( 1 + \)\(26\!\cdots\!32\)\( T + \)\(74\!\cdots\!77\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!71\)\( T^{4} + \)\(13\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \))
$71$ (\( 1 + \)\(92\!\cdots\!64\)\( T + \)\(12\!\cdots\!05\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!55\)\( T^{4} + \)\(10\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!96\)\( T + \)\(27\!\cdots\!45\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!95\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \))
$73$ (\( 1 - \)\(19\!\cdots\!94\)\( T + \)\(20\!\cdots\!03\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!91\)\( T^{4} - \)\(92\!\cdots\!46\)\( T^{5} + \)\(33\!\cdots\!73\)\( T^{6} \))(\( 1 - \)\(59\!\cdots\!86\)\( T + \)\(32\!\cdots\!23\)\( T^{2} - \)\(89\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!31\)\( T^{4} - \)\(29\!\cdots\!74\)\( T^{5} + \)\(33\!\cdots\!73\)\( T^{6} \))
$79$ (\( 1 + \)\(73\!\cdots\!80\)\( T + \)\(37\!\cdots\!77\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!43\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(15\!\cdots\!80\)\( T + \)\(11\!\cdots\!77\)\( T^{2} - \)\(54\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!43\)\( T^{4} - \)\(28\!\cdots\!80\)\( T^{5} + \)\(26\!\cdots\!79\)\( T^{6} \))
$83$ (\( 1 - \)\(18\!\cdots\!24\)\( T + \)\(81\!\cdots\!73\)\( T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + \)\(30\!\cdots\!71\)\( T^{4} - \)\(27\!\cdots\!96\)\( T^{5} + \)\(54\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(12\!\cdots\!76\)\( T + \)\(10\!\cdots\!73\)\( T^{2} - \)\(94\!\cdots\!92\)\( T^{3} + \)\(41\!\cdots\!71\)\( T^{4} - \)\(17\!\cdots\!04\)\( T^{5} + \)\(54\!\cdots\!83\)\( T^{6} \))
$89$ (\( 1 - \)\(27\!\cdots\!30\)\( T + \)\(99\!\cdots\!87\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!23\)\( T^{4} - \)\(45\!\cdots\!30\)\( T^{5} + \)\(67\!\cdots\!89\)\( T^{6} \))(\( 1 + \)\(39\!\cdots\!70\)\( T + \)\(10\!\cdots\!87\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!23\)\( T^{4} + \)\(65\!\cdots\!70\)\( T^{5} + \)\(67\!\cdots\!89\)\( T^{6} \))
$97$ (\( 1 + \)\(46\!\cdots\!38\)\( T + \)\(30\!\cdots\!87\)\( T^{2} + \)\(12\!\cdots\!24\)\( T^{3} + \)\(40\!\cdots\!31\)\( T^{4} + \)\(78\!\cdots\!22\)\( T^{5} + \)\(21\!\cdots\!97\)\( T^{6} \))(\( 1 + \)\(21\!\cdots\!62\)\( T + \)\(37\!\cdots\!87\)\( T^{2} + \)\(53\!\cdots\!76\)\( T^{3} + \)\(48\!\cdots\!31\)\( T^{4} + \)\(37\!\cdots\!78\)\( T^{5} + \)\(21\!\cdots\!97\)\( T^{6} \))
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