Properties

Label 3.38.a
Level 3
Weight 38
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(3))\).

Total New Old
Modular forms 13 7 6
Cusp forms 11 7 4
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.
\(+\)\(3\)
\(-\)\(4\)

Trace form

\( 7q + 126654q^{2} + 387420489q^{3} + 684575556340q^{4} - 13728547643694q^{5} + 289972613401830q^{6} + 1983925604451440q^{7} + 47100204248248488q^{8} + 1050662447078993847q^{9} + O(q^{10}) \) \( 7q + 126654q^{2} + 387420489q^{3} + 684575556340q^{4} - 13728547643694q^{5} + 289972613401830q^{6} + 1983925604451440q^{7} + 47100204248248488q^{8} + 1050662447078993847q^{9} + 10677439113740148564q^{10} + 43627012209334921596q^{11} + 2953056363509900916q^{12} - 116439107414805217342q^{13} - 1011900338090011919184q^{14} + 2142004542906729634254q^{15} + 87255901380513400366096q^{16} + 53933221344537261354222q^{17} + 19010085938906126671134q^{18} + 37431357216330735837572q^{19} - 7632340044421182766445256q^{20} + 4349838812892944922819792q^{21} + 19422800609832966452458824q^{22} - 23262632371810686242267160q^{23} + 90605263613645777821825896q^{24} + 70500389161904121000790441q^{25} - 190049008972894374663386556q^{26} + 58149737003040059690390169q^{27} + 436404251980738279231633760q^{28} + 4282387159715215775639357514q^{29} - 4153326304813313427219998844q^{30} - 2864027368599097964995766872q^{31} + 9236041216446580613557668000q^{32} - 666206143987247432978609916q^{33} + 12435643218207912669769044060q^{34} + 89869427129508625745958757920q^{35} + 102751118462092574391065977140q^{36} + 115863671844314605837317826490q^{37} - 1107633303229040631381110062344q^{38} + 85271595592601424874818976542q^{39} - 930670397641098462628460596752q^{40} + 1217769522923831842591596907158q^{41} - 1009658064856086695606942529360q^{42} + 1974001937268466063840199045564q^{43} + 7123114228038704575669086836208q^{44} - 2060581351737727564473739192974q^{45} - 19159636947469943799013210624752q^{46} - 7493630272642461617006238652512q^{47} + 8954664258664264102770633426576q^{48} + 17777681797108039345187920072047q^{49} + 6073272961727520596403935775714q^{50} + 41946140543332132836242206168626q^{51} + 229991532199618315922613932880152q^{52} - 241155436559598971130695435510670q^{53} + 43523333654665393476466329791430q^{54} - 453860871097216155007773977194104q^{55} + 679813569394113295052284624175040q^{56} - 436483358600513385501061741356804q^{57} - 177250208537205591084553136157468q^{58} - 231030154028789531327080142644452q^{59} + 193777584194928096043010367942456q^{60} - 915631926311408975897805450272158q^{61} - 3526030476634524366768610629595200q^{62} + 297776590056517422675871367184240q^{63} + 7489482758536410774431040832535104q^{64} - 5918935767911995666073259191789892q^{65} + 11001043672663058443826467735295592q^{66} + 20037212315258745158622400365789092q^{67} - 14964628486001657391340224710150232q^{68} + 4281679486048626419130782999386776q^{69} - 34853363080158496834151256770488800q^{70} - 11256928646966945694391516939711656q^{71} + 7069487979055025456312146725579048q^{72} + 13286862853326838922960192885928230q^{73} - 42636436263036741069505645420848684q^{74} + 46604114812831642145212466210638599q^{75} + 251585815873023986043650792581090448q^{76} - 317311456849407219291248056732618560q^{77} + 405690213954176805237105803184658548q^{78} + 117214277910546077013071739760375160q^{79} - 1573122528355469898112660077331279776q^{80} + 157698796814574220882881035123408487q^{81} + 439407919632883552581457950171236460q^{82} - 780620880363296415782883626608968828q^{83} + 1744656585209134598679469690579281504q^{84} + 2058649870565625761770386988960299492q^{85} - 3018614020714216403538494878879907064q^{86} + 1572921319640150885314260279379461462q^{87} + 2856972222974864440117459932508175328q^{88} - 1919002589120493935902265256833893578q^{89} + 1602626329682741114871197285917412244q^{90} + 3074815552050522621982938457870992928q^{91} - 22025423841918805162477656924353262240q^{92} + 8079908710240208253230295102126005400q^{93} + 14581692702710050867156722254196766752q^{94} - 19671761716261279025385552016851096456q^{95} + 24717005554576238245245842290026843552q^{96} + 3135834703288340273897746546516640462q^{97} - 44762639877851592030494418305208632178q^{98} + 6548180486657852927733977168615917116q^{99} + O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.38.a.a \(3\) \(26.014\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-310908\) \(-1162261467\) \(-9\!\cdots\!90\) \(-4\!\cdots\!44\) \(+\) \(q+(-103636-\beta _{1})q^{2}-3^{18}q^{3}+(112825533616+\cdots)q^{4}+\cdots\)
3.38.a.b \(4\) \(26.014\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(437562\) \(1549681956\) \(-4\!\cdots\!04\) \(66\!\cdots\!84\) \(-\) \(q+(109391-\beta _{1})q^{2}+3^{18}q^{3}+(86524834843+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 310908 T + 85252022016 T^{2} + 33372034010185728 T^{3} + \)\(11\!\cdots\!52\)\( T^{4} + \)\(58\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} \))(\( 1 - 437562 T + 197558681120 T^{2} - 85300907961581568 T^{3} + \)\(26\!\cdots\!16\)\( T^{4} - \)\(11\!\cdots\!96\)\( T^{5} + \)\(37\!\cdots\!80\)\( T^{6} - \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!56\)\( T^{8} \))
$3$ (\( ( 1 + 387420489 T )^{3} \))(\( ( 1 - 387420489 T )^{4} \))
$5$ (\( 1 + 9628717886790 T + \)\(16\!\cdots\!75\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} + \)\(50\!\cdots\!50\)\( T^{5} + \)\(38\!\cdots\!25\)\( T^{6} \))(\( 1 + 4099829756904 T + \)\(10\!\cdots\!00\)\( T^{2} - \)\(66\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!50\)\( T^{4} - \)\(48\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + 4621884343701744 T + \)\(49\!\cdots\!65\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(92\!\cdots\!55\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} \))(\( 1 - 6605809948153184 T + \)\(38\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(53\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} - \)\(42\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 - 22673303357139628620 T + \)\(11\!\cdots\!01\)\( T^{2} - \)\(15\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!71\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!11\)\( T^{6} \))(\( 1 - 20953708852195292976 T + \)\(92\!\cdots\!92\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!06\)\( T^{4} - \)\(43\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(82\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \))
$13$ (\( 1 + \)\(16\!\cdots\!10\)\( T + \)\(21\!\cdots\!91\)\( T^{2} + \)\(71\!\cdots\!68\)\( T^{3} + \)\(35\!\cdots\!03\)\( T^{4} + \)\(45\!\cdots\!90\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} \))(\( 1 - 51830892788989874168 T + \)\(47\!\cdots\!04\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(23\!\cdots\!16\)\( T^{7} + \)\(73\!\cdots\!21\)\( T^{8} \))
$17$ (\( 1 + \)\(27\!\cdots\!06\)\( T + \)\(72\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!11\)\( T^{4} + \)\(30\!\cdots\!74\)\( T^{5} + \)\(38\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(81\!\cdots\!28\)\( T + \)\(10\!\cdots\!52\)\( T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} - \)\(30\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \))
$19$ (\( 1 - \)\(58\!\cdots\!04\)\( T + \)\(32\!\cdots\!77\)\( T^{2} - \)\(15\!\cdots\!12\)\( T^{3} + \)\(66\!\cdots\!03\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(54\!\cdots\!32\)\( T + \)\(55\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!12\)\( T^{6} + \)\(47\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))
$23$ (\( 1 + \)\(17\!\cdots\!72\)\( T + \)\(68\!\cdots\!85\)\( T^{2} + \)\(74\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} + \)\(10\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \))(\( 1 + \)\(61\!\cdots\!88\)\( T + \)\(98\!\cdots\!68\)\( T^{2} + \)\(39\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!90\)\( T^{4} + \)\(95\!\cdots\!16\)\( T^{5} + \)\(57\!\cdots\!12\)\( T^{6} + \)\(86\!\cdots\!76\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} \))
$29$ (\( 1 - \)\(11\!\cdots\!78\)\( T + \)\(16\!\cdots\!47\)\( T^{2} + \)\(21\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!18\)\( T^{5} + \)\(21\!\cdots\!29\)\( T^{6} \))(\( 1 - \)\(41\!\cdots\!36\)\( T + \)\(10\!\cdots\!80\)\( T^{2} - \)\(16\!\cdots\!32\)\( T^{3} + \)\(21\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!88\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} - \)\(88\!\cdots\!44\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} \))
$31$ (\( 1 + \)\(11\!\cdots\!36\)\( T + \)\(91\!\cdots\!57\)\( T^{2} + \)\(41\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!27\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(89\!\cdots\!64\)\( T + \)\(72\!\cdots\!68\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(55\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} \))
$37$ (\( 1 - \)\(60\!\cdots\!66\)\( T + \)\(29\!\cdots\!95\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!15\)\( T^{4} - \)\(67\!\cdots\!74\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(55\!\cdots\!24\)\( T + \)\(22\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} - \)\(65\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))
$41$ (\( 1 - \)\(20\!\cdots\!34\)\( T + \)\(27\!\cdots\!47\)\( T^{2} - \)\(22\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!07\)\( T^{4} - \)\(46\!\cdots\!74\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} \))(\( 1 + \)\(86\!\cdots\!76\)\( T + \)\(17\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} + \)\(49\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!68\)\( T^{6} + \)\(90\!\cdots\!16\)\( T^{7} + \)\(49\!\cdots\!21\)\( T^{8} \))
$43$ (\( 1 - \)\(24\!\cdots\!44\)\( T + \)\(28\!\cdots\!53\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(78\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!80\)\( T + \)\(85\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(56\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 + \)\(11\!\cdots\!32\)\( T + \)\(10\!\cdots\!17\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(79\!\cdots\!79\)\( T^{4} + \)\(64\!\cdots\!08\)\( T^{5} + \)\(40\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(42\!\cdots\!20\)\( T + \)\(24\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} - \)\(75\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} - \)\(17\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} \))
$53$ (\( 1 + \)\(11\!\cdots\!82\)\( T + \)\(23\!\cdots\!15\)\( T^{2} + \)\(14\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} + \)\(45\!\cdots\!58\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!88\)\( T^{2} + \)\(19\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} + \)\(12\!\cdots\!16\)\( T^{5} + \)\(79\!\cdots\!72\)\( T^{6} + \)\(31\!\cdots\!36\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \))
$59$ (\( 1 - \)\(10\!\cdots\!36\)\( T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(73\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!83\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!88\)\( T + \)\(62\!\cdots\!12\)\( T^{2} - \)\(19\!\cdots\!84\)\( T^{3} - \)\(29\!\cdots\!66\)\( T^{4} - \)\(64\!\cdots\!96\)\( T^{5} + \)\(68\!\cdots\!32\)\( T^{6} + \)\(48\!\cdots\!92\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))
$61$ (\( 1 - \)\(29\!\cdots\!02\)\( T + \)\(91\!\cdots\!19\)\( T^{2} - \)\(92\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!99\)\( T^{4} - \)\(38\!\cdots\!82\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!46\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))
$67$ (\( 1 - \)\(38\!\cdots\!44\)\( T + \)\(85\!\cdots\!93\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!11\)\( T^{4} - \)\(51\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(16\!\cdots\!48\)\( T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} - \)\(80\!\cdots\!84\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))
$71$ (\( 1 + \)\(22\!\cdots\!44\)\( T + \)\(96\!\cdots\!85\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!35\)\( T^{4} + \)\(21\!\cdots\!64\)\( T^{5} + \)\(30\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(10\!\cdots\!88\)\( T + \)\(55\!\cdots\!68\)\( T^{2} - \)\(47\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(55\!\cdots\!08\)\( T^{6} - \)\(33\!\cdots\!48\)\( T^{7} + \)\(96\!\cdots\!61\)\( T^{8} \))
$73$ (\( 1 - \)\(32\!\cdots\!78\)\( T + \)\(10\!\cdots\!55\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(96\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!48\)\( T + \)\(28\!\cdots\!68\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!96\)\( T^{7} + \)\(59\!\cdots\!81\)\( T^{8} \))
$79$ (\( 1 - \)\(74\!\cdots\!40\)\( T + \)\(44\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!43\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(42\!\cdots\!20\)\( T + \)\(83\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} - \)\(27\!\cdots\!14\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!16\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(70\!\cdots\!61\)\( T^{8} \))
$83$ (\( 1 + \)\(31\!\cdots\!04\)\( T + \)\(10\!\cdots\!49\)\( T^{2} + \)\(54\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} + \)\(32\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \))(\( 1 + \)\(46\!\cdots\!24\)\( T + \)\(47\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(48\!\cdots\!40\)\( T^{6} + \)\(48\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \))
$89$ (\( 1 + \)\(16\!\cdots\!26\)\( T + \)\(69\!\cdots\!47\)\( T^{2} - \)\(72\!\cdots\!92\)\( T^{3} + \)\(93\!\cdots\!63\)\( T^{4} + \)\(28\!\cdots\!66\)\( T^{5} + \)\(24\!\cdots\!89\)\( T^{6} \))(\( 1 + \)\(31\!\cdots\!52\)\( T + \)\(35\!\cdots\!52\)\( T^{2} + \)\(89\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} + \)\(12\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!32\)\( T^{6} + \)\(75\!\cdots\!28\)\( T^{7} + \)\(32\!\cdots\!81\)\( T^{8} \))
$97$ (\( 1 - \)\(26\!\cdots\!14\)\( T + \)\(74\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!91\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(44\!\cdots\!48\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} \))
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