Properties

Label 3.43.b
Level 3
Weight 43
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 13
Newform subspaces 2
Sturm bound 14
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 43 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{43}(3, [\chi])\).

Total New Old
Modular forms 15 15 0
Cusp forms 13 13 0
Eisenstein series 2 2 0

Trace form

\( 13q + 6526851993q^{3} - 30601457050784q^{4} - 47900661722006688q^{6} + 104431479090834554q^{7} + 98939492749372360149q^{9} + O(q^{10}) \) \( 13q + 6526851993q^{3} - 30601457050784q^{4} - 47900661722006688q^{6} + 104431479090834554q^{7} + 98939492749372360149q^{9} - 1558337248111507093440q^{10} + 3371257617792639413472q^{12} - 253012333644251341412734q^{13} - 448128019361693763840960q^{15} + 49532793987439104163820032q^{16} + 467763252053696004788775360q^{18} + 1576909774199427781769570258q^{19} + 7572989610421557604517258514q^{21} - 4314702866032998614435578560q^{22} + 213252820822149659960046027264q^{24} - 1248191749023087121437353235035q^{25} + 3535242569745007589266902603777q^{27} - 11929994979570989528218827713344q^{28} + 62452866934923176992747679887680q^{30} - 42510819487048676995695358862134q^{31} + 43265764437015443353441835054400q^{33} - 263315485295738664634172504653056q^{34} - 1923971593950224017148630445513120q^{36} + 1526196720239071204471967871249074q^{37} - 7248612364199215982418683252126214q^{39} + 14264689156939495925288444850785280q^{40} - 2831182046378653470535212831685440q^{42} - 41240191555200066457703837814401374q^{43} + 171247471703495918987972144910848640q^{45} - 368252835390302858087002506036095616q^{46} + 190719216730956732645885104049043968q^{48} + 212869305691518808215922573851245175q^{49} - 26481340521925644924949289315966208q^{51} + 4621003571934453098144523281061984704q^{52} - 7762950205729260582930120693521404128q^{54} + 7333373720922934220673629457367543680q^{55} - 5917433808744215242767209354239887606q^{57} - 40828097874719028971304350423217887040q^{58} + 67409911106604936995672440638418037760q^{60} - 30214456684741097447409854535489547294q^{61} + 116836120779656750125400090217501760266q^{63} - 56388048458203571721647092899163381760q^{64} - 410927857618739095889759252898209949120q^{66} + 240564925468816186556265888196697026994q^{67} - 900977595682723671847728174824987261568q^{69} + 1299257312773585175235999395827621799040q^{70} - 1002862304292335428089280868557297904640q^{72} + 3596840906002472576188859342548993298666q^{73} + 7913245428648450142303694817241331316945q^{75} - 11577637952750293583108597010602259595840q^{76} - 1154864387471236146792590220220273542720q^{78} - 2394298619153593223068126360634187742102q^{79} - 17962382915172057496073817662807569190307q^{81} + 12117774709574398821091222390193736370560q^{82} - 122230594401868819650764739876303888650304q^{84} + 179892388106464049414097141341403281134080q^{85} - 108737363241821037303405571455850394882880q^{87} + 262137413065538961600854476553505008040960q^{88} - 246742532847586844558695609281104739341760q^{90} + 33297383399835377811307712929369792008388q^{91} - 265172693180256540460917211677794463781854q^{93} + 711306705503600048406114202903471029729024q^{94} - 1719405066055622809595776412409802129711104q^{96} + 1316788779790214809459143375755232261157274q^{97} - 4562402641239113129658705054315657608741760q^{99} + O(q^{100}) \)

Decomposition of \(S_{43}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.43.b.a \(1\) \(33.518\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-10460353203\) \(0\) \(14\!\cdots\!86\) \(q-3^{21}q^{3}+2^{42}q^{4}+146246101081752386q^{7}+\cdots\)
3.43.b.b \(12\) \(33.518\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(16987205196\) \(0\) \(-4\!\cdots\!32\) \(q+\beta _{1}q^{2}+(1415600433+546\beta _{1}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2097152 T )( 1 + 2097152 T ) \))(\( 1 - 8888527285680 T^{2} + \)\(70\!\cdots\!84\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{10} + \)\(58\!\cdots\!60\)\( T^{12} - \)\(23\!\cdots\!60\)\( T^{14} + \)\(86\!\cdots\!20\)\( T^{16} - \)\(28\!\cdots\!60\)\( T^{18} + \)\(98\!\cdots\!24\)\( T^{20} - \)\(24\!\cdots\!80\)\( T^{22} + \)\(52\!\cdots\!16\)\( T^{24} \))
$3$ (\( 1 + 10460353203 T \))(\( 1 - 16987205196 T + \)\(14\!\cdots\!38\)\( T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(38\!\cdots\!59\)\( T^{4} - \)\(33\!\cdots\!28\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!79\)\( T^{8} - \)\(32\!\cdots\!16\)\( T^{9} + \)\(21\!\cdots\!18\)\( T^{10} - \)\(26\!\cdots\!04\)\( T^{11} + \)\(17\!\cdots\!41\)\( T^{12} \))
$5$ (\( ( 1 - 476837158203125 T )( 1 + 476837158203125 T ) \))(\( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(25\!\cdots\!50\)\( T^{4} - \)\(61\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!75\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(57\!\cdots\!00\)\( T^{14} + \)\(27\!\cdots\!75\)\( T^{16} - \)\(84\!\cdots\!00\)\( T^{18} + \)\(18\!\cdots\!50\)\( T^{20} - \)\(23\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!25\)\( T^{24} \))
$7$ (\( 1 - 146246101081752386 T + \)\(31\!\cdots\!49\)\( T^{2} \))(\( ( 1 + 20907310995458916 T + \)\(81\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(41\!\cdots\!19\)\( T^{4} + \)\(51\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(16\!\cdots\!92\)\( T^{7} + \)\(40\!\cdots\!19\)\( T^{8} + \)\(36\!\cdots\!96\)\( T^{9} + \)\(76\!\cdots\!18\)\( T^{10} + \)\(61\!\cdots\!84\)\( T^{11} + \)\(92\!\cdots\!01\)\( T^{12} )^{2} \))
$11$ (\( ( 1 - \)\(74\!\cdots\!11\)\( T )( 1 + \)\(74\!\cdots\!11\)\( T ) \))(\( 1 - \)\(51\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!84\)\( T^{12} - \)\(54\!\cdots\!72\)\( T^{14} + \)\(19\!\cdots\!95\)\( T^{16} - \)\(53\!\cdots\!20\)\( T^{18} + \)\(10\!\cdots\!66\)\( T^{20} - \)\(12\!\cdots\!92\)\( T^{22} + \)\(72\!\cdots\!41\)\( T^{24} \))
$13$ (\( 1 + \)\(35\!\cdots\!26\)\( T + \)\(61\!\cdots\!69\)\( T^{2} \))(\( ( 1 - \)\(52\!\cdots\!96\)\( T + \)\(21\!\cdots\!38\)\( T^{2} - \)\(38\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!79\)\( T^{4} - \)\(31\!\cdots\!68\)\( T^{5} + \)\(18\!\cdots\!84\)\( T^{6} - \)\(19\!\cdots\!92\)\( T^{7} + \)\(93\!\cdots\!19\)\( T^{8} - \)\(87\!\cdots\!76\)\( T^{9} + \)\(29\!\cdots\!98\)\( T^{10} - \)\(44\!\cdots\!04\)\( T^{11} + \)\(51\!\cdots\!81\)\( T^{12} )^{2} \))
$17$ (\( ( 1 - \)\(69\!\cdots\!17\)\( T )( 1 + \)\(69\!\cdots\!17\)\( T ) \))(\( 1 - \)\(21\!\cdots\!20\)\( T^{2} + \)\(28\!\cdots\!94\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(69\!\cdots\!00\)\( T^{12} - \)\(30\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!35\)\( T^{16} - \)\(32\!\cdots\!40\)\( T^{18} + \)\(77\!\cdots\!14\)\( T^{20} - \)\(13\!\cdots\!20\)\( T^{22} + \)\(14\!\cdots\!21\)\( T^{24} \))
$19$ (\( 1 + \)\(16\!\cdots\!62\)\( T + \)\(51\!\cdots\!61\)\( T^{2} \))(\( ( 1 - \)\(87\!\cdots\!60\)\( T + \)\(16\!\cdots\!34\)\( T^{2} - \)\(96\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} - \)\(81\!\cdots\!20\)\( T^{5} + \)\(82\!\cdots\!80\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!94\)\( T^{10} - \)\(30\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!61\)\( T^{12} )^{2} \))
$23$ (\( ( 1 - \)\(39\!\cdots\!23\)\( T )( 1 + \)\(39\!\cdots\!23\)\( T ) \))(\( 1 - \)\(94\!\cdots\!40\)\( T^{2} + \)\(48\!\cdots\!14\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!75\)\( T^{8} - \)\(91\!\cdots\!80\)\( T^{10} + \)\(15\!\cdots\!40\)\( T^{12} - \)\(22\!\cdots\!80\)\( T^{14} + \)\(25\!\cdots\!75\)\( T^{16} - \)\(23\!\cdots\!80\)\( T^{18} + \)\(16\!\cdots\!54\)\( T^{20} - \)\(79\!\cdots\!40\)\( T^{22} + \)\(20\!\cdots\!41\)\( T^{24} \))
$29$ (\( ( 1 - \)\(51\!\cdots\!29\)\( T )( 1 + \)\(51\!\cdots\!29\)\( T ) \))(\( 1 - \)\(22\!\cdots\!52\)\( T^{2} + \)\(24\!\cdots\!46\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(88\!\cdots\!95\)\( T^{8} - \)\(34\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!04\)\( T^{12} - \)\(23\!\cdots\!52\)\( T^{14} + \)\(42\!\cdots\!95\)\( T^{16} - \)\(57\!\cdots\!20\)\( T^{18} + \)\(56\!\cdots\!66\)\( T^{20} - \)\(35\!\cdots\!52\)\( T^{22} + \)\(11\!\cdots\!81\)\( T^{24} \))
$31$ (\( 1 - \)\(40\!\cdots\!62\)\( T + \)\(43\!\cdots\!61\)\( T^{2} \))(\( ( 1 + \)\(41\!\cdots\!48\)\( T + \)\(22\!\cdots\!26\)\( T^{2} + \)\(57\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!95\)\( T^{4} + \)\(34\!\cdots\!08\)\( T^{5} + \)\(87\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!95\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} + \)\(79\!\cdots\!66\)\( T^{10} + \)\(63\!\cdots\!48\)\( T^{11} + \)\(66\!\cdots\!61\)\( T^{12} )^{2} \))
$37$ (\( 1 - \)\(16\!\cdots\!26\)\( T + \)\(73\!\cdots\!69\)\( T^{2} \))(\( ( 1 + \)\(50\!\cdots\!76\)\( T + \)\(14\!\cdots\!18\)\( T^{2} + \)\(83\!\cdots\!84\)\( T^{3} + \)\(99\!\cdots\!79\)\( T^{4} + \)\(13\!\cdots\!08\)\( T^{5} + \)\(64\!\cdots\!24\)\( T^{6} + \)\(98\!\cdots\!52\)\( T^{7} + \)\(53\!\cdots\!19\)\( T^{8} + \)\(32\!\cdots\!56\)\( T^{9} + \)\(42\!\cdots\!78\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{11} + \)\(15\!\cdots\!81\)\( T^{12} )^{2} \))
$41$ (\( ( 1 - \)\(73\!\cdots\!41\)\( T )( 1 + \)\(73\!\cdots\!41\)\( T ) \))(\( 1 - \)\(30\!\cdots\!12\)\( T^{2} + \)\(46\!\cdots\!26\)\( T^{4} - \)\(45\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!95\)\( T^{8} - \)\(20\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{12} - \)\(61\!\cdots\!12\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{16} - \)\(11\!\cdots\!20\)\( T^{18} + \)\(36\!\cdots\!66\)\( T^{20} - \)\(72\!\cdots\!12\)\( T^{22} + \)\(69\!\cdots\!61\)\( T^{24} \))
$43$ (\( 1 - \)\(30\!\cdots\!14\)\( T + \)\(40\!\cdots\!49\)\( T^{2} \))(\( ( 1 + \)\(35\!\cdots\!44\)\( T + \)\(24\!\cdots\!78\)\( T^{2} + \)\(65\!\cdots\!76\)\( T^{3} + \)\(24\!\cdots\!79\)\( T^{4} + \)\(50\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!84\)\( T^{6} + \)\(20\!\cdots\!48\)\( T^{7} + \)\(40\!\cdots\!79\)\( T^{8} + \)\(43\!\cdots\!24\)\( T^{9} + \)\(64\!\cdots\!78\)\( T^{10} + \)\(38\!\cdots\!56\)\( T^{11} + \)\(43\!\cdots\!01\)\( T^{12} )^{2} \))
$47$ (\( ( 1 - \)\(13\!\cdots\!47\)\( T )( 1 + \)\(13\!\cdots\!47\)\( T ) \))(\( 1 - \)\(12\!\cdots\!60\)\( T^{2} + \)\(79\!\cdots\!54\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!55\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{10} + \)\(31\!\cdots\!20\)\( T^{12} - \)\(50\!\cdots\!20\)\( T^{14} + \)\(67\!\cdots\!55\)\( T^{16} - \)\(70\!\cdots\!20\)\( T^{18} + \)\(52\!\cdots\!34\)\( T^{20} - \)\(24\!\cdots\!60\)\( T^{22} + \)\(54\!\cdots\!81\)\( T^{24} \))
$53$ (\( ( 1 - \)\(16\!\cdots\!53\)\( T )( 1 + \)\(16\!\cdots\!53\)\( T ) \))(\( 1 - \)\(79\!\cdots\!60\)\( T^{2} + \)\(43\!\cdots\!54\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(58\!\cdots\!55\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{10} + \)\(45\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!55\)\( T^{16} - \)\(57\!\cdots\!20\)\( T^{18} + \)\(99\!\cdots\!34\)\( T^{20} - \)\(12\!\cdots\!60\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))
$59$ (\( ( 1 - \)\(15\!\cdots\!59\)\( T )( 1 + \)\(15\!\cdots\!59\)\( T ) \))(\( 1 - \)\(14\!\cdots\!12\)\( T^{2} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(55\!\cdots\!20\)\( T^{6} + \)\(20\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!92\)\( T^{10} + \)\(16\!\cdots\!24\)\( T^{12} - \)\(36\!\cdots\!12\)\( T^{14} + \)\(66\!\cdots\!95\)\( T^{16} - \)\(98\!\cdots\!20\)\( T^{18} + \)\(11\!\cdots\!66\)\( T^{20} - \)\(85\!\cdots\!12\)\( T^{22} + \)\(32\!\cdots\!61\)\( T^{24} \))
$61$ (\( 1 - \)\(55\!\cdots\!22\)\( T + \)\(96\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(43\!\cdots\!08\)\( T + \)\(39\!\cdots\!86\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(73\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(89\!\cdots\!84\)\( T^{6} + \)\(19\!\cdots\!68\)\( T^{7} + \)\(68\!\cdots\!95\)\( T^{8} + \)\(11\!\cdots\!80\)\( T^{9} + \)\(33\!\cdots\!66\)\( T^{10} + \)\(35\!\cdots\!08\)\( T^{11} + \)\(80\!\cdots\!21\)\( T^{12} )^{2} \))
$67$ (\( 1 + \)\(39\!\cdots\!34\)\( T + \)\(49\!\cdots\!89\)\( T^{2} \))(\( ( 1 - \)\(31\!\cdots\!64\)\( T + \)\(26\!\cdots\!18\)\( T^{2} - \)\(67\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!39\)\( T^{4} - \)\(62\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!24\)\( T^{6} - \)\(30\!\cdots\!88\)\( T^{7} + \)\(74\!\cdots\!19\)\( T^{8} - \)\(82\!\cdots\!84\)\( T^{9} + \)\(15\!\cdots\!38\)\( T^{10} - \)\(95\!\cdots\!36\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} )^{2} \))
$71$ (\( ( 1 - \)\(75\!\cdots\!71\)\( T )( 1 + \)\(75\!\cdots\!71\)\( T ) \))(\( 1 - \)\(27\!\cdots\!52\)\( T^{2} + \)\(39\!\cdots\!46\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!95\)\( T^{8} - \)\(15\!\cdots\!92\)\( T^{10} + \)\(85\!\cdots\!04\)\( T^{12} - \)\(48\!\cdots\!52\)\( T^{14} + \)\(26\!\cdots\!95\)\( T^{16} - \)\(12\!\cdots\!20\)\( T^{18} + \)\(41\!\cdots\!66\)\( T^{20} - \)\(94\!\cdots\!52\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))
$73$ (\( 1 + \)\(11\!\cdots\!46\)\( T + \)\(18\!\cdots\!29\)\( T^{2} \))(\( ( 1 - \)\(23\!\cdots\!56\)\( T + \)\(81\!\cdots\!78\)\( T^{2} - \)\(97\!\cdots\!04\)\( T^{3} + \)\(19\!\cdots\!39\)\( T^{4} - \)\(13\!\cdots\!68\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} - \)\(23\!\cdots\!72\)\( T^{7} + \)\(65\!\cdots\!99\)\( T^{8} - \)\(58\!\cdots\!56\)\( T^{9} + \)\(89\!\cdots\!18\)\( T^{10} - \)\(47\!\cdots\!44\)\( T^{11} + \)\(36\!\cdots\!21\)\( T^{12} )^{2} \))
$79$ (\( 1 - \)\(14\!\cdots\!58\)\( T + \)\(50\!\cdots\!41\)\( T^{2} \))(\( ( 1 + \)\(82\!\cdots\!80\)\( T + \)\(30\!\cdots\!74\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!35\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(98\!\cdots\!35\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!14\)\( T^{10} + \)\(26\!\cdots\!80\)\( T^{11} + \)\(15\!\cdots\!41\)\( T^{12} )^{2} \))
$83$ (\( ( 1 - \)\(19\!\cdots\!83\)\( T )( 1 + \)\(19\!\cdots\!83\)\( T ) \))(\( 1 - \)\(26\!\cdots\!60\)\( T^{2} + \)\(37\!\cdots\!14\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!55\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{10} + \)\(65\!\cdots\!20\)\( T^{12} - \)\(23\!\cdots\!20\)\( T^{14} + \)\(66\!\cdots\!55\)\( T^{16} - \)\(14\!\cdots\!20\)\( T^{18} + \)\(24\!\cdots\!34\)\( T^{20} - \)\(27\!\cdots\!60\)\( T^{22} + \)\(16\!\cdots\!21\)\( T^{24} \))
$89$ (\( ( 1 - \)\(86\!\cdots\!89\)\( T )( 1 + \)\(86\!\cdots\!89\)\( T ) \))(\( 1 - \)\(57\!\cdots\!92\)\( T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(32\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!95\)\( T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(41\!\cdots\!84\)\( T^{12} - \)\(27\!\cdots\!72\)\( T^{14} + \)\(14\!\cdots\!95\)\( T^{16} - \)\(57\!\cdots\!20\)\( T^{18} + \)\(16\!\cdots\!66\)\( T^{20} - \)\(32\!\cdots\!92\)\( T^{22} + \)\(31\!\cdots\!41\)\( T^{24} \))
$97$ (\( 1 - \)\(22\!\cdots\!06\)\( T + \)\(27\!\cdots\!09\)\( T^{2} \))(\( ( 1 - \)\(54\!\cdots\!84\)\( T + \)\(10\!\cdots\!98\)\( T^{2} - \)\(26\!\cdots\!96\)\( T^{3} + \)\(43\!\cdots\!79\)\( T^{4} - \)\(49\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(13\!\cdots\!48\)\( T^{7} + \)\(33\!\cdots\!99\)\( T^{8} - \)\(57\!\cdots\!84\)\( T^{9} + \)\(63\!\cdots\!78\)\( T^{10} - \)\(90\!\cdots\!16\)\( T^{11} + \)\(46\!\cdots\!41\)\( T^{12} )^{2} \))
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