Properties

Label 9.38.a
Level 9
Weight 38
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 15
Newform subspaces 4
Sturm bound 38
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(38\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_0(9))\).

Total New Old
Modular forms 39 16 23
Cusp forms 35 15 20
Eisenstein series 4 1 3

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.
\(+\)\(6\)
\(-\)\(9\)

Trace form

\( 15q + 67746q^{2} + 785430810132q^{4} + 8198963257794q^{5} + 1530671576429640q^{7} - 13056161204613288q^{8} + O(q^{10}) \) \( 15q + 67746q^{2} + 785430810132q^{4} + 8198963257794q^{5} + 1530671576429640q^{7} - 13056161204613288q^{8} + 701810811167570964q^{10} - 16892975854486383540q^{11} + 52320102428576639658q^{13} - 2157518781494053266864q^{14} + 60188323265818471794192q^{16} + 35505852537394505978478q^{17} + 340103894093608822057596q^{19} + 5879902135370202309024456q^{20} + 5423184383818460315790024q^{22} + 49503812521744568187729960q^{23} + 197740759565853839268042441q^{25} + 28850718701503607618665212q^{26} - 289768883906346168359703840q^{28} - 3011713332589333357687728294q^{29} - 781483822639604171690107920q^{31} - 22636540887382406531598141600q^{32} + 34944844065537508001921479068q^{34} + 1478831265694675394820929280q^{35} - 48698215244285736812332707510q^{37} + 764165442780594139231937643144q^{38} - 15584835515266358737879195152q^{40} + 42713772542541291383087934678q^{41} - 357193640373124911022591078236q^{43} - 8456608498093768123832483825136q^{44} - 6101690305449575035049134096368q^{46} + 3241423397073527591847831069312q^{47} + 6505178018778518629376164804119q^{49} + 51936896861448473228503326404286q^{50} + 201968765402745817321803572497752q^{52} + 81355700301008161059189756616770q^{53} - 396692024744375825342851041563304q^{55} - 455988078172087222021525366964160q^{56} - 603179502508704160403830986424668q^{58} + 468992613634879660085979512345292q^{59} - 1514039574412423112792120285935182q^{61} + 1652905151731163411958039260712000q^{62} + 3759236298641461088639243569404480q^{64} + 10593915558478990030921473507246492q^{65} - 266882168285961617964776700445908q^{67} + 18057929447638014873272440437407832q^{68} - 36774376798864258069820421993439200q^{70} + 12003061736760778187081479098579672q^{71} - 57831362673923186462758017850739970q^{73} + 109748095358061943362910860700827756q^{74} - 123611336921910076486872284521676400q^{76} + 362439451050831948798822172417618560q^{77} - 269095987265518808553895680263862624q^{79} + 1823603543904832845123315675847913376q^{80} - 886552596245968138461548193728960340q^{82} + 1251081155754206345091629844210042228q^{83} - 1975648459900377977331028992833131908q^{85} + 4025042851501199044794491013596379000q^{86} - 2423730904070903005805695363984983072q^{88} + 3251871510831732053816609235628836438q^{89} - 8162730110137418314902634761148155888q^{91} + 24462997923659378849989105119795745440q^{92} - 9326919510839415053010162109476526560q^{94} + 29601754868160475604558651229979722456q^{95} - 25066318508284790409476541519381690138q^{97} + 54164241523199545963406990749090560978q^{98} + O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.38.a.a \(2\) \(78.043\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(194400\) \(0\) \(-5\!\cdots\!00\) \(-3\!\cdots\!00\) \(-\) \(q+(97200-\beta )q^{2}+(18860134912+\cdots)q^{4}+\cdots\)
9.38.a.b \(3\) \(78.043\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(310908\) \(0\) \(96\!\cdots\!90\) \(-4\!\cdots\!44\) \(-\) \(q+(103636+\beta _{1})q^{2}+(112825533616+\cdots)q^{4}+\cdots\)
9.38.a.c \(4\) \(78.043\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-437562\) \(0\) \(40\!\cdots\!04\) \(66\!\cdots\!84\) \(-\) \(q+(-109391+\beta _{1})q^{2}+(86524834843+\cdots)q^{4}+\cdots\)
9.38.a.d \(6\) \(78.043\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(0\) \(29\!\cdots\!00\) \(+\) \(q+\beta _{1}q^{2}+(10522497328+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 194400 T + 137474498560 T^{2} - 26718132554956800 T^{3} + \)\(18\!\cdots\!84\)\( T^{4} \))(\( 1 - 310908 T + 85252022016 T^{2} - 33372034010185728 T^{3} + \)\(11\!\cdots\!52\)\( T^{4} - \)\(58\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} \))(\( 1 + 437562 T + 197558681120 T^{2} + 85300907961581568 T^{3} + \)\(26\!\cdots\!16\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(37\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!56\)\( T^{8} \))(\( 1 + 380749368432 T^{2} + \)\(73\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(13\!\cdots\!92\)\( T^{10} + \)\(67\!\cdots\!04\)\( T^{12} \))
$3$ 1
$5$ (\( 1 + 5529584385900 T + \)\(30\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \))(\( 1 - 9628717886790 T + \)\(16\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} - \)\(50\!\cdots\!50\)\( T^{5} + \)\(38\!\cdots\!25\)\( T^{6} \))(\( 1 - 4099829756904 T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!50\)\( T^{4} + \)\(48\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!25\)\( T^{8} \))(\( 1 + \)\(21\!\cdots\!50\)\( T^{2} + \)\(20\!\cdots\!75\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(59\!\cdots\!50\)\( T^{10} + \)\(14\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))(\( 1 + 4621884343701744 T + \)\(49\!\cdots\!65\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(92\!\cdots\!55\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} \))(\( 1 - 6605809948153184 T + \)\(38\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(53\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} - \)\(42\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 1497594962732100 T + \)\(30\!\cdots\!21\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!47\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} )^{2} \))
$11$ (\( 1 - 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 + 22673303357139628620 T + \)\(11\!\cdots\!01\)\( T^{2} + \)\(15\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!71\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!11\)\( T^{6} \))(\( 1 + 20953708852195292976 T + \)\(92\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!06\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + \)\(82\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(13\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!15\)\( T^{4} + \)\(25\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(18\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))
$13$ (\( 1 - \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(16\!\cdots\!10\)\( T + \)\(21\!\cdots\!91\)\( T^{2} + \)\(71\!\cdots\!68\)\( T^{3} + \)\(35\!\cdots\!03\)\( T^{4} + \)\(45\!\cdots\!90\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} \))(\( 1 - 51830892788989874168 T + \)\(47\!\cdots\!04\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(23\!\cdots\!16\)\( T^{7} + \)\(73\!\cdots\!21\)\( T^{8} \))(\( ( 1 + \)\(18\!\cdots\!50\)\( T + \)\(14\!\cdots\!99\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!67\)\( T^{4} + \)\(48\!\cdots\!50\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} )^{2} \))
$17$ (\( 1 - \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(27\!\cdots\!06\)\( T + \)\(72\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!11\)\( T^{4} - \)\(30\!\cdots\!74\)\( T^{5} + \)\(38\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(81\!\cdots\!28\)\( T + \)\(10\!\cdots\!52\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} + \)\(30\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( 1 + \)\(58\!\cdots\!62\)\( T^{2} - \)\(25\!\cdots\!65\)\( T^{4} - \)\(74\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!85\)\( T^{8} + \)\(75\!\cdots\!42\)\( T^{10} + \)\(14\!\cdots\!89\)\( T^{12} \))
$19$ (\( 1 - \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(58\!\cdots\!04\)\( T + \)\(32\!\cdots\!77\)\( T^{2} - \)\(15\!\cdots\!12\)\( T^{3} + \)\(66\!\cdots\!03\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(54\!\cdots\!32\)\( T + \)\(55\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!12\)\( T^{6} + \)\(47\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))(\( ( 1 + \)\(35\!\cdots\!48\)\( T + \)\(41\!\cdots\!85\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(84\!\cdots\!15\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} )^{2} \))
$23$ (\( 1 - \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(17\!\cdots\!72\)\( T + \)\(68\!\cdots\!85\)\( T^{2} - \)\(74\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \))(\( 1 - \)\(61\!\cdots\!88\)\( T + \)\(98\!\cdots\!68\)\( T^{2} - \)\(39\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!90\)\( T^{4} - \)\(95\!\cdots\!16\)\( T^{5} + \)\(57\!\cdots\!12\)\( T^{6} - \)\(86\!\cdots\!76\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(91\!\cdots\!18\)\( T^{2} + \)\(37\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(20\!\cdots\!29\)\( T^{12} \))
$29$ (\( 1 - \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!78\)\( T + \)\(16\!\cdots\!47\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!23\)\( T^{4} + \)\(18\!\cdots\!18\)\( T^{5} + \)\(21\!\cdots\!29\)\( T^{6} \))(\( 1 + \)\(41\!\cdots\!36\)\( T + \)\(10\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(21\!\cdots\!78\)\( T^{4} + \)\(21\!\cdots\!88\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(88\!\cdots\!44\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(33\!\cdots\!54\)\( T^{2} + \)\(67\!\cdots\!15\)\( T^{4} + \)\(92\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{10} + \)\(44\!\cdots\!41\)\( T^{12} \))
$31$ (\( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!36\)\( T + \)\(91\!\cdots\!57\)\( T^{2} + \)\(41\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!27\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(89\!\cdots\!64\)\( T + \)\(72\!\cdots\!68\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(55\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} \))(\( ( 1 - \)\(91\!\cdots\!64\)\( T + \)\(24\!\cdots\!65\)\( T^{2} - \)\(55\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} )^{2} \))
$37$ (\( 1 + \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(60\!\cdots\!66\)\( T + \)\(29\!\cdots\!95\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!15\)\( T^{4} - \)\(67\!\cdots\!74\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(55\!\cdots\!24\)\( T + \)\(22\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} - \)\(65\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))(\( ( 1 + \)\(48\!\cdots\!50\)\( T + \)\(11\!\cdots\!51\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(53\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} )^{2} \))
$41$ (\( 1 - \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} - \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(20\!\cdots\!34\)\( T + \)\(27\!\cdots\!47\)\( T^{2} + \)\(22\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!07\)\( T^{4} + \)\(46\!\cdots\!74\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(86\!\cdots\!76\)\( T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(49\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!68\)\( T^{6} - \)\(90\!\cdots\!16\)\( T^{7} + \)\(49\!\cdots\!21\)\( T^{8} \))(\( 1 + \)\(59\!\cdots\!86\)\( T^{2} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(25\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \))
$43$ (\( 1 + \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!44\)\( T + \)\(28\!\cdots\!53\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(78\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!80\)\( T + \)\(85\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(56\!\cdots\!01\)\( T^{8} \))(\( ( 1 - \)\(11\!\cdots\!00\)\( T + \)\(55\!\cdots\!29\)\( T^{2} - \)\(69\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!47\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} )^{2} \))
$47$ (\( 1 + \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!32\)\( T + \)\(10\!\cdots\!17\)\( T^{2} - \)\(39\!\cdots\!60\)\( T^{3} + \)\(79\!\cdots\!79\)\( T^{4} - \)\(64\!\cdots\!08\)\( T^{5} + \)\(40\!\cdots\!03\)\( T^{6} \))(\( 1 + \)\(42\!\cdots\!20\)\( T + \)\(24\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(75\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(34\!\cdots\!22\)\( T^{2} + \)\(47\!\cdots\!35\)\( T^{4} + \)\(84\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!15\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(16\!\cdots\!09\)\( T^{12} \))
$53$ (\( 1 + \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!82\)\( T + \)\(23\!\cdots\!15\)\( T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(45\!\cdots\!58\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!88\)\( T^{2} - \)\(19\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(79\!\cdots\!72\)\( T^{6} - \)\(31\!\cdots\!36\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(25\!\cdots\!78\)\( T^{2} + \)\(31\!\cdots\!35\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} + \)\(40\!\cdots\!58\)\( T^{10} + \)\(61\!\cdots\!09\)\( T^{12} \))
$59$ (\( 1 - \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(10\!\cdots\!36\)\( T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(73\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!88\)\( T + \)\(62\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{3} - \)\(29\!\cdots\!66\)\( T^{4} + \)\(64\!\cdots\!96\)\( T^{5} + \)\(68\!\cdots\!32\)\( T^{6} - \)\(48\!\cdots\!92\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))(\( 1 + \)\(11\!\cdots\!14\)\( T^{2} + \)\(73\!\cdots\!15\)\( T^{4} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(80\!\cdots\!15\)\( T^{8} + \)\(14\!\cdots\!94\)\( T^{10} + \)\(13\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(29\!\cdots\!02\)\( T + \)\(91\!\cdots\!19\)\( T^{2} - \)\(92\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!99\)\( T^{4} - \)\(38\!\cdots\!82\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!46\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))(\( ( 1 + \)\(35\!\cdots\!34\)\( T + \)\(34\!\cdots\!15\)\( T^{2} + \)\(79\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} + \)\(45\!\cdots\!94\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} )^{2} \))
$67$ (\( 1 + \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(38\!\cdots\!44\)\( T + \)\(85\!\cdots\!93\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!11\)\( T^{4} - \)\(51\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(16\!\cdots\!48\)\( T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} - \)\(80\!\cdots\!84\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))(\( ( 1 + \)\(47\!\cdots\!00\)\( T + \)\(10\!\cdots\!81\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!87\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} )^{2} \))
$71$ (\( 1 - \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(22\!\cdots\!44\)\( T + \)\(96\!\cdots\!85\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!35\)\( T^{4} - \)\(21\!\cdots\!64\)\( T^{5} + \)\(30\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!88\)\( T + \)\(55\!\cdots\!68\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(55\!\cdots\!08\)\( T^{6} + \)\(33\!\cdots\!48\)\( T^{7} + \)\(96\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(11\!\cdots\!46\)\( T^{2} + \)\(59\!\cdots\!15\)\( T^{4} + \)\(20\!\cdots\!20\)\( T^{6} + \)\(58\!\cdots\!15\)\( T^{8} + \)\(10\!\cdots\!06\)\( T^{10} + \)\(95\!\cdots\!41\)\( T^{12} \))
$73$ (\( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(32\!\cdots\!78\)\( T + \)\(10\!\cdots\!55\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(96\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!48\)\( T + \)\(28\!\cdots\!68\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!96\)\( T^{7} + \)\(59\!\cdots\!81\)\( T^{8} \))(\( ( 1 + \)\(45\!\cdots\!50\)\( T + \)\(22\!\cdots\!59\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!27\)\( T^{4} + \)\(34\!\cdots\!50\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} )^{2} \))
$79$ (\( 1 - \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(74\!\cdots\!40\)\( T + \)\(44\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!43\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(42\!\cdots\!20\)\( T + \)\(83\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} - \)\(27\!\cdots\!14\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!16\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(70\!\cdots\!61\)\( T^{8} \))(\( ( 1 + \)\(32\!\cdots\!32\)\( T + \)\(80\!\cdots\!85\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!15\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} )^{2} \))
$83$ (\( 1 - \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(31\!\cdots\!04\)\( T + \)\(10\!\cdots\!49\)\( T^{2} - \)\(54\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(32\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \))(\( 1 - \)\(46\!\cdots\!24\)\( T + \)\(47\!\cdots\!60\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(48\!\cdots\!40\)\( T^{6} - \)\(48\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \))(\( 1 + \)\(37\!\cdots\!38\)\( T^{2} + \)\(62\!\cdots\!35\)\( T^{4} + \)\(69\!\cdots\!40\)\( T^{6} + \)\(63\!\cdots\!15\)\( T^{8} + \)\(39\!\cdots\!58\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} \))
$89$ (\( 1 - \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!26\)\( T + \)\(69\!\cdots\!47\)\( T^{2} + \)\(72\!\cdots\!92\)\( T^{3} + \)\(93\!\cdots\!63\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(24\!\cdots\!89\)\( T^{6} \))(\( 1 - \)\(31\!\cdots\!52\)\( T + \)\(35\!\cdots\!52\)\( T^{2} - \)\(89\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!32\)\( T^{6} - \)\(75\!\cdots\!28\)\( T^{7} + \)\(32\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(51\!\cdots\!74\)\( T^{2} + \)\(13\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!15\)\( T^{8} + \)\(16\!\cdots\!94\)\( T^{10} + \)\(58\!\cdots\!21\)\( T^{12} \))
$97$ (\( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(26\!\cdots\!14\)\( T + \)\(74\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!91\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(44\!\cdots\!48\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} \))(\( ( 1 + \)\(17\!\cdots\!50\)\( T + \)\(19\!\cdots\!11\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!07\)\( T^{4} + \)\(17\!\cdots\!50\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} )^{2} \))
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