Properties

Label 9.42.a.c
Level 9
Weight 42
Character orbit 9.a
Self dual yes
Analytic conductor 95.825
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(95.8245034108\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 196497525461 x^{2} + 10360343667016365 x + 6095744045744274504000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{10}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 17455 + \beta_{1} ) q^{2} + ( 1338237117038 - 439610 \beta_{1} + \beta_{2} ) q^{4} + ( -29634266666058 + 51443969 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} ) q^{5} + ( 37564083753994332 - 35063520317 \beta_{1} + 15946 \beta_{2} - 4779 \beta_{3} ) q^{7} + ( -1569906771202510460 + 418345277780 \beta_{1} - 1435842 \beta_{2} + 45248 \beta_{3} ) q^{8} +O(q^{10})\) \( q +(17455 + \beta_{1}) q^{2} +(1338237117038 - 439610 \beta_{1} + \beta_{2}) q^{4} +(-29634266666058 + 51443969 \beta_{1} - 18 \beta_{2} + 7 \beta_{3}) q^{5} +(37564083753994332 - 35063520317 \beta_{1} + 15946 \beta_{2} - 4779 \beta_{3}) q^{7} +(-1569906771202510460 + 418345277780 \beta_{1} - 1435842 \beta_{2} + 45248 \beta_{3}) q^{8} +(\)\(18\!\cdots\!14\)\( - 74836662113542 \beta_{1} + 181536704 \beta_{2} + 9725184 \beta_{3}) q^{10} +(-\)\(18\!\cdots\!76\)\( - 445969526243582 \beta_{1} + 1110357468 \beta_{2} + 53582606 \beta_{3}) q^{11} +(-\)\(22\!\cdots\!02\)\( + 4459831718836882 \beta_{1} - 36533853764 \beta_{2} - 137466882 \beta_{3}) q^{13} +(-\)\(12\!\cdots\!88\)\( + 72341024860883240 \beta_{1} - 127523051712 \beta_{2} - 6474043648 \beta_{3}) q^{14} +(-\)\(14\!\cdots\!60\)\( - 2357279389979362536 \beta_{1} + 374759622948 \beta_{2} + 3159305856 \beta_{3}) q^{16} +(\)\(95\!\cdots\!14\)\( + 6485295754548475050 \beta_{1} + 5106626451468 \beta_{2} - 63847564634 \beta_{3}) q^{17} +(\)\(65\!\cdots\!40\)\( - 81391276309975697418 \beta_{1} + 18349548635316 \beta_{2} + 1793065426362 \beta_{3}) q^{19} +(-\)\(19\!\cdots\!16\)\( + \)\(29\!\cdots\!48\)\( \beta_{1} - 60280906667526 \beta_{2} + 7463869435904 \beta_{3}) q^{20} +(-\)\(15\!\cdots\!88\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} - 693592447070336 \beta_{2} + 130918855592448 \beta_{3}) q^{22} +(\)\(38\!\cdots\!12\)\( - \)\(31\!\cdots\!10\)\( \beta_{1} + 1690662527110188 \beta_{2} + 164397073048502 \beta_{3}) q^{23} +(\)\(29\!\cdots\!15\)\( + \)\(15\!\cdots\!80\)\( \beta_{1} + 14362136572628440 \beta_{2} - 153575990203860 \beta_{3}) q^{25} +(\)\(15\!\cdots\!06\)\( - \)\(43\!\cdots\!66\)\( \beta_{1} + 38653402731981696 \beta_{2} - 1860062750533120 \beta_{3}) q^{26} +(\)\(17\!\cdots\!40\)\( - \)\(21\!\cdots\!84\)\( \beta_{1} + 60584751373846056 \beta_{2} - 5008765360803840 \beta_{3}) q^{28} +(\)\(25\!\cdots\!70\)\( + \)\(27\!\cdots\!39\)\( \beta_{1} + 283149395406383706 \beta_{2} + 8521223156703453 \beta_{3}) q^{29} +(\)\(23\!\cdots\!88\)\( - \)\(10\!\cdots\!89\)\( \beta_{1} + 283781381122053778 \beta_{2} - 31160787502593735 \beta_{3}) q^{31} +(-\)\(49\!\cdots\!48\)\( - \)\(93\!\cdots\!24\)\( \beta_{1} + 477444389324619384 \beta_{2} - 77787427755697408 \beta_{3}) q^{32} +(\)\(23\!\cdots\!62\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + 374885565815596416 \beta_{2} + 134931654118937088 \beta_{3}) q^{34} +(-\)\(51\!\cdots\!24\)\( - \)\(87\!\cdots\!58\)\( \beta_{1} - 9946665426813493644 \beta_{2} + 235208506978322586 \beta_{3}) q^{35} +(\)\(50\!\cdots\!82\)\( + \)\(20\!\cdots\!04\)\( \beta_{1} - 45237321564510086448 \beta_{2} + 2776647317048496360 \beta_{3}) q^{37} +(-\)\(28\!\cdots\!76\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} - 70941509482557361536 \beta_{2} + 3530034438768692736 \beta_{3}) q^{38} +(\)\(64\!\cdots\!80\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} + 75523271032104212620 \beta_{2} - 12875416735073477760 \beta_{3}) q^{40} +(-\)\(58\!\cdots\!30\)\( + \)\(46\!\cdots\!38\)\( \beta_{1} + \)\(17\!\cdots\!48\)\( \beta_{2} - 27650179365112504830 \beta_{3}) q^{41} +(\)\(98\!\cdots\!60\)\( + \)\(90\!\cdots\!14\)\( \beta_{1} + \)\(65\!\cdots\!00\)\( \beta_{2} - 41372446718770080354 \beta_{3}) q^{43} +(\)\(46\!\cdots\!88\)\( - \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(15\!\cdots\!32\)\( \beta_{2} + 47906740054628933632 \beta_{3}) q^{44} +(-\)\(11\!\cdots\!96\)\( + \)\(70\!\cdots\!32\)\( \beta_{1} - \)\(21\!\cdots\!52\)\( \beta_{2} + \)\(32\!\cdots\!52\)\( \beta_{3}) q^{46} +(\)\(22\!\cdots\!72\)\( + \)\(69\!\cdots\!38\)\( \beta_{1} - \)\(26\!\cdots\!96\)\( \beta_{2} + 68913360574891567378 \beta_{3}) q^{47} +(-\)\(82\!\cdots\!35\)\( + \)\(48\!\cdots\!24\)\( \beta_{1} + \)\(69\!\cdots\!52\)\( \beta_{2} - \)\(24\!\cdots\!44\)\( \beta_{3}) q^{49} +(\)\(54\!\cdots\!05\)\( + \)\(37\!\cdots\!35\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} + \)\(41\!\cdots\!80\)\( \beta_{3}) q^{50} +(-\)\(14\!\cdots\!36\)\( + \)\(68\!\cdots\!76\)\( \beta_{1} - \)\(31\!\cdots\!50\)\( \beta_{2} - \)\(74\!\cdots\!08\)\( \beta_{3}) q^{52} +(-\)\(23\!\cdots\!70\)\( - \)\(31\!\cdots\!69\)\( \beta_{1} - \)\(38\!\cdots\!54\)\( \beta_{2} - \)\(46\!\cdots\!55\)\( \beta_{3}) q^{53} +(\)\(32\!\cdots\!16\)\( + \)\(42\!\cdots\!92\)\( \beta_{1} + \)\(11\!\cdots\!16\)\( \beta_{2} - \)\(60\!\cdots\!44\)\( \beta_{3}) q^{55} +(-\)\(48\!\cdots\!04\)\( + \)\(17\!\cdots\!24\)\( \beta_{1} - \)\(74\!\cdots\!32\)\( \beta_{2} + \)\(94\!\cdots\!24\)\( \beta_{3}) q^{56} +(\)\(96\!\cdots\!66\)\( + \)\(43\!\cdots\!46\)\( \beta_{1} + \)\(12\!\cdots\!08\)\( \beta_{2} + \)\(25\!\cdots\!80\)\( \beta_{3}) q^{58} +(\)\(45\!\cdots\!32\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} + \)\(11\!\cdots\!04\)\( \beta_{2} + \)\(76\!\cdots\!08\)\( \beta_{3}) q^{59} +(\)\(13\!\cdots\!30\)\( + \)\(60\!\cdots\!84\)\( \beta_{1} + \)\(66\!\cdots\!52\)\( \beta_{2} + \)\(10\!\cdots\!76\)\( \beta_{3}) q^{61} +(-\)\(35\!\cdots\!60\)\( + \)\(30\!\cdots\!40\)\( \beta_{1} - \)\(17\!\cdots\!80\)\( \beta_{2} - \)\(34\!\cdots\!96\)\( \beta_{3}) q^{62} +(-\)\(97\!\cdots\!96\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} - \)\(34\!\cdots\!04\)\( \beta_{2} - \)\(10\!\cdots\!92\)\( \beta_{3}) q^{64} +(\)\(24\!\cdots\!48\)\( - \)\(79\!\cdots\!74\)\( \beta_{1} - \)\(85\!\cdots\!52\)\( \beta_{2} - \)\(49\!\cdots\!82\)\( \beta_{3}) q^{65} +(-\)\(18\!\cdots\!48\)\( + \)\(59\!\cdots\!68\)\( \beta_{1} + \)\(43\!\cdots\!64\)\( \beta_{2} - \)\(95\!\cdots\!36\)\( \beta_{3}) q^{67} +(\)\(22\!\cdots\!08\)\( + \)\(36\!\cdots\!28\)\( \beta_{1} + \)\(26\!\cdots\!98\)\( \beta_{2} + \)\(36\!\cdots\!68\)\( \beta_{3}) q^{68} +(-\)\(31\!\cdots\!28\)\( - \)\(58\!\cdots\!76\)\( \beta_{1} + \)\(49\!\cdots\!32\)\( \beta_{2} - \)\(95\!\cdots\!08\)\( \beta_{3}) q^{70} +(\)\(21\!\cdots\!68\)\( - \)\(98\!\cdots\!38\)\( \beta_{1} - \)\(31\!\cdots\!48\)\( \beta_{2} - \)\(12\!\cdots\!22\)\( \beta_{3}) q^{71} +(-\)\(11\!\cdots\!22\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(61\!\cdots\!40\)\( \beta_{2} + \)\(26\!\cdots\!20\)\( \beta_{3}) q^{73} +(\)\(74\!\cdots\!10\)\( - \)\(90\!\cdots\!50\)\( \beta_{1} + \)\(11\!\cdots\!60\)\( \beta_{2} + \)\(21\!\cdots\!36\)\( \beta_{3}) q^{74} +(\)\(28\!\cdots\!12\)\( - \)\(24\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!04\)\( \beta_{2} - \)\(18\!\cdots\!48\)\( \beta_{3}) q^{76} +(-\)\(20\!\cdots\!16\)\( - \)\(29\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!80\)\( \beta_{2} + \)\(51\!\cdots\!72\)\( \beta_{3}) q^{77} +(\)\(35\!\cdots\!36\)\( + \)\(33\!\cdots\!59\)\( \beta_{1} - \)\(14\!\cdots\!62\)\( \beta_{2} - \)\(51\!\cdots\!91\)\( \beta_{3}) q^{79} +(-\)\(38\!\cdots\!88\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(38\!\cdots\!08\)\( \beta_{2} - \)\(32\!\cdots\!88\)\( \beta_{3}) q^{80} +(\)\(15\!\cdots\!90\)\( - \)\(41\!\cdots\!58\)\( \beta_{1} - \)\(56\!\cdots\!08\)\( \beta_{2} - \)\(33\!\cdots\!16\)\( \beta_{3}) q^{82} +(-\)\(38\!\cdots\!40\)\( + \)\(56\!\cdots\!98\)\( \beta_{1} + \)\(28\!\cdots\!40\)\( \beta_{2} + \)\(53\!\cdots\!30\)\( \beta_{3}) q^{83} +(-\)\(71\!\cdots\!56\)\( + \)\(76\!\cdots\!58\)\( \beta_{1} + \)\(11\!\cdots\!24\)\( \beta_{2} + \)\(14\!\cdots\!74\)\( \beta_{3}) q^{85} +(\)\(32\!\cdots\!32\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} - \)\(40\!\cdots\!84\)\( \beta_{2} - \)\(32\!\cdots\!56\)\( \beta_{3}) q^{86} +(-\)\(33\!\cdots\!80\)\( + \)\(45\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2} - \)\(14\!\cdots\!12\)\( \beta_{3}) q^{88} +(\)\(98\!\cdots\!30\)\( + \)\(15\!\cdots\!24\)\( \beta_{1} - \)\(80\!\cdots\!20\)\( \beta_{2} + \)\(34\!\cdots\!52\)\( \beta_{3}) q^{89} +(-\)\(22\!\cdots\!52\)\( + \)\(56\!\cdots\!94\)\( \beta_{1} + \)\(76\!\cdots\!76\)\( \beta_{2} + \)\(26\!\cdots\!02\)\( \beta_{3}) q^{91} +(\)\(16\!\cdots\!20\)\( - \)\(98\!\cdots\!96\)\( \beta_{1} + \)\(10\!\cdots\!44\)\( \beta_{2} + \)\(26\!\cdots\!28\)\( \beta_{3}) q^{92} +(\)\(24\!\cdots\!16\)\( - \)\(37\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!96\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3}) q^{94} +(-\)\(19\!\cdots\!80\)\( + \)\(19\!\cdots\!60\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(38\!\cdots\!00\)\( \beta_{3}) q^{95} +(\)\(91\!\cdots\!78\)\( - \)\(20\!\cdots\!16\)\( \beta_{1} - \)\(26\!\cdots\!96\)\( \beta_{2} - \)\(68\!\cdots\!64\)\( \beta_{3}) q^{97} +(\)\(17\!\cdots\!47\)\( - \)\(28\!\cdots\!63\)\( \beta_{1} - \)\(60\!\cdots\!68\)\( \beta_{2} - \)\(58\!\cdots\!20\)\( \beta_{3}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 69822q^{2} + 5352947588932q^{4} - 118536963776280q^{5} + 150256264888927136q^{7} - 6279626248119395784q^{8} + O(q^{10}) \) \( 4q + 69822q^{2} + 5352947588932q^{4} - 118536963776280q^{5} + 150256264888927136q^{7} - 6279626248119395784q^{8} + \)\(72\!\cdots\!40\)\(q^{10} - \)\(72\!\cdots\!56\)\(q^{11} - \)\(88\!\cdots\!08\)\(q^{13} - \)\(49\!\cdots\!68\)\(q^{14} - \)\(59\!\cdots\!00\)\(q^{16} + \)\(38\!\cdots\!88\)\(q^{17} + \)\(26\!\cdots\!48\)\(q^{19} - \)\(78\!\cdots\!60\)\(q^{20} - \)\(63\!\cdots\!76\)\(q^{22} + \)\(15\!\cdots\!32\)\(q^{23} + \)\(11\!\cdots\!00\)\(q^{25} + \)\(62\!\cdots\!52\)\(q^{26} + \)\(68\!\cdots\!12\)\(q^{28} + \)\(10\!\cdots\!64\)\(q^{29} + \)\(92\!\cdots\!04\)\(q^{31} - \)\(19\!\cdots\!56\)\(q^{32} + \)\(92\!\cdots\!04\)\(q^{34} - \)\(20\!\cdots\!40\)\(q^{35} + \)\(20\!\cdots\!56\)\(q^{37} - \)\(11\!\cdots\!28\)\(q^{38} + \)\(25\!\cdots\!00\)\(q^{40} - \)\(23\!\cdots\!04\)\(q^{41} + \)\(39\!\cdots\!60\)\(q^{43} + \)\(18\!\cdots\!04\)\(q^{44} - \)\(44\!\cdots\!16\)\(q^{46} + \)\(88\!\cdots\!20\)\(q^{47} - \)\(33\!\cdots\!80\)\(q^{49} + \)\(21\!\cdots\!50\)\(q^{50} - \)\(59\!\cdots\!08\)\(q^{52} - \)\(95\!\cdots\!28\)\(q^{53} + \)\(12\!\cdots\!60\)\(q^{55} - \)\(19\!\cdots\!20\)\(q^{56} + \)\(38\!\cdots\!16\)\(q^{58} + \)\(18\!\cdots\!08\)\(q^{59} + \)\(53\!\cdots\!40\)\(q^{61} - \)\(14\!\cdots\!52\)\(q^{62} - \)\(38\!\cdots\!92\)\(q^{64} + \)\(97\!\cdots\!80\)\(q^{65} - \)\(73\!\cdots\!28\)\(q^{67} + \)\(89\!\cdots\!24\)\(q^{68} - \)\(12\!\cdots\!80\)\(q^{70} + \)\(84\!\cdots\!52\)\(q^{71} - \)\(44\!\cdots\!32\)\(q^{73} + \)\(29\!\cdots\!12\)\(q^{74} + \)\(11\!\cdots\!72\)\(q^{76} - \)\(83\!\cdots\!52\)\(q^{77} + \)\(14\!\cdots\!80\)\(q^{79} - \)\(15\!\cdots\!80\)\(q^{80} + \)\(61\!\cdots\!12\)\(q^{82} - \)\(15\!\cdots\!04\)\(q^{83} - \)\(28\!\cdots\!60\)\(q^{85} + \)\(12\!\cdots\!24\)\(q^{86} - \)\(13\!\cdots\!96\)\(q^{88} + \)\(39\!\cdots\!72\)\(q^{89} - \)\(88\!\cdots\!16\)\(q^{91} + \)\(65\!\cdots\!44\)\(q^{92} + \)\(98\!\cdots\!32\)\(q^{94} - \)\(78\!\cdots\!00\)\(q^{95} + \)\(36\!\cdots\!52\)\(q^{97} + \)\(68\!\cdots\!22\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 196497525461 x^{2} + 10360343667016365 x + 6095744045744274504000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 36 \nu^{2} + 2847108 \nu - 3536956170084 \)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{3} + 6696918 \nu^{2} - 3100608741285 \nu - 448170152274909880 \)\()/5656\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 474518 \beta_{1} + 3536955695566\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(11312 \beta_{3} - 372051 \beta_{2} + 1210081143513 \beta_{1} - 419586565404959011\)\()/54\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−433547.
−161109.
263663.
330995.
−2.58383e6 0 4.47715e12 −2.55548e14 0 1.98945e17 −5.88630e18 0 6.60294e20
1.2 −949201. 0 −1.29804e12 1.14706e14 0 −7.22302e16 3.31942e18 0 −1.08879e20
1.3 1.59943e6 0 3.59152e11 −3.20915e14 0 2.35480e17 −2.94274e18 0 −5.13282e20
1.4 2.00342e6 0 1.81468e12 3.43221e14 0 −2.11939e17 −7.69998e17 0 6.87617e20
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.42.a.c 4
3.b odd 2 1 3.42.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.42.a.b 4 3.b odd 2 1
9.42.a.c 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 69822 T_{2}^{3} - \)\(70\!\cdots\!28\)\( T_{2}^{2} + \)\(24\!\cdots\!64\)\( T_{2} + \)\(78\!\cdots\!44\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 69822 T + 1724010272480 T^{2} + 2024128434727084032 T^{3} + \)\(57\!\cdots\!56\)\( T^{4} + \)\(44\!\cdots\!64\)\( T^{5} + \)\(83\!\cdots\!20\)\( T^{6} - \)\(74\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!16\)\( T^{8} \)
$3$ 1
$5$ \( 1 + 118536963776280 T + \)\(39\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!50\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(81\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 150256264888927136 T + \)\(11\!\cdots\!52\)\( T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!30\)\( T^{4} - \)\(59\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} - \)\(13\!\cdots\!48\)\( T^{7} + \)\(39\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(72\!\cdots\!56\)\( T + \)\(21\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!26\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!12\)\( T^{6} + \)\(89\!\cdots\!36\)\( T^{7} + \)\(61\!\cdots\!41\)\( T^{8} \)
$13$ \( 1 + \)\(88\!\cdots\!08\)\( T + \)\(84\!\cdots\!44\)\( T^{2} - \)\(83\!\cdots\!84\)\( T^{3} + \)\(45\!\cdots\!50\)\( T^{4} - \)\(39\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!36\)\( T^{6} + \)\(91\!\cdots\!76\)\( T^{7} + \)\(48\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 - \)\(38\!\cdots\!88\)\( T + \)\(11\!\cdots\!72\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(46\!\cdots\!26\)\( T^{4} - \)\(67\!\cdots\!60\)\( T^{5} + \)\(91\!\cdots\!08\)\( T^{6} - \)\(84\!\cdots\!44\)\( T^{7} + \)\(62\!\cdots\!21\)\( T^{8} \)
$19$ \( 1 - \)\(26\!\cdots\!48\)\( T + \)\(75\!\cdots\!12\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(25\!\cdots\!34\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(54\!\cdots\!32\)\( T^{6} - \)\(50\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!21\)\( T^{8} \)
$23$ \( 1 - \)\(15\!\cdots\!32\)\( T + \)\(19\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{5} + \)\(91\!\cdots\!92\)\( T^{6} - \)\(47\!\cdots\!44\)\( T^{7} + \)\(21\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 - \)\(10\!\cdots\!64\)\( T + \)\(27\!\cdots\!60\)\( T^{2} - \)\(26\!\cdots\!08\)\( T^{3} + \)\(32\!\cdots\!98\)\( T^{4} - \)\(23\!\cdots\!32\)\( T^{5} + \)\(22\!\cdots\!60\)\( T^{6} - \)\(77\!\cdots\!96\)\( T^{7} + \)\(68\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 - \)\(92\!\cdots\!04\)\( T + \)\(77\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!72\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(55\!\cdots\!32\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(55\!\cdots\!92\)\( T^{2} - \)\(86\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!70\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \)
$41$ \( 1 + \)\(23\!\cdots\!04\)\( T + \)\(51\!\cdots\!08\)\( T^{2} + \)\(70\!\cdots\!72\)\( T^{3} + \)\(99\!\cdots\!94\)\( T^{4} + \)\(94\!\cdots\!52\)\( T^{5} + \)\(91\!\cdots\!48\)\( T^{6} + \)\(55\!\cdots\!84\)\( T^{7} + \)\(31\!\cdots\!61\)\( T^{8} \)
$43$ \( 1 - \)\(39\!\cdots\!60\)\( T + \)\(29\!\cdots\!00\)\( T^{2} - \)\(85\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!98\)\( T^{4} - \)\(79\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} - \)\(32\!\cdots\!20\)\( T^{7} + \)\(77\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(88\!\cdots\!20\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} - \)\(41\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 + \)\(95\!\cdots\!28\)\( T + \)\(13\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(85\!\cdots\!10\)\( T^{4} + \)\(54\!\cdots\!16\)\( T^{5} + \)\(33\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!56\)\( T^{7} + \)\(60\!\cdots\!81\)\( T^{8} \)
$59$ \( 1 - \)\(18\!\cdots\!08\)\( T + \)\(66\!\cdots\!52\)\( T^{2} - \)\(24\!\cdots\!16\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} - \)\(99\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!61\)\( T^{8} \)
$61$ \( 1 - \)\(53\!\cdots\!40\)\( T + \)\(67\!\cdots\!56\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!76\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!41\)\( T^{8} \)
$67$ \( 1 + \)\(73\!\cdots\!28\)\( T + \)\(45\!\cdots\!12\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!06\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} + \)\(29\!\cdots\!64\)\( T^{7} + \)\(29\!\cdots\!21\)\( T^{8} \)
$71$ \( 1 - \)\(84\!\cdots\!52\)\( T + \)\(22\!\cdots\!48\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} - \)\(42\!\cdots\!72\)\( T^{7} + \)\(40\!\cdots\!81\)\( T^{8} \)
$73$ \( 1 + \)\(44\!\cdots\!32\)\( T + \)\(51\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(39\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!72\)\( T^{6} + \)\(69\!\cdots\!44\)\( T^{7} + \)\(38\!\cdots\!41\)\( T^{8} \)
$79$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(22\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(91\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \)
$83$ \( 1 + \)\(15\!\cdots\!04\)\( T + \)\(99\!\cdots\!00\)\( T^{2} + \)\(72\!\cdots\!24\)\( T^{3} + \)\(39\!\cdots\!46\)\( T^{4} + \)\(34\!\cdots\!92\)\( T^{5} + \)\(23\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!48\)\( T^{7} + \)\(53\!\cdots\!21\)\( T^{8} \)
$89$ \( 1 - \)\(39\!\cdots\!72\)\( T + \)\(90\!\cdots\!12\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!94\)\( T^{4} - \)\(11\!\cdots\!76\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} - \)\(23\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!41\)\( T^{8} \)
$97$ \( 1 - \)\(36\!\cdots\!52\)\( T + \)\(23\!\cdots\!52\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} - \)\(59\!\cdots\!94\)\( T^{4} + \)\(91\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(86\!\cdots\!96\)\( T^{7} + \)\(67\!\cdots\!81\)\( T^{8} \)
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