Properties

Label 9.38.a.c
Level 9
Weight 38
Character orbit 9.a
Self dual yes
Analytic conductor 78.043
Analytic rank 0
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0426343121\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + ( -109391 + \beta_{1} ) q^{2} + ( 86524834843 - 191939 \beta_{1} + \beta_{3} ) q^{4} + ( 1024957187485 + 503509 \beta_{1} - \beta_{2} + 28 \beta_{3} ) q^{5} + ( 1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3} ) q^{7} + ( -35121079860249342 + 103049124558 \beta_{1} + 5440 \beta_{2} - 299578 \beta_{3} ) q^{8} +O(q^{10})\) \( q +(-109391 + \beta_{1}) q^{2} +(86524834843 - 191939 \beta_{1} + \beta_{3}) q^{4} +(1024957187485 + 503509 \beta_{1} - \beta_{2} + 28 \beta_{3}) q^{5} +(1651450994333561 + 2985430279 \beta_{1} + 405 \beta_{2} + 20404 \beta_{3}) q^{7} +(-35121079860249342 + 103049124558 \beta_{1} + 5440 \beta_{2} - 299578 \beta_{3}) q^{8} +(-5380189282634790 + 4867048785914 \beta_{1} + 290304 \beta_{2} - 15949312 \beta_{3}) q^{10} +(-5238439959104483494 + 25492139539658 \beta_{1} + 1387870 \beta_{2} + 26831288 \beta_{3}) q^{11} +(12957972542954522468 - 498690534219638 \beta_{1} + 14789790 \beta_{2} + 865098424 \beta_{3}) q^{13} +(\)\(45\!\cdots\!20\)\( + 4216580839272024 \beta_{1} + 55114240 \beta_{2} + 6231930368 \beta_{3}) q^{14} +(\)\(13\!\cdots\!28\)\( - 58713905237284428 \beta_{1} - 2380337280 \beta_{2} + 70964221188 \beta_{3}) q^{16} +(-\)\(20\!\cdots\!68\)\( + 26353299406777634 \beta_{1} - 6152361370 \beta_{2} + 41466582232 \beta_{3}) q^{17} +(-\)\(13\!\cdots\!30\)\( + 796715087148526350 \beta_{1} - 23805860310 \beta_{2} + 691505159016 \beta_{3}) q^{19} +(\)\(89\!\cdots\!90\)\( - 2683989303798924654 \beta_{1} + 10617389056 \beta_{2} + 6636900927482 \beta_{3}) q^{20} +(\)\(59\!\cdots\!56\)\( - 3659504875238983628 \beta_{1} - 45541647360 \beta_{2} + 41255532921856 \beta_{3}) q^{22} +(\)\(15\!\cdots\!30\)\( - 16703658020546342574 \beta_{1} + 178310885110 \beta_{2} + 114016465995032 \beta_{3}) q^{23} +(\)\(23\!\cdots\!15\)\( - \)\(10\!\cdots\!64\)\( \beta_{1} + 6346460995596 \beta_{2} - 254265946315088 \beta_{3}) q^{25} +(-\)\(10\!\cdots\!10\)\( + \)\(17\!\cdots\!62\)\( \beta_{1} + 2665381043200 \beta_{2} - 393049925745664 \beta_{3}) q^{26} +(\)\(61\!\cdots\!04\)\( + \)\(55\!\cdots\!60\)\( \beta_{1} - 29365958246400 \beta_{2} + 1482154037997144 \beta_{3}) q^{28} +(-\)\(10\!\cdots\!69\)\( - \)\(14\!\cdots\!09\)\( \beta_{1} - 23642784642315 \beta_{2} + 2583712585520436 \beta_{3}) q^{29} +(\)\(22\!\cdots\!29\)\( + \)\(56\!\cdots\!67\)\( \beta_{1} - 144867143474775 \beta_{2} - 6432129197675132 \beta_{3}) q^{31} +(-\)\(91\!\cdots\!96\)\( + \)\(14\!\cdots\!96\)\( \beta_{1} - 33174084381440 \beta_{2} - 57167919940477864 \beta_{3}) q^{32} +(\)\(78\!\cdots\!14\)\( - \)\(16\!\cdots\!50\)\( \beta_{1} + 1074505638620160 \beta_{2} - 60791269671865344 \beta_{3}) q^{34} +(-\)\(10\!\cdots\!50\)\( + \)\(42\!\cdots\!10\)\( \beta_{1} + 126089450324010 \beta_{2} + 175002471366658920 \beta_{3}) q^{35} +(\)\(13\!\cdots\!14\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} + 556310835696600 \beta_{2} + 508236733739231328 \beta_{3}) q^{37} +(\)\(18\!\cdots\!40\)\( - \)\(10\!\cdots\!80\)\( \beta_{1} + 7046615894062080 \beta_{2} + 402356896847195136 \beta_{3}) q^{38} +(-\)\(66\!\cdots\!80\)\( + \)\(13\!\cdots\!08\)\( \beta_{1} - 5259366714736512 \beta_{2} - 1063635595164787364 \beta_{3}) q^{40} +(\)\(21\!\cdots\!68\)\( + \)\(50\!\cdots\!30\)\( \beta_{1} - 46408136639113950 \beta_{2} + 101394467803539528 \beta_{3}) q^{41} +(-\)\(12\!\cdots\!18\)\( + \)\(36\!\cdots\!66\)\( \beta_{1} - 106767768332859570 \beta_{2} + 3642447885337803640 \beta_{3}) q^{43} +(-\)\(70\!\cdots\!20\)\( + \)\(84\!\cdots\!24\)\( \beta_{1} + 39966717409034240 \beta_{2} - 12399904291854550988 \beta_{3}) q^{44} +(-\)\(37\!\cdots\!96\)\( + \)\(18\!\cdots\!52\)\( \beta_{1} + 595645525841955840 \beta_{2} - 26579969078865144832 \beta_{3}) q^{46} +(-\)\(10\!\cdots\!42\)\( - \)\(12\!\cdots\!70\)\( \beta_{1} + 580153156762703090 \beta_{2} + 13645452734161870216 \beta_{3}) q^{47} +(\)\(10\!\cdots\!93\)\( + \)\(55\!\cdots\!36\)\( \beta_{1} - 335873647667195280 \beta_{2} + 40861585860796998592 \beta_{3}) q^{49} +(-\)\(25\!\cdots\!85\)\( - \)\(23\!\cdots\!69\)\( \beta_{1} - 2258916821970397184 \beta_{2} + 3778457216840394752 \beta_{3}) q^{50} +(\)\(46\!\cdots\!42\)\( - \)\(10\!\cdots\!10\)\( \beta_{1} - 4538664793591971840 \beta_{2} + \)\(13\!\cdots\!30\)\( \beta_{3}) q^{52} +(\)\(31\!\cdots\!59\)\( + \)\(45\!\cdots\!55\)\( \beta_{1} + 1922334687103569825 \beta_{2} + \)\(40\!\cdots\!04\)\( \beta_{3}) q^{53} +(-\)\(81\!\cdots\!60\)\( + \)\(16\!\cdots\!96\)\( \beta_{1} + 8408678824795228056 \beta_{2} + 22892580198991444832 \beta_{3}) q^{55} +(-\)\(12\!\cdots\!84\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + 4540106882371079680 \beta_{2} - \)\(85\!\cdots\!52\)\( \beta_{3}) q^{56} +(-\)\(18\!\cdots\!22\)\( - \)\(56\!\cdots\!82\)\( \beta_{1} + 17317722461316364800 \beta_{2} - \)\(20\!\cdots\!48\)\( \beta_{3}) q^{58} +(\)\(33\!\cdots\!36\)\( - \)\(15\!\cdots\!44\)\( \beta_{1} - 3830895234533430040 \beta_{2} - \)\(24\!\cdots\!76\)\( \beta_{3}) q^{59} +(-\)\(30\!\cdots\!62\)\( - \)\(70\!\cdots\!84\)\( \beta_{1} - 56929543616748524580 \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3}) q^{61} +(\)\(94\!\cdots\!24\)\( + \)\(89\!\cdots\!08\)\( \beta_{1} - 15001434910129364480 \beta_{2} + \)\(43\!\cdots\!40\)\( \beta_{3}) q^{62} +(\)\(21\!\cdots\!64\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} + 20735073056676072960 \beta_{2} + \)\(10\!\cdots\!96\)\( \beta_{3}) q^{64} +(-\)\(39\!\cdots\!70\)\( - \)\(30\!\cdots\!78\)\( \beta_{1} - 42166817812091905458 \beta_{2} + \)\(13\!\cdots\!24\)\( \beta_{3}) q^{65} +(\)\(40\!\cdots\!44\)\( + \)\(87\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!80\)\( \beta_{2} - \)\(11\!\cdots\!24\)\( \beta_{3}) q^{67} +(-\)\(15\!\cdots\!98\)\( - \)\(28\!\cdots\!54\)\( \beta_{1} + \)\(36\!\cdots\!40\)\( \beta_{2} - \)\(13\!\cdots\!98\)\( \beta_{3}) q^{68} +(\)\(20\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + \)\(93\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!80\)\( \beta_{3}) q^{70} +(-\)\(27\!\cdots\!38\)\( - \)\(10\!\cdots\!26\)\( \beta_{1} - \)\(11\!\cdots\!90\)\( \beta_{2} - \)\(95\!\cdots\!68\)\( \beta_{3}) q^{71} +(-\)\(48\!\cdots\!86\)\( + \)\(22\!\cdots\!88\)\( \beta_{1} - \)\(19\!\cdots\!00\)\( \beta_{2} - \)\(24\!\cdots\!60\)\( \beta_{3}) q^{73} +(\)\(22\!\cdots\!94\)\( + \)\(74\!\cdots\!58\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2} + \)\(65\!\cdots\!20\)\( \beta_{3}) q^{74} +(-\)\(23\!\cdots\!00\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(44\!\cdots\!40\)\( \beta_{2} - \)\(14\!\cdots\!16\)\( \beta_{3}) q^{76} +(\)\(64\!\cdots\!64\)\( + \)\(17\!\cdots\!28\)\( \beta_{1} - \)\(20\!\cdots\!40\)\( \beta_{2} - \)\(10\!\cdots\!00\)\( \beta_{3}) q^{77} +(\)\(10\!\cdots\!01\)\( - \)\(96\!\cdots\!09\)\( \beta_{1} - \)\(13\!\cdots\!95\)\( \beta_{2} + \)\(69\!\cdots\!28\)\( \beta_{3}) q^{79} +(\)\(23\!\cdots\!40\)\( - \)\(55\!\cdots\!44\)\( \beta_{1} - \)\(65\!\cdots\!84\)\( \beta_{2} + \)\(49\!\cdots\!52\)\( \beta_{3}) q^{80} +(\)\(83\!\cdots\!42\)\( + \)\(19\!\cdots\!98\)\( \beta_{1} + \)\(69\!\cdots\!20\)\( \beta_{2} - \)\(12\!\cdots\!12\)\( \beta_{3}) q^{82} +(\)\(11\!\cdots\!62\)\( + \)\(11\!\cdots\!98\)\( \beta_{1} + \)\(74\!\cdots\!50\)\( \beta_{2} + \)\(12\!\cdots\!60\)\( \beta_{3}) q^{83} +(\)\(46\!\cdots\!30\)\( - \)\(33\!\cdots\!98\)\( \beta_{1} + \)\(52\!\cdots\!22\)\( \beta_{2} - \)\(28\!\cdots\!16\)\( \beta_{3}) q^{85} +(\)\(90\!\cdots\!12\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} + \)\(34\!\cdots\!80\)\( \beta_{2} - \)\(14\!\cdots\!24\)\( \beta_{3}) q^{86} +(\)\(10\!\cdots\!84\)\( - \)\(26\!\cdots\!16\)\( \beta_{1} - \)\(66\!\cdots\!60\)\( \beta_{2} + \)\(46\!\cdots\!84\)\( \beta_{3}) q^{88} +(\)\(78\!\cdots\!22\)\( + \)\(10\!\cdots\!72\)\( \beta_{1} - \)\(24\!\cdots\!60\)\( \beta_{2} - \)\(46\!\cdots\!00\)\( \beta_{3}) q^{89} +(\)\(65\!\cdots\!10\)\( - \)\(71\!\cdots\!62\)\( \beta_{1} - \)\(97\!\cdots\!10\)\( \beta_{2} - \)\(44\!\cdots\!44\)\( \beta_{3}) q^{91} +(\)\(41\!\cdots\!44\)\( - \)\(66\!\cdots\!84\)\( \beta_{1} - \)\(25\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!56\)\( \beta_{3}) q^{92} +(-\)\(14\!\cdots\!48\)\( + \)\(90\!\cdots\!68\)\( \beta_{1} - \)\(58\!\cdots\!20\)\( \beta_{2} + \)\(50\!\cdots\!16\)\( \beta_{3}) q^{94} +(\)\(22\!\cdots\!40\)\( + \)\(11\!\cdots\!76\)\( \beta_{1} + \)\(45\!\cdots\!36\)\( \beta_{2} - \)\(20\!\cdots\!08\)\( \beta_{3}) q^{95} +(\)\(11\!\cdots\!34\)\( + \)\(93\!\cdots\!72\)\( \beta_{1} + \)\(73\!\cdots\!80\)\( \beta_{2} - \)\(65\!\cdots\!64\)\( \beta_{3}) q^{97} +(\)\(10\!\cdots\!21\)\( + \)\(11\!\cdots\!25\)\( \beta_{1} + \)\(26\!\cdots\!00\)\( \beta_{2} + \)\(47\!\cdots\!28\)\( \beta_{3}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} + O(q^{10}) \) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} - 21511023001649316q^{10} - 20953708852195292976q^{11} + 51830892788989874168q^{13} + \)\(18\!\cdots\!12\)\(q^{14} + \)\(55\!\cdots\!40\)\(q^{16} - \)\(81\!\cdots\!28\)\(q^{17} - \)\(54\!\cdots\!32\)\(q^{19} + \)\(35\!\cdots\!76\)\(q^{20} + \)\(23\!\cdots\!76\)\(q^{22} + \)\(61\!\cdots\!88\)\(q^{23} + \)\(95\!\cdots\!16\)\(q^{25} - \)\(42\!\cdots\!88\)\(q^{26} + \)\(24\!\cdots\!48\)\(q^{28} - \)\(41\!\cdots\!36\)\(q^{29} + \)\(89\!\cdots\!64\)\(q^{31} - \)\(36\!\cdots\!84\)\(q^{32} + \)\(31\!\cdots\!24\)\(q^{34} - \)\(42\!\cdots\!40\)\(q^{35} + \)\(55\!\cdots\!24\)\(q^{37} + \)\(73\!\cdots\!68\)\(q^{38} - \)\(26\!\cdots\!52\)\(q^{40} + \)\(86\!\cdots\!76\)\(q^{41} - \)\(50\!\cdots\!80\)\(q^{43} - \)\(28\!\cdots\!36\)\(q^{44} - \)\(14\!\cdots\!96\)\(q^{46} - \)\(42\!\cdots\!20\)\(q^{47} + \)\(40\!\cdots\!20\)\(q^{49} - \)\(10\!\cdots\!14\)\(q^{50} + \)\(18\!\cdots\!68\)\(q^{52} + \)\(12\!\cdots\!88\)\(q^{53} - \)\(32\!\cdots\!24\)\(q^{55} - \)\(49\!\cdots\!80\)\(q^{56} - \)\(75\!\cdots\!56\)\(q^{58} + \)\(13\!\cdots\!88\)\(q^{59} - \)\(12\!\cdots\!60\)\(q^{61} + \)\(37\!\cdots\!92\)\(q^{62} + \)\(85\!\cdots\!28\)\(q^{64} - \)\(15\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!48\)\(q^{67} - \)\(63\!\cdots\!84\)\(q^{68} + \)\(82\!\cdots\!60\)\(q^{70} - \)\(10\!\cdots\!88\)\(q^{71} - \)\(19\!\cdots\!48\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} - \)\(95\!\cdots\!68\)\(q^{76} + \)\(25\!\cdots\!92\)\(q^{77} + \)\(42\!\cdots\!20\)\(q^{79} + \)\(95\!\cdots\!36\)\(q^{80} + \)\(33\!\cdots\!48\)\(q^{82} + \)\(46\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!12\)\(q^{85} + \)\(36\!\cdots\!04\)\(q^{86} + \)\(42\!\cdots\!56\)\(q^{88} + \)\(31\!\cdots\!52\)\(q^{89} + \)\(26\!\cdots\!24\)\(q^{91} + \)\(16\!\cdots\!16\)\(q^{92} - \)\(57\!\cdots\!48\)\(q^{94} + \)\(89\!\cdots\!56\)\(q^{95} + \)\(44\!\cdots\!48\)\(q^{97} + \)\(43\!\cdots\!78\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 11777633936 x^{2} - 35120319927360 x + 11967042111800832000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( 27 \nu^{3} - 128691 \nu^{2} - 262552030440 \nu + 46478406852080 \)\()/680\)
\(\beta_{3}\)\(=\)\( 36 \nu^{2} - 161070 \nu - 211997370590 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 26845 \beta_{1} + 211997397435\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(14299 \beta_{3} + 2720 \beta_{2} + 175418543615 \beta_{1} + 2845612193201705\)\()/108\)

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
−101317.
−35434.6
31743.2
105009.
−717293. 0 3.77070e11 9.63758e12 0 1.74370e15 −1.71886e17 0 −6.91297e18
1.2 −322000. 0 −3.37552e10 −1.53382e13 0 2.48687e15 5.51245e16 0 4.93889e18
1.3 81067.1 0 −1.30867e11 7.16603e12 0 −5.96869e15 −2.17508e16 0 5.80929e17
1.4 520663. 0 1.33651e11 2.63441e12 0 8.34393e15 −1.97212e15 0 1.37164e18
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 9.38.a.c 4
3.b odd 2 1 3.38.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.38.a.b 4 3.b odd 2 1
9.38.a.c 4 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 437562 T_{2}^{3} - 352197132768 T_{2}^{2} - \)\(95\!\cdots\!24\)\( T_{2} + \)\(97\!\cdots\!04\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke Characteristic Polynomials

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