# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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