# Properties

 Label 9.44.a.a Level $9$ Weight $44$ Character orbit 9.a Self dual yes Analytic conductor $105.399$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$44$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$105.399355811$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 908401710 x + 974756489742$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{13}\cdot 3^{5}\cdot 5\cdot 11$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1619008 + \beta_{1} ) q^{2} + ( -593686009856 - 3392488 \beta_{1} + 4 \beta_{2} ) q^{4} + ( 169251107035090 - 87483970 \beta_{1} + 1317 \beta_{2} ) q^{5} + ( -544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2} ) q^{7} + ( -3732086898157617152 - 606520958464 \beta_{1} - 19428096 \beta_{2} ) q^{8} +O(q^{10})$$ $$q +(-1619008 + \beta_{1}) q^{2} +(-593686009856 - 3392488 \beta_{1} + 4 \beta_{2}) q^{4} +(169251107035090 - 87483970 \beta_{1} + 1317 \beta_{2}) q^{5} +(-544389767902363096 + 407863957886 \beta_{1} + 1184197 \beta_{2}) q^{7} +(-3732086898157617152 - 606520958464 \beta_{1} - 19428096 \beta_{2}) q^{8} +(-$$$$76\!\cdots\!40$$$$+ 1235354107077970 \beta_{1} - 2278729792 \beta_{2}) q^{10} +($$$$92\!\cdots\!40$$$$- 7617306526252796 \beta_{1} + 14933373558 \beta_{2}) q^{11} +(-$$$$33\!\cdots\!50$$$$- 202202396925505444 \beta_{1} + 641923722778 \beta_{2}) q^{13} +($$$$31\!\cdots\!76$$$$- 448634534786039896 \beta_{1} - 102843306048 \beta_{2}) q^{14} +($$$$78\!\cdots\!12$$$$+ 13746042266337042432 \beta_{1} - 9157309919232 \beta_{2}) q^{16} +($$$$53\!\cdots\!94$$$$- 90683027194690706820 \beta_{1} - 201871787631414 \beta_{2}) q^{17} +(-$$$$10\!\cdots\!88$$$$+$$$$40\!\cdots\!04$$$$\beta_{1} - 2147995640621886 \beta_{2}) q^{19} +($$$$66\!\cdots\!20$$$$-$$$$37\!\cdots\!60$$$$\beta_{1} - 3305756267279544 \beta_{2}) q^{20} +(-$$$$57\!\cdots\!08$$$$+$$$$33\!\cdots\!40$$$$\beta_{1} - 52339689282150272 \beta_{2}) q^{22} +(-$$$$38\!\cdots\!68$$$$+$$$$60\!\cdots\!28$$$$\beta_{1} - 32453105180134026 \beta_{2}) q^{23} +($$$$60\!\cdots\!75$$$$-$$$$14\!\cdots\!00$$$$\beta_{1} - 758897906364611660 \beta_{2}) q^{25} +(-$$$$59\!\cdots\!12$$$$+$$$$47\!\cdots\!90$$$$\beta_{1} - 1748929988964422784 \beta_{2}) q^{26} +(-$$$$28\!\cdots\!48$$$$+$$$$29\!\cdots\!48$$$$\beta_{1} - 12060227383697492832 \beta_{2}) q^{28} +($$$$94\!\cdots\!86$$$$+$$$$12\!\cdots\!42$$$$\beta_{1} - 20892595961060032401 \beta_{2}) q^{29} +(-$$$$18\!\cdots\!12$$$$+$$$$35\!\cdots\!42$$$$\beta_{1} +$$$$13\!\cdots\!53$$$$\beta_{2}) q^{31} +($$$$96\!\cdots\!96$$$$-$$$$17\!\cdots\!16$$$$\beta_{1} +$$$$23\!\cdots\!48$$$$\beta_{2}) q^{32} +(-$$$$59\!\cdots\!92$$$$+$$$$74\!\cdots\!34$$$$\beta_{1} - 67083608408202293376 \beta_{2}) q^{34} +($$$$12\!\cdots\!40$$$$+$$$$46\!\cdots\!80$$$$\beta_{1} -$$$$16\!\cdots\!58$$$$\beta_{2}) q^{35} +(-$$$$25\!\cdots\!26$$$$-$$$$15\!\cdots\!64$$$$\beta_{1} -$$$$37\!\cdots\!36$$$$\beta_{2}) q^{37} +($$$$39\!\cdots\!36$$$$-$$$$32\!\cdots\!48$$$$\beta_{1} +$$$$47\!\cdots\!12$$$$\beta_{2}) q^{38} +(-$$$$25\!\cdots\!00$$$$+$$$$15\!\cdots\!00$$$$\beta_{1} +$$$$98\!\cdots\!80$$$$\beta_{2}) q^{40} +($$$$50\!\cdots\!22$$$$+$$$$21\!\cdots\!68$$$$\beta_{1} +$$$$13\!\cdots\!78$$$$\beta_{2}) q^{41} +(-$$$$16\!\cdots\!96$$$$+$$$$14\!\cdots\!44$$$$\beta_{1} +$$$$35\!\cdots\!70$$$$\beta_{2}) q^{43} +($$$$19\!\cdots\!24$$$$-$$$$85\!\cdots\!20$$$$\beta_{1} +$$$$77\!\cdots\!88$$$$\beta_{2}) q^{44} +($$$$40\!\cdots\!88$$$$-$$$$16\!\cdots\!48$$$$\beta_{1} +$$$$29\!\cdots\!48$$$$\beta_{2}) q^{46} +(-$$$$17\!\cdots\!12$$$$+$$$$41\!\cdots\!68$$$$\beta_{1} -$$$$50\!\cdots\!22$$$$\beta_{2}) q^{47} +($$$$39\!\cdots\!01$$$$+$$$$23\!\cdots\!36$$$$\beta_{1} -$$$$14\!\cdots\!52$$$$\beta_{2}) q^{49} +(-$$$$17\!\cdots\!00$$$$+$$$$33\!\cdots\!75$$$$\beta_{1} +$$$$53\!\cdots\!60$$$$\beta_{2}) q^{50} +($$$$65\!\cdots\!36$$$$-$$$$85\!\cdots\!20$$$$\beta_{1} -$$$$12\!\cdots\!40$$$$\beta_{2}) q^{52} +(-$$$$10\!\cdots\!78$$$$-$$$$57\!\cdots\!02$$$$\beta_{1} -$$$$38\!\cdots\!57$$$$\beta_{2}) q^{53} +($$$$24\!\cdots\!20$$$$-$$$$95\!\cdots\!60$$$$\beta_{1} +$$$$20\!\cdots\!16$$$$\beta_{2}) q^{55} +(-$$$$21\!\cdots\!40$$$$-$$$$77\!\cdots\!00$$$$\beta_{1} +$$$$19\!\cdots\!28$$$$\beta_{2}) q^{56} +($$$$55\!\cdots\!88$$$$-$$$$27\!\cdots\!14$$$$\beta_{1} +$$$$81\!\cdots\!04$$$$\beta_{2}) q^{58} +($$$$59\!\cdots\!92$$$$-$$$$17\!\cdots\!36$$$$\beta_{1} +$$$$22\!\cdots\!36$$$$\beta_{2}) q^{59} +(-$$$$41\!\cdots\!06$$$$-$$$$87\!\cdots\!96$$$$\beta_{1} -$$$$17\!\cdots\!68$$$$\beta_{2}) q^{61} +($$$$22\!\cdots\!52$$$$+$$$$14\!\cdots\!48$$$$\beta_{1} -$$$$62\!\cdots\!40$$$$\beta_{2}) q^{62} +(-$$$$32\!\cdots\!12$$$$+$$$$17\!\cdots\!40$$$$\beta_{1} -$$$$33\!\cdots\!36$$$$\beta_{2}) q^{64} +($$$$85\!\cdots\!80$$$$-$$$$20\!\cdots\!40$$$$\beta_{1} -$$$$89\!\cdots\!26$$$$\beta_{2}) q^{65} +($$$$43\!\cdots\!16$$$$-$$$$72\!\cdots\!40$$$$\beta_{1} +$$$$22\!\cdots\!08$$$$\beta_{2}) q^{67} +($$$$90\!\cdots\!56$$$$+$$$$26\!\cdots\!08$$$$\beta_{1} +$$$$21\!\cdots\!84$$$$\beta_{2}) q^{68} +($$$$62\!\cdots\!60$$$$-$$$$76\!\cdots\!80$$$$\beta_{1} +$$$$43\!\cdots\!08$$$$\beta_{2}) q^{70} +(-$$$$74\!\cdots\!72$$$$-$$$$47\!\cdots\!60$$$$\beta_{1} +$$$$13\!\cdots\!02$$$$\beta_{2}) q^{71} +($$$$10\!\cdots\!18$$$$+$$$$89\!\cdots\!92$$$$\beta_{1} -$$$$57\!\cdots\!40$$$$\beta_{2}) q^{73} +(-$$$$42\!\cdots\!44$$$$-$$$$25\!\cdots\!46$$$$\beta_{1} -$$$$52\!\cdots\!60$$$$\beta_{2}) q^{74} +(-$$$$15\!\cdots\!68$$$$+$$$$95\!\cdots\!24$$$$\beta_{1} -$$$$11\!\cdots\!36$$$$\beta_{2}) q^{76} +(-$$$$53\!\cdots\!48$$$$+$$$$68\!\cdots\!28$$$$\beta_{1} -$$$$22\!\cdots\!80$$$$\beta_{2}) q^{77} +(-$$$$14\!\cdots\!60$$$$+$$$$10\!\cdots\!50$$$$\beta_{1} -$$$$24\!\cdots\!83$$$$\beta_{2}) q^{79} +(-$$$$17\!\cdots\!60$$$$+$$$$14\!\cdots\!80$$$$\beta_{1} +$$$$15\!\cdots\!72$$$$\beta_{2}) q^{80} +(-$$$$69\!\cdots\!32$$$$+$$$$47\!\cdots\!02$$$$\beta_{1} +$$$$66\!\cdots\!64$$$$\beta_{2}) q^{82} +(-$$$$29\!\cdots\!56$$$$+$$$$90\!\cdots\!92$$$$\beta_{1} +$$$$21\!\cdots\!90$$$$\beta_{2}) q^{83} +(-$$$$20\!\cdots\!40$$$$-$$$$98\!\cdots\!80$$$$\beta_{1} +$$$$23\!\cdots\!38$$$$\beta_{2}) q^{85} +($$$$35\!\cdots\!40$$$$-$$$$19\!\cdots\!16$$$$\beta_{1} +$$$$51\!\cdots\!56$$$$\beta_{2}) q^{86} +(-$$$$28\!\cdots\!28$$$$+$$$$11\!\cdots\!24$$$$\beta_{1} +$$$$51\!\cdots\!28$$$$\beta_{2}) q^{88} +(-$$$$21\!\cdots\!02$$$$-$$$$24\!\cdots\!04$$$$\beta_{1} +$$$$14\!\cdots\!20$$$$\beta_{2}) q^{89} +($$$$45\!\cdots\!08$$$$+$$$$23\!\cdots\!72$$$$\beta_{1} -$$$$15\!\cdots\!34$$$$\beta_{2}) q^{91} +(-$$$$12\!\cdots\!24$$$$+$$$$36\!\cdots\!24$$$$\beta_{1} -$$$$81\!\cdots\!12$$$$\beta_{2}) q^{92} +($$$$26\!\cdots\!40$$$$-$$$$12\!\cdots\!32$$$$\beta_{1} +$$$$24\!\cdots\!64$$$$\beta_{2}) q^{94} +(-$$$$31\!\cdots\!00$$$$+$$$$59\!\cdots\!00$$$$\beta_{1} +$$$$14\!\cdots\!60$$$$\beta_{2}) q^{95} +($$$$20\!\cdots\!26$$$$+$$$$21\!\cdots\!80$$$$\beta_{1} -$$$$24\!\cdots\!92$$$$\beta_{2}) q^{97} +($$$$67\!\cdots\!20$$$$-$$$$10\!\cdots\!59$$$$\beta_{1} +$$$$30\!\cdots\!16$$$$\beta_{2}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 4857024q^{2} - 1781058029568q^{4} + 507753321105270q^{5} - 1633169303707089288q^{7} - 11196260694472851456q^{8} + O(q^{10})$$ $$3q - 4857024q^{2} - 1781058029568q^{4} + 507753321105270q^{5} - 1633169303707089288q^{7} - 11196260694472851456q^{8} -$$$$22\!\cdots\!20$$$$q^{10} +$$$$27\!\cdots\!20$$$$q^{11} -$$$$99\!\cdots\!50$$$$q^{13} +$$$$94\!\cdots\!28$$$$q^{14} +$$$$23\!\cdots\!36$$$$q^{16} +$$$$16\!\cdots\!82$$$$q^{17} -$$$$32\!\cdots\!64$$$$q^{19} +$$$$19\!\cdots\!60$$$$q^{20} -$$$$17\!\cdots\!24$$$$q^{22} -$$$$11\!\cdots\!04$$$$q^{23} +$$$$18\!\cdots\!25$$$$q^{25} -$$$$17\!\cdots\!36$$$$q^{26} -$$$$84\!\cdots\!44$$$$q^{28} +$$$$28\!\cdots\!58$$$$q^{29} -$$$$56\!\cdots\!36$$$$q^{31} +$$$$29\!\cdots\!88$$$$q^{32} -$$$$17\!\cdots\!76$$$$q^{34} +$$$$36\!\cdots\!20$$$$q^{35} -$$$$77\!\cdots\!78$$$$q^{37} +$$$$11\!\cdots\!08$$$$q^{38} -$$$$75\!\cdots\!00$$$$q^{40} +$$$$15\!\cdots\!66$$$$q^{41} -$$$$50\!\cdots\!88$$$$q^{43} +$$$$58\!\cdots\!72$$$$q^{44} +$$$$12\!\cdots\!64$$$$q^{46} -$$$$51\!\cdots\!36$$$$q^{47} +$$$$11\!\cdots\!03$$$$q^{49} -$$$$53\!\cdots\!00$$$$q^{50} +$$$$19\!\cdots\!08$$$$q^{52} -$$$$31\!\cdots\!34$$$$q^{53} +$$$$72\!\cdots\!60$$$$q^{55} -$$$$64\!\cdots\!20$$$$q^{56} +$$$$16\!\cdots\!64$$$$q^{58} +$$$$17\!\cdots\!76$$$$q^{59} -$$$$12\!\cdots\!18$$$$q^{61} +$$$$68\!\cdots\!56$$$$q^{62} -$$$$97\!\cdots\!36$$$$q^{64} +$$$$25\!\cdots\!40$$$$q^{65} +$$$$12\!\cdots\!48$$$$q^{67} +$$$$27\!\cdots\!68$$$$q^{68} +$$$$18\!\cdots\!80$$$$q^{70} -$$$$22\!\cdots\!16$$$$q^{71} +$$$$31\!\cdots\!54$$$$q^{73} -$$$$12\!\cdots\!32$$$$q^{74} -$$$$45\!\cdots\!04$$$$q^{76} -$$$$15\!\cdots\!44$$$$q^{77} -$$$$43\!\cdots\!80$$$$q^{79} -$$$$51\!\cdots\!80$$$$q^{80} -$$$$20\!\cdots\!96$$$$q^{82} -$$$$89\!\cdots\!68$$$$q^{83} -$$$$60\!\cdots\!20$$$$q^{85} +$$$$10\!\cdots\!20$$$$q^{86} -$$$$86\!\cdots\!84$$$$q^{88} -$$$$65\!\cdots\!06$$$$q^{89} +$$$$13\!\cdots\!24$$$$q^{91} -$$$$37\!\cdots\!72$$$$q^{92} +$$$$78\!\cdots\!20$$$$q^{94} -$$$$93\!\cdots\!00$$$$q^{95} +$$$$61\!\cdots\!78$$$$q^{97} +$$$$20\!\cdots\!60$$$$q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 908401710 x + 974756489742$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$96 \nu - 32$$ $$\beta_{2}$$ $$=$$ $$2304 \nu^{2} + 3705792 \nu - 1395306262592$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 32$$$$)/96$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 38602 \beta_{1} + 1395305027328$$$$)/2304$$

## 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
 −30662.1 1074.41 29588.6
−4.56260e6 0 1.20212e13 1.29230e15 0 −9.66721e17 −1.47149e19 0 −5.89624e21
1.2 −1.51590e6 0 −6.49815e12 −1.66864e15 0 −2.14679e18 2.31845e19 0 2.52949e21
1.3 1.22147e6 0 −7.30411e12 8.84097e14 0 1.48034e18 −1.96659e19 0 1.07990e21
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 9.44.a.a 3
3.b odd 2 1 3.44.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.44.a.a 3 3.b odd 2 1
9.44.a.a 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 4857024 T_{2}^{2} - 508269450240 T_{2} -$$8448203181300645888

'>$$84\!\cdots\!88$$ acting on $$S_{44}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4857024 T + 25880009616384 T^{2} + 76997466648892932096 T^{3} +$$$$22\!\cdots\!72$$$$T^{4} +$$$$37\!\cdots\!36$$$$T^{5} +$$$$68\!\cdots\!12$$$$T^{6}$$
$3$ 1
$5$ $$1 - 507753321105270 T +$$$$92\!\cdots\!75$$$$T^{2} +$$$$75\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!75$$$$T^{4} -$$$$65\!\cdots\!50$$$$T^{5} +$$$$14\!\cdots\!25$$$$T^{6}$$
$7$ $$1 + 1633169303707089288 T +$$$$40\!\cdots\!85$$$$T^{2} +$$$$40\!\cdots\!48$$$$T^{3} +$$$$87\!\cdots\!55$$$$T^{4} +$$$$77\!\cdots\!12$$$$T^{5} +$$$$10\!\cdots\!07$$$$T^{6}$$
$11$ $$1 -$$$$27\!\cdots\!20$$$$T +$$$$12\!\cdots\!61$$$$T^{2} -$$$$34\!\cdots\!16$$$$T^{3} +$$$$75\!\cdots\!91$$$$T^{4} -$$$$10\!\cdots\!20$$$$T^{5} +$$$$21\!\cdots\!91$$$$T^{6}$$
$13$ $$1 +$$$$99\!\cdots\!50$$$$T +$$$$17\!\cdots\!59$$$$T^{2} +$$$$12\!\cdots\!56$$$$T^{3} +$$$$14\!\cdots\!23$$$$T^{4} +$$$$62\!\cdots\!50$$$$T^{5} +$$$$49\!\cdots\!73$$$$T^{6}$$
$17$ $$1 -$$$$16\!\cdots\!82$$$$T +$$$$12\!\cdots\!47$$$$T^{2} -$$$$46\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!11$$$$T^{4} -$$$$10\!\cdots\!58$$$$T^{5} +$$$$53\!\cdots\!97$$$$T^{6}$$
$19$ $$1 +$$$$32\!\cdots\!64$$$$T +$$$$24\!\cdots\!37$$$$T^{2} +$$$$51\!\cdots\!52$$$$T^{3} +$$$$23\!\cdots\!83$$$$T^{4} +$$$$30\!\cdots\!84$$$$T^{5} +$$$$91\!\cdots\!79$$$$T^{6}$$
$23$ $$1 +$$$$11\!\cdots\!04$$$$T +$$$$79\!\cdots\!65$$$$T^{2} +$$$$85\!\cdots\!56$$$$T^{3} +$$$$28\!\cdots\!55$$$$T^{4} +$$$$14\!\cdots\!56$$$$T^{5} +$$$$46\!\cdots\!63$$$$T^{6}$$
$29$ $$1 -$$$$28\!\cdots\!58$$$$T +$$$$58\!\cdots\!27$$$$T^{2} +$$$$63\!\cdots\!76$$$$T^{3} +$$$$45\!\cdots\!03$$$$T^{4} -$$$$16\!\cdots\!18$$$$T^{5} +$$$$44\!\cdots\!69$$$$T^{6}$$
$31$ $$1 +$$$$56\!\cdots\!36$$$$T +$$$$27\!\cdots\!97$$$$T^{2} +$$$$31\!\cdots\!52$$$$T^{3} +$$$$36\!\cdots\!27$$$$T^{4} +$$$$10\!\cdots\!16$$$$T^{5} +$$$$24\!\cdots\!71$$$$T^{6}$$
$37$ $$1 +$$$$77\!\cdots\!78$$$$T +$$$$61\!\cdots\!75$$$$T^{2} +$$$$40\!\cdots\!48$$$$T^{3} +$$$$16\!\cdots\!75$$$$T^{4} +$$$$56\!\cdots\!02$$$$T^{5} +$$$$19\!\cdots\!77$$$$T^{6}$$
$41$ $$1 -$$$$15\!\cdots\!66$$$$T +$$$$14\!\cdots\!07$$$$T^{2} -$$$$79\!\cdots\!12$$$$T^{3} +$$$$31\!\cdots\!47$$$$T^{4} -$$$$75\!\cdots\!06$$$$T^{5} +$$$$11\!\cdots\!61$$$$T^{6}$$
$43$ $$1 +$$$$50\!\cdots\!88$$$$T +$$$$13\!\cdots\!17$$$$T^{2} +$$$$21\!\cdots\!80$$$$T^{3} +$$$$23\!\cdots\!19$$$$T^{4} +$$$$15\!\cdots\!12$$$$T^{5} +$$$$52\!\cdots\!43$$$$T^{6}$$
$47$ $$1 +$$$$51\!\cdots\!36$$$$T +$$$$63\!\cdots\!93$$$$T^{2} +$$$$14\!\cdots\!20$$$$T^{3} +$$$$50\!\cdots\!39$$$$T^{4} +$$$$32\!\cdots\!44$$$$T^{5} +$$$$50\!\cdots\!67$$$$T^{6}$$
$53$ $$1 +$$$$31\!\cdots\!34$$$$T +$$$$11\!\cdots\!15$$$$T^{2} +$$$$18\!\cdots\!76$$$$T^{3} +$$$$16\!\cdots\!55$$$$T^{4} +$$$$61\!\cdots\!86$$$$T^{5} +$$$$27\!\cdots\!33$$$$T^{6}$$
$59$ $$1 -$$$$17\!\cdots\!76$$$$T +$$$$51\!\cdots\!17$$$$T^{2} -$$$$51\!\cdots\!08$$$$T^{3} +$$$$72\!\cdots\!43$$$$T^{4} -$$$$34\!\cdots\!16$$$$T^{5} +$$$$27\!\cdots\!39$$$$T^{6}$$
$61$ $$1 +$$$$12\!\cdots\!18$$$$T +$$$$74\!\cdots\!59$$$$T^{2} +$$$$22\!\cdots\!04$$$$T^{3} +$$$$43\!\cdots\!79$$$$T^{4} +$$$$42\!\cdots\!98$$$$T^{5} +$$$$20\!\cdots\!41$$$$T^{6}$$
$67$ $$1 -$$$$12\!\cdots\!48$$$$T +$$$$60\!\cdots\!57$$$$T^{2} -$$$$84\!\cdots\!44$$$$T^{3} +$$$$20\!\cdots\!91$$$$T^{4} -$$$$14\!\cdots\!12$$$$T^{5} +$$$$36\!\cdots\!47$$$$T^{6}$$
$71$ $$1 +$$$$22\!\cdots\!16$$$$T +$$$$28\!\cdots\!85$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!35$$$$T^{4} +$$$$36\!\cdots\!36$$$$T^{5} +$$$$64\!\cdots\!31$$$$T^{6}$$
$73$ $$1 -$$$$31\!\cdots\!54$$$$T +$$$$67\!\cdots\!75$$$$T^{2} -$$$$89\!\cdots\!76$$$$T^{3} +$$$$88\!\cdots\!75$$$$T^{4} -$$$$55\!\cdots\!06$$$$T^{5} +$$$$23\!\cdots\!13$$$$T^{6}$$
$79$ $$1 +$$$$43\!\cdots\!80$$$$T +$$$$10\!\cdots\!17$$$$T^{2} +$$$$34\!\cdots\!40$$$$T^{3} +$$$$42\!\cdots\!63$$$$T^{4} +$$$$68\!\cdots\!80$$$$T^{5} +$$$$62\!\cdots\!19$$$$T^{6}$$
$83$ $$1 +$$$$89\!\cdots\!68$$$$T +$$$$32\!\cdots\!21$$$$T^{2} +$$$$24\!\cdots\!88$$$$T^{3} +$$$$10\!\cdots\!27$$$$T^{4} +$$$$98\!\cdots\!92$$$$T^{5} +$$$$36\!\cdots\!03$$$$T^{6}$$
$89$ $$1 +$$$$65\!\cdots\!06$$$$T +$$$$14\!\cdots\!07$$$$T^{2} -$$$$34\!\cdots\!72$$$$T^{3} +$$$$97\!\cdots\!83$$$$T^{4} +$$$$28\!\cdots\!66$$$$T^{5} +$$$$29\!\cdots\!09$$$$T^{6}$$
$97$ $$1 -$$$$61\!\cdots\!78$$$$T +$$$$84\!\cdots\!47$$$$T^{2} -$$$$32\!\cdots\!64$$$$T^{3} +$$$$22\!\cdots\!31$$$$T^{4} -$$$$44\!\cdots\!62$$$$T^{5} +$$$$19\!\cdots\!17$$$$T^{6}$$