# Properties

 Label 2.50.a.a Level $2$ Weight $50$ Character orbit 2.a Self dual yes Analytic conductor $30.413$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2$$ Weight: $$k$$ $$=$$ $$50$$ Character orbit: $$[\chi]$$ $$=$$ 2.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.4132410198$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 22129540960032$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{7}\cdot 3^{3}\cdot 5^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 43200\sqrt{88518163840129}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -16777216 q^{2} + ( 140525537796 - \beta ) q^{3} + 281474976710656 q^{4} + ( 41587265591271750 - 349020 \beta ) q^{5} + ( -2357627301119655936 + 16777216 \beta ) q^{6} + ( -216194632642999109368 + 1176353838 \beta ) q^{7} -4722366482869645213696 q^{8} + ( -54355764372760160092467 - 281051075592 \beta ) q^{9} +O(q^{10})$$ $$q -16777216 q^{2} +(140525537796 - \beta) q^{3} +281474976710656 q^{4} +(41587265591271750 - 349020 \beta) q^{5} +(-2357627301119655936 + 16777216 \beta) q^{6} +(-$$$$21\!\cdots\!68$$$$+ 1176353838 \beta) q^{7} -$$$$47\!\cdots\!96$$$$q^{8} +(-$$$$54\!\cdots\!67$$$$- 281051075592 \beta) q^{9} +(-$$$$69\!\cdots\!00$$$$+ 5855583928320 \beta) q^{10} +(-$$$$12\!\cdots\!68$$$$+ 91137694171293 \beta) q^{11} +($$$$39\!\cdots\!76$$$$- 281474976710656 \beta) q^{12} +(-$$$$83\!\cdots\!14$$$$+ 4871108754480708 \beta) q^{13} +($$$$36\!\cdots\!88$$$$- 19735942432555008 \beta) q^{14} +($$$$63\!\cdots\!00$$$$- 90633488792831670 \beta) q^{15} +$$$$79\!\cdots\!36$$$$q^{16} +($$$$62\!\cdots\!82$$$$+ 1739956737232606392 \beta) q^{17} +($$$$91\!\cdots\!72$$$$+ 4715254602239311872 \beta) q^{18} +($$$$15\!\cdots\!60$$$$- 15619643210757982437 \beta) q^{19} +($$$$11\!\cdots\!00$$$$- 98240396371553157120 \beta) q^{20} +(-$$$$22\!\cdots\!28$$$$+$$$$38\!\cdots\!16$$$$\beta) q^{21} +($$$$21\!\cdots\!88$$$$-$$$$15\!\cdots\!88$$$$\beta) q^{22} +(-$$$$14\!\cdots\!64$$$$+$$$$49\!\cdots\!14$$$$\beta) q^{23} +(-$$$$66\!\cdots\!16$$$$+$$$$47\!\cdots\!96$$$$\beta) q^{24} +($$$$40\!\cdots\!75$$$$-$$$$29\!\cdots\!00$$$$\beta) q^{25} +($$$$14\!\cdots\!24$$$$-$$$$81\!\cdots\!28$$$$\beta) q^{26} +($$$$51\!\cdots\!00$$$$+$$$$25\!\cdots\!18$$$$\beta) q^{27} +(-$$$$60\!\cdots\!08$$$$+$$$$33\!\cdots\!28$$$$\beta) q^{28} +(-$$$$59\!\cdots\!90$$$$-$$$$12\!\cdots\!04$$$$\beta) q^{29} +(-$$$$10\!\cdots\!00$$$$+$$$$15\!\cdots\!20$$$$\beta) q^{30} +(-$$$$35\!\cdots\!48$$$$-$$$$35\!\cdots\!44$$$$\beta) q^{31} -$$$$13\!\cdots\!76$$$$q^{32} +(-$$$$15\!\cdots\!28$$$$+$$$$14\!\cdots\!96$$$$\beta) q^{33} +(-$$$$10\!\cdots\!12$$$$-$$$$29\!\cdots\!72$$$$\beta) q^{34} +(-$$$$76\!\cdots\!00$$$$+$$$$12\!\cdots\!60$$$$\beta) q^{35} +(-$$$$15\!\cdots\!52$$$$-$$$$79\!\cdots\!52$$$$\beta) q^{36} +(-$$$$22\!\cdots\!58$$$$-$$$$60\!\cdots\!08$$$$\beta) q^{37} +(-$$$$26\!\cdots\!60$$$$+$$$$26\!\cdots\!92$$$$\beta) q^{38} +(-$$$$92\!\cdots\!44$$$$+$$$$15\!\cdots\!82$$$$\beta) q^{39} +(-$$$$19\!\cdots\!00$$$$+$$$$16\!\cdots\!20$$$$\beta) q^{40} +($$$$34\!\cdots\!82$$$$-$$$$38\!\cdots\!76$$$$\beta) q^{41} +($$$$37\!\cdots\!48$$$$-$$$$64\!\cdots\!56$$$$\beta) q^{42} +($$$$11\!\cdots\!16$$$$-$$$$24\!\cdots\!63$$$$\beta) q^{43} +(-$$$$35\!\cdots\!08$$$$+$$$$25\!\cdots\!08$$$$\beta) q^{44} +($$$$13\!\cdots\!50$$$$+$$$$72\!\cdots\!40$$$$\beta) q^{45} +($$$$24\!\cdots\!24$$$$-$$$$82\!\cdots\!24$$$$\beta) q^{46} +(-$$$$59\!\cdots\!48$$$$+$$$$19\!\cdots\!48$$$$\beta) q^{47} +($$$$11\!\cdots\!56$$$$-$$$$79\!\cdots\!36$$$$\beta) q^{48} +($$$$18\!\cdots\!17$$$$-$$$$50\!\cdots\!68$$$$\beta) q^{49} +(-$$$$68\!\cdots\!00$$$$+$$$$48\!\cdots\!00$$$$\beta) q^{50} +(-$$$$20\!\cdots\!28$$$$-$$$$37\!\cdots\!50$$$$\beta) q^{51} +(-$$$$23\!\cdots\!84$$$$+$$$$13\!\cdots\!48$$$$\beta) q^{52} +(-$$$$19\!\cdots\!34$$$$+$$$$10\!\cdots\!28$$$$\beta) q^{53} +(-$$$$86\!\cdots\!00$$$$-$$$$42\!\cdots\!88$$$$\beta) q^{54} +(-$$$$53\!\cdots\!00$$$$+$$$$42\!\cdots\!10$$$$\beta) q^{55} +($$$$10\!\cdots\!28$$$$-$$$$55\!\cdots\!48$$$$\beta) q^{56} +($$$$48\!\cdots\!60$$$$-$$$$18\!\cdots\!12$$$$\beta) q^{57} +($$$$99\!\cdots\!40$$$$+$$$$21\!\cdots\!64$$$$\beta) q^{58} +($$$$10\!\cdots\!20$$$$+$$$$71\!\cdots\!77$$$$\beta) q^{59} +($$$$17\!\cdots\!00$$$$-$$$$25\!\cdots\!20$$$$\beta) q^{60} +($$$$28\!\cdots\!02$$$$-$$$$15\!\cdots\!48$$$$\beta) q^{61} +($$$$59\!\cdots\!68$$$$+$$$$59\!\cdots\!04$$$$\beta) q^{62} +(-$$$$42\!\cdots\!44$$$$-$$$$31\!\cdots\!90$$$$\beta) q^{63} +$$$$22\!\cdots\!16$$$$q^{64} +(-$$$$31\!\cdots\!00$$$$+$$$$49\!\cdots\!80$$$$\beta) q^{65} +($$$$25\!\cdots\!48$$$$-$$$$23\!\cdots\!36$$$$\beta) q^{66} +(-$$$$56\!\cdots\!08$$$$-$$$$89\!\cdots\!41$$$$\beta) q^{67} +($$$$17\!\cdots\!92$$$$+$$$$48\!\cdots\!52$$$$\beta) q^{68} +(-$$$$10\!\cdots\!44$$$$+$$$$21\!\cdots\!08$$$$\beta) q^{69} +($$$$12\!\cdots\!00$$$$-$$$$20\!\cdots\!60$$$$\beta) q^{70} +(-$$$$12\!\cdots\!08$$$$-$$$$42\!\cdots\!78$$$$\beta) q^{71} +($$$$25\!\cdots\!32$$$$+$$$$13\!\cdots\!32$$$$\beta) q^{72} +(-$$$$48\!\cdots\!94$$$$+$$$$61\!\cdots\!28$$$$\beta) q^{73} +($$$$37\!\cdots\!28$$$$+$$$$10\!\cdots\!28$$$$\beta) q^{74} +($$$$53\!\cdots\!00$$$$-$$$$81\!\cdots\!75$$$$\beta) q^{75} +($$$$44\!\cdots\!60$$$$-$$$$43\!\cdots\!72$$$$\beta) q^{76} +($$$$17\!\cdots\!24$$$$-$$$$21\!\cdots\!08$$$$\beta) q^{77} +($$$$15\!\cdots\!04$$$$-$$$$25\!\cdots\!12$$$$\beta) q^{78} +(-$$$$84\!\cdots\!80$$$$+$$$$65\!\cdots\!48$$$$\beta) q^{79} +($$$$32\!\cdots\!00$$$$-$$$$27\!\cdots\!20$$$$\beta) q^{80} +(-$$$$28\!\cdots\!39$$$$+$$$$97\!\cdots\!64$$$$\beta) q^{81} +(-$$$$58\!\cdots\!12$$$$+$$$$63\!\cdots\!16$$$$\beta) q^{82} +(-$$$$12\!\cdots\!64$$$$-$$$$30\!\cdots\!25$$$$\beta) q^{83} +(-$$$$63\!\cdots\!68$$$$+$$$$10\!\cdots\!96$$$$\beta) q^{84} +(-$$$$74\!\cdots\!00$$$$-$$$$14\!\cdots\!40$$$$\beta) q^{85} +(-$$$$19\!\cdots\!56$$$$+$$$$40\!\cdots\!08$$$$\beta) q^{86} +($$$$12\!\cdots\!60$$$$+$$$$41\!\cdots\!06$$$$\beta) q^{87} +($$$$59\!\cdots\!28$$$$-$$$$43\!\cdots\!28$$$$\beta) q^{88} +($$$$50\!\cdots\!90$$$$+$$$$84\!\cdots\!40$$$$\beta) q^{89} +(-$$$$23\!\cdots\!00$$$$-$$$$12\!\cdots\!40$$$$\beta) q^{90} +($$$$11\!\cdots\!52$$$$-$$$$20\!\cdots\!76$$$$\beta) q^{91} +(-$$$$40\!\cdots\!84$$$$+$$$$13\!\cdots\!84$$$$\beta) q^{92} +($$$$90\!\cdots\!92$$$$+$$$$30\!\cdots\!24$$$$\beta) q^{93} +($$$$99\!\cdots\!68$$$$-$$$$32\!\cdots\!68$$$$\beta) q^{94} +($$$$15\!\cdots\!00$$$$-$$$$61\!\cdots\!50$$$$\beta) q^{95} +(-$$$$18\!\cdots\!96$$$$+$$$$13\!\cdots\!76$$$$\beta) q^{96} +($$$$43\!\cdots\!62$$$$+$$$$97\!\cdots\!40$$$$\beta) q^{97} +(-$$$$30\!\cdots\!72$$$$+$$$$85\!\cdots\!88$$$$\beta) q^{98} +(-$$$$41\!\cdots\!44$$$$-$$$$46\!\cdots\!75$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 33554432q^{2} + 281051075592q^{3} + 562949953421312q^{4} + 83174531182543500q^{5} - 4715254602239311872q^{6} - 432389265285998218736q^{7} - 9444732965739290427392q^{8} - 108711528745520320184934q^{9} + O(q^{10})$$ $$2q - 33554432q^{2} + 281051075592q^{3} + 562949953421312q^{4} + 83174531182543500q^{5} - 4715254602239311872q^{6} -$$$$43\!\cdots\!36$$$$q^{7} -$$$$94\!\cdots\!92$$$$q^{8} -$$$$10\!\cdots\!34$$$$q^{9} -$$$$13\!\cdots\!00$$$$q^{10} -$$$$25\!\cdots\!36$$$$q^{11} +$$$$79\!\cdots\!52$$$$q^{12} -$$$$16\!\cdots\!28$$$$q^{13} +$$$$72\!\cdots\!76$$$$q^{14} +$$$$12\!\cdots\!00$$$$q^{15} +$$$$15\!\cdots\!72$$$$q^{16} +$$$$12\!\cdots\!64$$$$q^{17} +$$$$18\!\cdots\!44$$$$q^{18} +$$$$31\!\cdots\!20$$$$q^{19} +$$$$23\!\cdots\!00$$$$q^{20} -$$$$44\!\cdots\!56$$$$q^{21} +$$$$42\!\cdots\!76$$$$q^{22} -$$$$28\!\cdots\!28$$$$q^{23} -$$$$13\!\cdots\!32$$$$q^{24} +$$$$81\!\cdots\!50$$$$q^{25} +$$$$28\!\cdots\!48$$$$q^{26} +$$$$10\!\cdots\!00$$$$q^{27} -$$$$12\!\cdots\!16$$$$q^{28} -$$$$11\!\cdots\!80$$$$q^{29} -$$$$21\!\cdots\!00$$$$q^{30} -$$$$71\!\cdots\!96$$$$q^{31} -$$$$26\!\cdots\!52$$$$q^{32} -$$$$30\!\cdots\!56$$$$q^{33} -$$$$20\!\cdots\!24$$$$q^{34} -$$$$15\!\cdots\!00$$$$q^{35} -$$$$30\!\cdots\!04$$$$q^{36} -$$$$45\!\cdots\!16$$$$q^{37} -$$$$53\!\cdots\!20$$$$q^{38} -$$$$18\!\cdots\!88$$$$q^{39} -$$$$39\!\cdots\!00$$$$q^{40} +$$$$69\!\cdots\!64$$$$q^{41} +$$$$75\!\cdots\!96$$$$q^{42} +$$$$23\!\cdots\!32$$$$q^{43} -$$$$70\!\cdots\!16$$$$q^{44} +$$$$27\!\cdots\!00$$$$q^{45} +$$$$48\!\cdots\!48$$$$q^{46} -$$$$11\!\cdots\!96$$$$q^{47} +$$$$22\!\cdots\!12$$$$q^{48} +$$$$36\!\cdots\!34$$$$q^{49} -$$$$13\!\cdots\!00$$$$q^{50} -$$$$40\!\cdots\!56$$$$q^{51} -$$$$47\!\cdots\!68$$$$q^{52} -$$$$39\!\cdots\!68$$$$q^{53} -$$$$17\!\cdots\!00$$$$q^{54} -$$$$10\!\cdots\!00$$$$q^{55} +$$$$20\!\cdots\!56$$$$q^{56} +$$$$96\!\cdots\!20$$$$q^{57} +$$$$19\!\cdots\!80$$$$q^{58} +$$$$21\!\cdots\!40$$$$q^{59} +$$$$35\!\cdots\!00$$$$q^{60} +$$$$57\!\cdots\!04$$$$q^{61} +$$$$11\!\cdots\!36$$$$q^{62} -$$$$85\!\cdots\!88$$$$q^{63} +$$$$44\!\cdots\!32$$$$q^{64} -$$$$63\!\cdots\!00$$$$q^{65} +$$$$51\!\cdots\!96$$$$q^{66} -$$$$11\!\cdots\!16$$$$q^{67} +$$$$34\!\cdots\!84$$$$q^{68} -$$$$20\!\cdots\!88$$$$q^{69} +$$$$25\!\cdots\!00$$$$q^{70} -$$$$25\!\cdots\!16$$$$q^{71} +$$$$51\!\cdots\!64$$$$q^{72} -$$$$97\!\cdots\!88$$$$q^{73} +$$$$75\!\cdots\!56$$$$q^{74} +$$$$10\!\cdots\!00$$$$q^{75} +$$$$89\!\cdots\!20$$$$q^{76} +$$$$35\!\cdots\!48$$$$q^{77} +$$$$30\!\cdots\!08$$$$q^{78} -$$$$16\!\cdots\!60$$$$q^{79} +$$$$65\!\cdots\!00$$$$q^{80} -$$$$56\!\cdots\!78$$$$q^{81} -$$$$11\!\cdots\!24$$$$q^{82} -$$$$24\!\cdots\!28$$$$q^{83} -$$$$12\!\cdots\!36$$$$q^{84} -$$$$14\!\cdots\!00$$$$q^{85} -$$$$38\!\cdots\!12$$$$q^{86} +$$$$24\!\cdots\!20$$$$q^{87} +$$$$11\!\cdots\!56$$$$q^{88} +$$$$10\!\cdots\!80$$$$q^{89} -$$$$46\!\cdots\!00$$$$q^{90} +$$$$22\!\cdots\!04$$$$q^{91} -$$$$81\!\cdots\!68$$$$q^{92} +$$$$18\!\cdots\!84$$$$q^{93} +$$$$19\!\cdots\!36$$$$q^{94} +$$$$31\!\cdots\!00$$$$q^{95} -$$$$37\!\cdots\!92$$$$q^{96} +$$$$86\!\cdots\!24$$$$q^{97} -$$$$61\!\cdots\!44$$$$q^{98} -$$$$83\!\cdots\!88$$$$q^{99} + O(q^{100})$$

## 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
 4.70421e6 −4.70420e6
−1.67772e7 −2.65918e11 2.81475e14 −1.00270e17 4.46136e18 2.61926e20 −4.72237e21 −1.68587e23 1.68224e24
1.2 −1.67772e7 5.46969e11 2.81475e14 1.83444e17 −9.17661e18 −6.94316e20 −4.72237e21 5.98756e22 −3.07768e24
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## 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 2.50.a.a 2
4.b odd 2 1 16.50.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.50.a.a 2 1.a even 1 1 trivial
16.50.a.a 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 281051075592 T_{3} -$$145448711312147320422384

'>$$14\!\cdots\!84$$ acting on $$S_{50}^{\mathrm{new}}(\Gamma_0(2))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 16777216 T )^{2}$$
$3$ $$1 - 281051075592 T +$$$$33\!\cdots\!82$$$$T^{2} -$$$$67\!\cdots\!36$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$
$5$ $$1 - 83174531182543500 T +$$$$17\!\cdots\!50$$$$T^{2} -$$$$14\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!25$$$$T^{4}$$
$7$ $$1 +$$$$43\!\cdots\!36$$$$T +$$$$33\!\cdots\!38$$$$T^{2} +$$$$11\!\cdots\!52$$$$T^{3} +$$$$66\!\cdots\!49$$$$T^{4}$$
$11$ $$1 +$$$$25\!\cdots\!36$$$$T +$$$$76\!\cdots\!06$$$$T^{2} +$$$$26\!\cdots\!76$$$$T^{3} +$$$$11\!\cdots\!81$$$$T^{4}$$
$13$ $$1 +$$$$16\!\cdots\!28$$$$T +$$$$44\!\cdots\!42$$$$T^{2} +$$$$64\!\cdots\!44$$$$T^{3} +$$$$14\!\cdots\!29$$$$T^{4}$$
$17$ $$1 -$$$$12\!\cdots\!64$$$$T +$$$$38\!\cdots\!18$$$$T^{2} -$$$$24\!\cdots\!08$$$$T^{3} +$$$$38\!\cdots\!09$$$$T^{4}$$
$19$ $$1 -$$$$31\!\cdots\!20$$$$T +$$$$11\!\cdots\!58$$$$T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$20\!\cdots\!41$$$$T^{4}$$
$23$ $$1 +$$$$28\!\cdots\!28$$$$T +$$$$86\!\cdots\!22$$$$T^{2} +$$$$15\!\cdots\!64$$$$T^{3} +$$$$28\!\cdots\!69$$$$T^{4}$$
$29$ $$1 +$$$$11\!\cdots\!80$$$$T +$$$$99\!\cdots\!38$$$$T^{2} +$$$$53\!\cdots\!20$$$$T^{3} +$$$$20\!\cdots\!61$$$$T^{4}$$
$31$ $$1 +$$$$71\!\cdots\!96$$$$T +$$$$34\!\cdots\!46$$$$T^{2} +$$$$84\!\cdots\!16$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4}$$
$37$ $$1 +$$$$45\!\cdots\!16$$$$T +$$$$12\!\cdots\!18$$$$T^{2} +$$$$31\!\cdots\!32$$$$T^{3} +$$$$48\!\cdots\!29$$$$T^{4}$$
$41$ $$1 -$$$$69\!\cdots\!64$$$$T +$$$$31\!\cdots\!46$$$$T^{2} -$$$$74\!\cdots\!04$$$$T^{3} +$$$$11\!\cdots\!21$$$$T^{4}$$
$43$ $$1 -$$$$23\!\cdots\!32$$$$T +$$$$35\!\cdots\!42$$$$T^{2} -$$$$25\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!49$$$$T^{4}$$
$47$ $$1 +$$$$11\!\cdots\!96$$$$T +$$$$10\!\cdots\!38$$$$T^{2} +$$$$10\!\cdots\!32$$$$T^{3} +$$$$73\!\cdots\!89$$$$T^{4}$$
$53$ $$1 +$$$$39\!\cdots\!68$$$$T +$$$$99\!\cdots\!22$$$$T^{2} +$$$$12\!\cdots\!44$$$$T^{3} +$$$$95\!\cdots\!89$$$$T^{4}$$
$59$ $$1 -$$$$21\!\cdots\!40$$$$T +$$$$46\!\cdots\!78$$$$T^{2} -$$$$12\!\cdots\!60$$$$T^{3} +$$$$34\!\cdots\!21$$$$T^{4}$$
$61$ $$1 -$$$$57\!\cdots\!04$$$$T +$$$$27\!\cdots\!86$$$$T^{2} -$$$$17\!\cdots\!64$$$$T^{3} +$$$$91\!\cdots\!81$$$$T^{4}$$
$67$ $$1 +$$$$11\!\cdots\!16$$$$T +$$$$79\!\cdots\!58$$$$T^{2} +$$$$34\!\cdots\!52$$$$T^{3} +$$$$90\!\cdots\!09$$$$T^{4}$$
$71$ $$1 +$$$$25\!\cdots\!16$$$$T +$$$$89\!\cdots\!26$$$$T^{2} +$$$$13\!\cdots\!96$$$$T^{3} +$$$$26\!\cdots\!61$$$$T^{4}$$
$73$ $$1 +$$$$97\!\cdots\!88$$$$T +$$$$57\!\cdots\!62$$$$T^{2} +$$$$19\!\cdots\!44$$$$T^{3} +$$$$40\!\cdots\!69$$$$T^{4}$$
$79$ $$1 +$$$$16\!\cdots\!60$$$$T +$$$$12\!\cdots\!38$$$$T^{2} +$$$$16\!\cdots\!40$$$$T^{3} +$$$$92\!\cdots\!61$$$$T^{4}$$
$83$ $$1 +$$$$24\!\cdots\!28$$$$T +$$$$64\!\cdots\!02$$$$T^{2} +$$$$26\!\cdots\!84$$$$T^{3} +$$$$11\!\cdots\!09$$$$T^{4}$$
$89$ $$1 -$$$$10\!\cdots\!80$$$$T +$$$$80\!\cdots\!18$$$$T^{2} -$$$$33\!\cdots\!20$$$$T^{3} +$$$$10\!\cdots\!81$$$$T^{4}$$
$97$ $$1 -$$$$86\!\cdots\!24$$$$T +$$$$29\!\cdots\!78$$$$T^{2} -$$$$19\!\cdots\!08$$$$T^{3} +$$$$50\!\cdots\!89$$$$T^{4}$$