# Properties

 Label 2.38.a.b Level $2$ Weight $38$ Character orbit 2.a Self dual yes Analytic conductor $17.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2$$ Weight: $$k$$ $$=$$ $$38$$ Character orbit: $$[\chi]$$ $$=$$ 2.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3428076249$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 55893200460$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}\cdot 3\cdot 5$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 1920\sqrt{223572801841}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 262144 q^{2} + ( -250843404 - \beta ) q^{3} + 68719476736 q^{4} + ( 2091300429750 - 2700 \beta ) q^{5} + ( -65757093298176 - 262144 \beta ) q^{6} + ( -1756339478768728 - 5318082 \beta ) q^{7} + 18014398509481984 q^{8} + ( 436817284145972253 + 501686808 \beta ) q^{9} +O(q^{10})$$ $$q +262144 q^{2} +(-250843404 - \beta) q^{3} +68719476736 q^{4} +(2091300429750 - 2700 \beta) q^{5} +(-65757093298176 - 262144 \beta) q^{6} +(-1756339478768728 - 5318082 \beta) q^{7} +18014398509481984 q^{8} +(436817284145972253 + 501686808 \beta) q^{9} +(548221859856384000 - 707788800 \beta) q^{10} +(12829972861280186652 - 7107001587 \beta) q^{11} +(-17237827465557049344 - 68719476736 \beta) q^{12} +(-75190417767106126114 + 695679299028 \beta) q^{13} +(-$$$$46\!\cdots\!32$$$$- 1394103287808 \beta) q^{14} +($$$$17\!\cdots\!00$$$$- 1414023238950 \beta) q^{15} +$$$$47\!\cdots\!96$$$$q^{16} +($$$$82\!\cdots\!82$$$$+ 14083120550232 \beta) q^{17} +($$$$11\!\cdots\!32$$$$+ 131514186596352 \beta) q^{18} +($$$$15\!\cdots\!00$$$$- 556602440642757 \beta) q^{19} +($$$$14\!\cdots\!00$$$$- 185542587187200 \beta) q^{20} +($$$$48\!\cdots\!12$$$$+ 3090345270399856 \beta) q^{21} +($$$$33\!\cdots\!88$$$$- 1863057824022528 \beta) q^{22} +($$$$11\!\cdots\!96$$$$- 723601869769926 \beta) q^{23} +(-$$$$45\!\cdots\!36$$$$- 18014398509481984 \beta) q^{24} +(-$$$$62\!\cdots\!25$$$$- 11293022320650000 \beta) q^{25} +(-$$$$19\!\cdots\!16$$$$+ 182368154164396032 \beta) q^{26} +(-$$$$41\!\cdots\!60$$$$- 112378204915589322 \beta) q^{27} +(-$$$$12\!\cdots\!08$$$$- 365455812279140352 \beta) q^{28} +(-$$$$77\!\cdots\!10$$$$+ 99413651854516356 \beta) q^{29} +($$$$44\!\cdots\!00$$$$- 370677707951308800 \beta) q^{30} +($$$$24\!\cdots\!72$$$$+ 3512250494601498936 \beta) q^{31} +$$$$12\!\cdots\!24$$$$q^{32} +($$$$26\!\cdots\!92$$$$- 11047228390963704504 \beta) q^{33} +($$$$21\!\cdots\!08$$$$+ 3691805553520017408 \beta) q^{34} +($$$$81\!\cdots\!00$$$$- 6379590579370173900 \beta) q^{35} +($$$$30\!\cdots\!08$$$$+ 34475654931114098688 \beta) q^{36} +(-$$$$64\!\cdots\!78$$$$+ 64367517886888263012 \beta) q^{37} +($$$$41\!\cdots\!00$$$$-$$$$14\!\cdots\!08$$$$\beta) q^{38} +(-$$$$55\!\cdots\!44$$$$- 99316145693411285198 \beta) q^{39} +($$$$37\!\cdots\!00$$$$- 48638875975601356800 \beta) q^{40} +(-$$$$55\!\cdots\!78$$$$+$$$$43\!\cdots\!24$$$$\beta) q^{41} +($$$$12\!\cdots\!28$$$$+$$$$81\!\cdots\!64$$$$\beta) q^{42} +($$$$52\!\cdots\!76$$$$-$$$$14\!\cdots\!03$$$$\beta) q^{43} +($$$$88\!\cdots\!72$$$$-$$$$48\!\cdots\!32$$$$\beta) q^{44} +(-$$$$20\!\cdots\!50$$$$-$$$$13\!\cdots\!00$$$$\beta) q^{45} +($$$$30\!\cdots\!24$$$$-$$$$18\!\cdots\!44$$$$\beta) q^{46} +($$$$80\!\cdots\!52$$$$+$$$$38\!\cdots\!88$$$$\beta) q^{47} +(-$$$$11\!\cdots\!84$$$$-$$$$47\!\cdots\!96$$$$\beta) q^{48} +($$$$78\!\cdots\!77$$$$+$$$$18\!\cdots\!92$$$$\beta) q^{49} +(-$$$$16\!\cdots\!00$$$$-$$$$29\!\cdots\!00$$$$\beta) q^{50} +(-$$$$32\!\cdots\!28$$$$-$$$$86\!\cdots\!10$$$$\beta) q^{51} +(-$$$$51\!\cdots\!04$$$$+$$$$47\!\cdots\!08$$$$\beta) q^{52} +($$$$18\!\cdots\!86$$$$+$$$$11\!\cdots\!88$$$$\beta) q^{53} +(-$$$$10\!\cdots\!40$$$$-$$$$29\!\cdots\!68$$$$\beta) q^{54} +($$$$42\!\cdots\!00$$$$-$$$$49\!\cdots\!50$$$$\beta) q^{55} +(-$$$$31\!\cdots\!52$$$$-$$$$95\!\cdots\!88$$$$\beta) q^{56} +($$$$41\!\cdots\!00$$$$-$$$$18\!\cdots\!72$$$$\beta) q^{57} +(-$$$$20\!\cdots\!40$$$$+$$$$26\!\cdots\!64$$$$\beta) q^{58} +($$$$63\!\cdots\!80$$$$+$$$$85\!\cdots\!57$$$$\beta) q^{59} +($$$$11\!\cdots\!00$$$$-$$$$97\!\cdots\!00$$$$\beta) q^{60} +(-$$$$65\!\cdots\!58$$$$+$$$$67\!\cdots\!52$$$$\beta) q^{61} +($$$$63\!\cdots\!68$$$$+$$$$92\!\cdots\!84$$$$\beta) q^{62} +(-$$$$29\!\cdots\!84$$$$-$$$$32\!\cdots\!70$$$$\beta) q^{63} +$$$$32\!\cdots\!56$$$$q^{64} +(-$$$$17\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$\beta) q^{65} +($$$$69\!\cdots\!48$$$$-$$$$28\!\cdots\!76$$$$\beta) q^{66} +($$$$19\!\cdots\!52$$$$+$$$$92\!\cdots\!59$$$$\beta) q^{67} +($$$$56\!\cdots\!52$$$$+$$$$96\!\cdots\!52$$$$\beta) q^{68} +(-$$$$23\!\cdots\!84$$$$-$$$$11\!\cdots\!92$$$$\beta) q^{69} +($$$$21\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$\beta) q^{70} +($$$$71\!\cdots\!12$$$$+$$$$13\!\cdots\!62$$$$\beta) q^{71} +($$$$78\!\cdots\!52$$$$+$$$$90\!\cdots\!72$$$$\beta) q^{72} +($$$$29\!\cdots\!86$$$$-$$$$21\!\cdots\!32$$$$\beta) q^{73} +(-$$$$16\!\cdots\!32$$$$+$$$$16\!\cdots\!28$$$$\beta) q^{74} +($$$$24\!\cdots\!00$$$$+$$$$65\!\cdots\!25$$$$\beta) q^{75} +($$$$10\!\cdots\!00$$$$-$$$$38\!\cdots\!52$$$$\beta) q^{76} +($$$$86\!\cdots\!44$$$$-$$$$55\!\cdots\!28$$$$\beta) q^{77} +(-$$$$14\!\cdots\!36$$$$-$$$$26\!\cdots\!12$$$$\beta) q^{78} +(-$$$$84\!\cdots\!40$$$$-$$$$15\!\cdots\!32$$$$\beta) q^{79} +($$$$98\!\cdots\!00$$$$-$$$$12\!\cdots\!00$$$$\beta) q^{80} +(-$$$$12\!\cdots\!99$$$$+$$$$21\!\cdots\!44$$$$\beta) q^{81} +(-$$$$14\!\cdots\!32$$$$+$$$$11\!\cdots\!56$$$$\beta) q^{82} +($$$$32\!\cdots\!56$$$$-$$$$30\!\cdots\!65$$$$\beta) q^{83} +($$$$33\!\cdots\!32$$$$+$$$$21\!\cdots\!16$$$$\beta) q^{84} +($$$$14\!\cdots\!00$$$$-$$$$19\!\cdots\!00$$$$\beta) q^{85} +($$$$13\!\cdots\!44$$$$-$$$$37\!\cdots\!32$$$$\beta) q^{86} +(-$$$$62\!\cdots\!60$$$$+$$$$52\!\cdots\!86$$$$\beta) q^{87} +($$$$23\!\cdots\!68$$$$-$$$$12\!\cdots\!08$$$$\beta) q^{88} +($$$$34\!\cdots\!90$$$$+$$$$15\!\cdots\!40$$$$\beta) q^{89} +(-$$$$53\!\cdots\!00$$$$-$$$$34\!\cdots\!00$$$$\beta) q^{90} +(-$$$$29\!\cdots\!08$$$$-$$$$82\!\cdots\!36$$$$\beta) q^{91} +($$$$80\!\cdots\!56$$$$-$$$$49\!\cdots\!36$$$$\beta) q^{92} +(-$$$$29\!\cdots\!88$$$$-$$$$11\!\cdots\!16$$$$\beta) q^{93} +($$$$21\!\cdots\!88$$$$+$$$$10\!\cdots\!72$$$$\beta) q^{94} +($$$$15\!\cdots\!00$$$$-$$$$15\!\cdots\!50$$$$\beta) q^{95} +(-$$$$31\!\cdots\!96$$$$-$$$$12\!\cdots\!24$$$$\beta) q^{96} +($$$$29\!\cdots\!22$$$$+$$$$20\!\cdots\!80$$$$\beta) q^{97} +($$$$20\!\cdots\!88$$$$+$$$$48\!\cdots\!48$$$$\beta) q^{98} +($$$$26\!\cdots\!56$$$$+$$$$33\!\cdots\!05$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 524288q^{2} - 501686808q^{3} + 137438953472q^{4} + 4182600859500q^{5} - 131514186596352q^{6} - 3512678957537456q^{7} + 36028797018963968q^{8} + 873634568291944506q^{9} + O(q^{10})$$ $$2q + 524288q^{2} - 501686808q^{3} + 137438953472q^{4} + 4182600859500q^{5} - 131514186596352q^{6} - 3512678957537456q^{7} + 36028797018963968q^{8} + 873634568291944506q^{9} + 1096443719712768000q^{10} + 25659945722560373304q^{11} - 34475654931114098688q^{12} -$$$$15\!\cdots\!28$$$$q^{13} -$$$$92\!\cdots\!64$$$$q^{14} +$$$$34\!\cdots\!00$$$$q^{15} +$$$$94\!\cdots\!92$$$$q^{16} +$$$$16\!\cdots\!64$$$$q^{17} +$$$$22\!\cdots\!64$$$$q^{18} +$$$$31\!\cdots\!00$$$$q^{19} +$$$$28\!\cdots\!00$$$$q^{20} +$$$$96\!\cdots\!24$$$$q^{21} +$$$$67\!\cdots\!76$$$$q^{22} +$$$$23\!\cdots\!92$$$$q^{23} -$$$$90\!\cdots\!72$$$$q^{24} -$$$$12\!\cdots\!50$$$$q^{25} -$$$$39\!\cdots\!32$$$$q^{26} -$$$$82\!\cdots\!20$$$$q^{27} -$$$$24\!\cdots\!16$$$$q^{28} -$$$$15\!\cdots\!20$$$$q^{29} +$$$$89\!\cdots\!00$$$$q^{30} +$$$$48\!\cdots\!44$$$$q^{31} +$$$$24\!\cdots\!48$$$$q^{32} +$$$$52\!\cdots\!84$$$$q^{33} +$$$$43\!\cdots\!16$$$$q^{34} +$$$$16\!\cdots\!00$$$$q^{35} +$$$$60\!\cdots\!16$$$$q^{36} -$$$$12\!\cdots\!56$$$$q^{37} +$$$$82\!\cdots\!00$$$$q^{38} -$$$$11\!\cdots\!88$$$$q^{39} +$$$$75\!\cdots\!00$$$$q^{40} -$$$$11\!\cdots\!56$$$$q^{41} +$$$$25\!\cdots\!56$$$$q^{42} +$$$$10\!\cdots\!52$$$$q^{43} +$$$$17\!\cdots\!44$$$$q^{44} -$$$$40\!\cdots\!00$$$$q^{45} +$$$$61\!\cdots\!48$$$$q^{46} +$$$$16\!\cdots\!04$$$$q^{47} -$$$$23\!\cdots\!68$$$$q^{48} +$$$$15\!\cdots\!54$$$$q^{49} -$$$$32\!\cdots\!00$$$$q^{50} -$$$$64\!\cdots\!56$$$$q^{51} -$$$$10\!\cdots\!08$$$$q^{52} +$$$$37\!\cdots\!72$$$$q^{53} -$$$$21\!\cdots\!80$$$$q^{54} +$$$$85\!\cdots\!00$$$$q^{55} -$$$$63\!\cdots\!04$$$$q^{56} +$$$$83\!\cdots\!00$$$$q^{57} -$$$$40\!\cdots\!80$$$$q^{58} +$$$$12\!\cdots\!60$$$$q^{59} +$$$$23\!\cdots\!00$$$$q^{60} -$$$$13\!\cdots\!16$$$$q^{61} +$$$$12\!\cdots\!36$$$$q^{62} -$$$$59\!\cdots\!68$$$$q^{63} +$$$$64\!\cdots\!12$$$$q^{64} -$$$$34\!\cdots\!00$$$$q^{65} +$$$$13\!\cdots\!96$$$$q^{66} +$$$$38\!\cdots\!04$$$$q^{67} +$$$$11\!\cdots\!04$$$$q^{68} -$$$$46\!\cdots\!68$$$$q^{69} +$$$$42\!\cdots\!00$$$$q^{70} +$$$$14\!\cdots\!24$$$$q^{71} +$$$$15\!\cdots\!04$$$$q^{72} +$$$$58\!\cdots\!72$$$$q^{73} -$$$$33\!\cdots\!64$$$$q^{74} +$$$$49\!\cdots\!00$$$$q^{75} +$$$$21\!\cdots\!00$$$$q^{76} +$$$$17\!\cdots\!88$$$$q^{77} -$$$$29\!\cdots\!72$$$$q^{78} -$$$$16\!\cdots\!80$$$$q^{79} +$$$$19\!\cdots\!00$$$$q^{80} -$$$$24\!\cdots\!98$$$$q^{81} -$$$$29\!\cdots\!64$$$$q^{82} +$$$$65\!\cdots\!12$$$$q^{83} +$$$$66\!\cdots\!64$$$$q^{84} +$$$$28\!\cdots\!00$$$$q^{85} +$$$$27\!\cdots\!88$$$$q^{86} -$$$$12\!\cdots\!20$$$$q^{87} +$$$$46\!\cdots\!36$$$$q^{88} +$$$$69\!\cdots\!80$$$$q^{89} -$$$$10\!\cdots\!00$$$$q^{90} -$$$$58\!\cdots\!16$$$$q^{91} +$$$$16\!\cdots\!12$$$$q^{92} -$$$$59\!\cdots\!76$$$$q^{93} +$$$$42\!\cdots\!76$$$$q^{94} +$$$$31\!\cdots\!00$$$$q^{95} -$$$$62\!\cdots\!92$$$$q^{96} +$$$$58\!\cdots\!44$$$$q^{97} +$$$$41\!\cdots\!76$$$$q^{98} +$$$$53\!\cdots\!12$$$$q^{99} + O(q^{100})$$

## 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
 236418. −236417.
262144. −1.15869e9 6.87195e10 −3.59875e11 −3.03743e14 −6.58432e15 1.80144e16 8.92270e17 −9.43392e16
1.2 262144. 6.57000e8 6.87195e10 4.54248e12 1.72228e14 3.07164e15 1.80144e16 −1.86355e16 1.19078e18
 $$n$$: 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 2.38.a.b 2
3.b odd 2 1 18.38.a.d 2
4.b odd 2 1 16.38.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.38.a.b 2 1.a even 1 1 trivial
16.38.a.c 2 4.b odd 2 1
18.38.a.d 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 501686808 T_{3} -$$761256363376355184

'>$$76\!\cdots\!84$$ acting on $$S_{38}^{\mathrm{new}}(\Gamma_0(2))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 262144 T )^{2}$$
$3$ $$1 + 501686808 T + 139311448405639542 T^{2} +$$$$22\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!69$$$$T^{4}$$
$5$ $$1 - 4182600859500 T +$$$$14\!\cdots\!50$$$$T^{2} -$$$$30\!\cdots\!00$$$$T^{3} +$$$$52\!\cdots\!25$$$$T^{4}$$
$7$ $$1 + 3512678957537456 T +$$$$16\!\cdots\!98$$$$T^{2} +$$$$65\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 25659945722560373304 T +$$$$80\!\cdots\!46$$$$T^{2} -$$$$87\!\cdots\!84$$$$T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$13$ $$1 +$$$$15\!\cdots\!28$$$$T -$$$$64\!\cdots\!38$$$$T^{2} +$$$$24\!\cdots\!24$$$$T^{3} +$$$$27\!\cdots\!89$$$$T^{4}$$
$17$ $$1 -$$$$16\!\cdots\!64$$$$T +$$$$13\!\cdots\!78$$$$T^{2} -$$$$55\!\cdots\!28$$$$T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$19$ $$1 -$$$$31\!\cdots\!00$$$$T +$$$$18\!\cdots\!78$$$$T^{2} -$$$$64\!\cdots\!00$$$$T^{3} +$$$$42\!\cdots\!21$$$$T^{4}$$
$23$ $$1 -$$$$23\!\cdots\!92$$$$T +$$$$62\!\cdots\!22$$$$T^{2} -$$$$56\!\cdots\!76$$$$T^{3} +$$$$58\!\cdots\!09$$$$T^{4}$$
$29$ $$1 +$$$$15\!\cdots\!20$$$$T +$$$$25\!\cdots\!18$$$$T^{2} +$$$$19\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!81$$$$T^{4}$$
$31$ $$1 -$$$$48\!\cdots\!44$$$$T +$$$$20\!\cdots\!06$$$$T^{2} -$$$$73\!\cdots\!84$$$$T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$37$ $$1 +$$$$12\!\cdots\!56$$$$T +$$$$21\!\cdots\!18$$$$T^{2} +$$$$13\!\cdots\!52$$$$T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$41$ $$1 +$$$$11\!\cdots\!56$$$$T +$$$$10\!\cdots\!46$$$$T^{2} +$$$$52\!\cdots\!36$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4}$$
$43$ $$1 -$$$$10\!\cdots\!52$$$$T +$$$$41\!\cdots\!62$$$$T^{2} -$$$$28\!\cdots\!36$$$$T^{3} +$$$$75\!\cdots\!49$$$$T^{4}$$
$47$ $$1 -$$$$16\!\cdots\!04$$$$T +$$$$13\!\cdots\!78$$$$T^{2} -$$$$11\!\cdots\!48$$$$T^{3} +$$$$54\!\cdots\!69$$$$T^{4}$$
$53$ $$1 -$$$$37\!\cdots\!72$$$$T +$$$$21\!\cdots\!22$$$$T^{2} -$$$$23\!\cdots\!36$$$$T^{3} +$$$$39\!\cdots\!69$$$$T^{4}$$
$59$ $$1 -$$$$12\!\cdots\!60$$$$T +$$$$10\!\cdots\!38$$$$T^{2} -$$$$42\!\cdots\!40$$$$T^{3} +$$$$11\!\cdots\!61$$$$T^{4}$$
$61$ $$1 +$$$$13\!\cdots\!16$$$$T +$$$$23\!\cdots\!06$$$$T^{2} +$$$$14\!\cdots\!36$$$$T^{3} +$$$$13\!\cdots\!41$$$$T^{4}$$
$67$ $$1 -$$$$38\!\cdots\!04$$$$T +$$$$31\!\cdots\!58$$$$T^{2} -$$$$14\!\cdots\!08$$$$T^{3} +$$$$13\!\cdots\!29$$$$T^{4}$$
$71$ $$1 -$$$$14\!\cdots\!24$$$$T +$$$$67\!\cdots\!26$$$$T^{2} -$$$$44\!\cdots\!84$$$$T^{3} +$$$$98\!\cdots\!81$$$$T^{4}$$
$73$ $$1 -$$$$58\!\cdots\!72$$$$T +$$$$13\!\cdots\!02$$$$T^{2} -$$$$51\!\cdots\!16$$$$T^{3} +$$$$76\!\cdots\!09$$$$T^{4}$$
$79$ $$1 +$$$$16\!\cdots\!80$$$$T +$$$$39\!\cdots\!18$$$$T^{2} +$$$$27\!\cdots\!20$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4}$$
$83$ $$1 -$$$$65\!\cdots\!12$$$$T +$$$$22\!\cdots\!82$$$$T^{2} -$$$$65\!\cdots\!76$$$$T^{3} +$$$$10\!\cdots\!29$$$$T^{4}$$
$89$ $$1 -$$$$69\!\cdots\!80$$$$T +$$$$91\!\cdots\!58$$$$T^{2} -$$$$93\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!41$$$$T^{4}$$
$97$ $$1 -$$$$58\!\cdots\!44$$$$T +$$$$69\!\cdots\!58$$$$T^{2} -$$$$19\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!69$$$$T^{4}$$