# Properties

 Label 2.38.a.a Level $2$ Weight $38$ Character orbit 2.a Self dual yes Analytic conductor $17.343$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 756643680$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}\cdot 3^{3}\cdot 5$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 17280\sqrt{3026574721}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -262144 q^{2} + ( 211535604 - \beta ) q^{3} + 68719476736 q^{4} + ( -6753765277770 + 3444 \beta ) q^{5} + ( -55452789374976 + 262144 \beta ) q^{6} + ( 1553087081481128 + 5126718 \beta ) q^{7} -18014398509481984 q^{8} + ( 498193775039693853 - 423071208 \beta ) q^{9} +O(q^{10})$$ $$q -262144 q^{2} +(211535604 - \beta) q^{3} +68719476736 q^{4} +(-6753765277770 + 3444 \beta) q^{5} +(-55452789374976 + 262144 \beta) q^{6} +(1553087081481128 + 5126718 \beta) q^{7} -18014398509481984 q^{8} +(498193775039693853 - 423071208 \beta) q^{9} +(1770459044975738880 - 902823936 \beta) q^{10} +(2001439330770516252 + 25266324237 \beta) q^{11} +(14536616017913708544 - 68719476736 \beta) q^{12} +(-$$$$20\!\cdots\!86$$$$+ 40178550228 \beta) q^{13} +(-$$$$40\!\cdots\!32$$$$- 1343938363392 \beta) q^{14} +(-$$$$45\!\cdots\!80$$$$+ 7482293897946 \beta) q^{15} +$$$$47\!\cdots\!96$$$$q^{16} +($$$$46\!\cdots\!18$$$$- 73067502067368 \beta) q^{17} +(-$$$$13\!\cdots\!32$$$$+ 110905578749952 \beta) q^{18} +(-$$$$75\!\cdots\!00$$$$+ 83542782562107 \beta) q^{19} +(-$$$$46\!\cdots\!20$$$$+ 236669877878784 \beta) q^{20} +(-$$$$43\!\cdots\!88$$$$- 468603692813456 \beta) q^{21} +(-$$$$52\!\cdots\!88$$$$- 6623415300784128 \beta) q^{22} +($$$$24\!\cdots\!04$$$$+ 13722480149762874 \beta) q^{23} +(-$$$$38\!\cdots\!36$$$$+ 18014398509481984 \beta) q^{24} +(-$$$$16\!\cdots\!25$$$$- 46519935233279760 \beta) q^{25} +($$$$54\!\cdots\!84$$$$- 10532565870968832 \beta) q^{26} +($$$$39\!\cdots\!60$$$$- 137404492667986122 \beta) q^{27} +($$$$10\!\cdots\!08$$$$+ 352305378333032448 \beta) q^{28} +($$$$58\!\cdots\!90$$$$+ 872730741021530244 \beta) q^{29} +($$$$11\!\cdots\!20$$$$- 1961438451583156224 \beta) q^{30} +(-$$$$27\!\cdots\!28$$$$- 872718796832206536 \beta) q^{31} -$$$$12\!\cdots\!24$$$$q^{32} +(-$$$$22\!\cdots\!92$$$$+ 3343287827563117896 \beta) q^{33} +(-$$$$12\!\cdots\!92$$$$+ 19154207261948116992 \beta) q^{34} +($$$$54\!\cdots\!40$$$$- 29275818108697454028 \beta) q^{35} +($$$$34\!\cdots\!08$$$$- 29073232035827417088 \beta) q^{36} +(-$$$$57\!\cdots\!22$$$$+ 28974975230710445412 \beta) q^{37} +($$$$19\!\cdots\!00$$$$- 21900239191960977408 \beta) q^{38} +(-$$$$80\!\cdots\!44$$$$+$$$$21\!\cdots\!98$$$$\beta) q^{39} +($$$$12\!\cdots\!80$$$$- 62041588466655952896 \beta) q^{40} +($$$$23\!\cdots\!22$$$$-$$$$33\!\cdots\!24$$$$\beta) q^{41} +($$$$11\!\cdots\!72$$$$+$$$$12\!\cdots\!64$$$$\beta) q^{42} +(-$$$$15\!\cdots\!76$$$$-$$$$13\!\cdots\!03$$$$\beta) q^{43} +($$$$13\!\cdots\!72$$$$+$$$$17\!\cdots\!32$$$$\beta) q^{44} +(-$$$$46\!\cdots\!10$$$$+$$$$45\!\cdots\!92$$$$\beta) q^{45} +(-$$$$65\!\cdots\!76$$$$-$$$$35\!\cdots\!56$$$$\beta) q^{46} +(-$$$$56\!\cdots\!52$$$$-$$$$75\!\cdots\!12$$$$\beta) q^{47} +($$$$99\!\cdots\!84$$$$-$$$$47\!\cdots\!96$$$$\beta) q^{48} +($$$$76\!\cdots\!77$$$$+$$$$15\!\cdots\!08$$$$\beta) q^{49} +($$$$43\!\cdots\!00$$$$+$$$$12\!\cdots\!40$$$$\beta) q^{50} +($$$$66\!\cdots\!72$$$$-$$$$15\!\cdots\!90$$$$\beta) q^{51} +(-$$$$14\!\cdots\!96$$$$+$$$$27\!\cdots\!08$$$$\beta) q^{52} +($$$$26\!\cdots\!14$$$$-$$$$10\!\cdots\!12$$$$\beta) q^{53} +(-$$$$10\!\cdots\!40$$$$+$$$$36\!\cdots\!68$$$$\beta) q^{54} +($$$$65\!\cdots\!60$$$$-$$$$16\!\cdots\!02$$$$\beta) q^{55} +(-$$$$27\!\cdots\!52$$$$-$$$$92\!\cdots\!12$$$$\beta) q^{56} +(-$$$$23\!\cdots\!00$$$$+$$$$76\!\cdots\!28$$$$\beta) q^{57} +(-$$$$15\!\cdots\!60$$$$-$$$$22\!\cdots\!36$$$$\beta) q^{58} +($$$$29\!\cdots\!80$$$$-$$$$81\!\cdots\!07$$$$\beta) q^{59} +(-$$$$31\!\cdots\!80$$$$+$$$$51\!\cdots\!56$$$$\beta) q^{60} +($$$$47\!\cdots\!42$$$$-$$$$77\!\cdots\!52$$$$\beta) q^{61} +($$$$72\!\cdots\!32$$$$+$$$$22\!\cdots\!84$$$$\beta) q^{62} +(-$$$$11\!\cdots\!16$$$$+$$$$18\!\cdots\!30$$$$\beta) q^{63} +$$$$32\!\cdots\!56$$$$q^{64} +($$$$15\!\cdots\!20$$$$-$$$$99\!\cdots\!44$$$$\beta) q^{65} +($$$$58\!\cdots\!48$$$$-$$$$87\!\cdots\!24$$$$\beta) q^{66} +($$$$16\!\cdots\!48$$$$+$$$$19\!\cdots\!59$$$$\beta) q^{67} +($$$$31\!\cdots\!48$$$$-$$$$50\!\cdots\!48$$$$\beta) q^{68} +(-$$$$11\!\cdots\!84$$$$+$$$$41\!\cdots\!92$$$$\beta) q^{69} +(-$$$$14\!\cdots\!60$$$$+$$$$76\!\cdots\!32$$$$\beta) q^{70} +(-$$$$18\!\cdots\!88$$$$+$$$$11\!\cdots\!38$$$$\beta) q^{71} +(-$$$$89\!\cdots\!52$$$$+$$$$76\!\cdots\!72$$$$\beta) q^{72} +(-$$$$53\!\cdots\!86$$$$-$$$$38\!\cdots\!32$$$$\beta) q^{73} +($$$$15\!\cdots\!68$$$$-$$$$75\!\cdots\!28$$$$\beta) q^{74} +($$$$38\!\cdots\!00$$$$+$$$$65\!\cdots\!85$$$$\beta) q^{75} +(-$$$$51\!\cdots\!00$$$$+$$$$57\!\cdots\!52$$$$\beta) q^{76} +($$$$12\!\cdots\!56$$$$+$$$$49\!\cdots\!72$$$$\beta) q^{77} +($$$$21\!\cdots\!36$$$$-$$$$56\!\cdots\!12$$$$\beta) q^{78} +($$$$20\!\cdots\!60$$$$+$$$$10\!\cdots\!32$$$$\beta) q^{79} +(-$$$$31\!\cdots\!20$$$$+$$$$16\!\cdots\!24$$$$\beta) q^{80} +(-$$$$17\!\cdots\!99$$$$-$$$$23\!\cdots\!44$$$$\beta) q^{81} +(-$$$$62\!\cdots\!68$$$$+$$$$88\!\cdots\!56$$$$\beta) q^{82} +($$$$86\!\cdots\!44$$$$-$$$$17\!\cdots\!65$$$$\beta) q^{83} +(-$$$$29\!\cdots\!68$$$$-$$$$32\!\cdots\!16$$$$\beta) q^{84} +(-$$$$23\!\cdots\!60$$$$+$$$$49\!\cdots\!52$$$$\beta) q^{85} +($$$$41\!\cdots\!44$$$$+$$$$35\!\cdots\!32$$$$\beta) q^{86} +(-$$$$66\!\cdots\!40$$$$-$$$$40\!\cdots\!14$$$$\beta) q^{87} +(-$$$$36\!\cdots\!68$$$$-$$$$45\!\cdots\!08$$$$\beta) q^{88} +(-$$$$38\!\cdots\!10$$$$+$$$$65\!\cdots\!60$$$$\beta) q^{89} +($$$$12\!\cdots\!40$$$$-$$$$11\!\cdots\!48$$$$\beta) q^{90} +(-$$$$13\!\cdots\!08$$$$-$$$$10\!\cdots\!64$$$$\beta) q^{91} +($$$$17\!\cdots\!44$$$$+$$$$94\!\cdots\!64$$$$\beta) q^{92} +($$$$20\!\cdots\!88$$$$+$$$$25\!\cdots\!84$$$$\beta) q^{93} +($$$$14\!\cdots\!88$$$$+$$$$19\!\cdots\!28$$$$\beta) q^{94} +($$$$53\!\cdots\!00$$$$-$$$$31\!\cdots\!90$$$$\beta) q^{95} +(-$$$$26\!\cdots\!96$$$$+$$$$12\!\cdots\!24$$$$\beta) q^{96} +(-$$$$32\!\cdots\!22$$$$-$$$$57\!\cdots\!20$$$$\beta) q^{97} +(-$$$$19\!\cdots\!88$$$$-$$$$41\!\cdots\!52$$$$\beta) q^{98} +(-$$$$86\!\cdots\!44$$$$+$$$$11\!\cdots\!45$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 524288q^{2} + 423071208q^{3} + 137438953472q^{4} - 13507530555540q^{5} - 110905578749952q^{6} + 3106174162962256q^{7} - 36028797018963968q^{8} + 996387550079387706q^{9} + O(q^{10})$$ $$2q - 524288q^{2} + 423071208q^{3} + 137438953472q^{4} - 13507530555540q^{5} - 110905578749952q^{6} + 3106174162962256q^{7} - 36028797018963968q^{8} + 996387550079387706q^{9} + 3540918089951477760q^{10} + 4002878661541032504q^{11} + 29073232035827417088q^{12} -$$$$41\!\cdots\!72$$$$q^{13} -$$$$81\!\cdots\!64$$$$q^{14} -$$$$90\!\cdots\!60$$$$q^{15} +$$$$94\!\cdots\!92$$$$q^{16} +$$$$92\!\cdots\!36$$$$q^{17} -$$$$26\!\cdots\!64$$$$q^{18} -$$$$15\!\cdots\!00$$$$q^{19} -$$$$92\!\cdots\!40$$$$q^{20} -$$$$86\!\cdots\!76$$$$q^{21} -$$$$10\!\cdots\!76$$$$q^{22} +$$$$49\!\cdots\!08$$$$q^{23} -$$$$76\!\cdots\!72$$$$q^{24} -$$$$32\!\cdots\!50$$$$q^{25} +$$$$10\!\cdots\!68$$$$q^{26} +$$$$78\!\cdots\!20$$$$q^{27} +$$$$21\!\cdots\!16$$$$q^{28} +$$$$11\!\cdots\!80$$$$q^{29} +$$$$23\!\cdots\!40$$$$q^{30} -$$$$55\!\cdots\!56$$$$q^{31} -$$$$24\!\cdots\!48$$$$q^{32} -$$$$44\!\cdots\!84$$$$q^{33} -$$$$24\!\cdots\!84$$$$q^{34} +$$$$10\!\cdots\!80$$$$q^{35} +$$$$68\!\cdots\!16$$$$q^{36} -$$$$11\!\cdots\!44$$$$q^{37} +$$$$39\!\cdots\!00$$$$q^{38} -$$$$16\!\cdots\!88$$$$q^{39} +$$$$24\!\cdots\!60$$$$q^{40} +$$$$47\!\cdots\!44$$$$q^{41} +$$$$22\!\cdots\!44$$$$q^{42} -$$$$31\!\cdots\!52$$$$q^{43} +$$$$27\!\cdots\!44$$$$q^{44} -$$$$93\!\cdots\!20$$$$q^{45} -$$$$13\!\cdots\!52$$$$q^{46} -$$$$11\!\cdots\!04$$$$q^{47} +$$$$19\!\cdots\!68$$$$q^{48} +$$$$15\!\cdots\!54$$$$q^{49} +$$$$86\!\cdots\!00$$$$q^{50} +$$$$13\!\cdots\!44$$$$q^{51} -$$$$28\!\cdots\!92$$$$q^{52} +$$$$53\!\cdots\!28$$$$q^{53} -$$$$20\!\cdots\!80$$$$q^{54} +$$$$13\!\cdots\!20$$$$q^{55} -$$$$55\!\cdots\!04$$$$q^{56} -$$$$46\!\cdots\!00$$$$q^{57} -$$$$30\!\cdots\!20$$$$q^{58} +$$$$58\!\cdots\!60$$$$q^{59} -$$$$62\!\cdots\!60$$$$q^{60} +$$$$94\!\cdots\!84$$$$q^{61} +$$$$14\!\cdots\!64$$$$q^{62} -$$$$23\!\cdots\!32$$$$q^{63} +$$$$64\!\cdots\!12$$$$q^{64} +$$$$30\!\cdots\!40$$$$q^{65} +$$$$11\!\cdots\!96$$$$q^{66} +$$$$32\!\cdots\!96$$$$q^{67} +$$$$63\!\cdots\!96$$$$q^{68} -$$$$23\!\cdots\!68$$$$q^{69} -$$$$28\!\cdots\!20$$$$q^{70} -$$$$36\!\cdots\!76$$$$q^{71} -$$$$17\!\cdots\!04$$$$q^{72} -$$$$10\!\cdots\!72$$$$q^{73} +$$$$30\!\cdots\!36$$$$q^{74} +$$$$77\!\cdots\!00$$$$q^{75} -$$$$10\!\cdots\!00$$$$q^{76} +$$$$24\!\cdots\!12$$$$q^{77} +$$$$42\!\cdots\!72$$$$q^{78} +$$$$41\!\cdots\!20$$$$q^{79} -$$$$63\!\cdots\!40$$$$q^{80} -$$$$34\!\cdots\!98$$$$q^{81} -$$$$12\!\cdots\!36$$$$q^{82} +$$$$17\!\cdots\!88$$$$q^{83} -$$$$59\!\cdots\!36$$$$q^{84} -$$$$46\!\cdots\!20$$$$q^{85} +$$$$83\!\cdots\!88$$$$q^{86} -$$$$13\!\cdots\!80$$$$q^{87} -$$$$72\!\cdots\!36$$$$q^{88} -$$$$77\!\cdots\!20$$$$q^{89} +$$$$24\!\cdots\!80$$$$q^{90} -$$$$27\!\cdots\!16$$$$q^{91} +$$$$34\!\cdots\!88$$$$q^{92} +$$$$40\!\cdots\!76$$$$q^{93} +$$$$29\!\cdots\!76$$$$q^{94} +$$$$10\!\cdots\!00$$$$q^{95} -$$$$52\!\cdots\!92$$$$q^{96} -$$$$65\!\cdots\!44$$$$q^{97} -$$$$39\!\cdots\!76$$$$q^{98} -$$$$17\!\cdots\!88$$$$q^{99} + O(q^{100})$$

## 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
 27507.7 −27506.7
−262144. −7.39112e8 6.87195e10 −3.47974e12 1.93754e14 6.42679e15 −1.80144e16 9.60023e16 9.12192e17
1.2 −262144. 1.16218e9 6.87195e10 −1.00278e13 −3.04659e14 −3.32061e15 −1.80144e16 9.00385e17 2.62873e18
 $$n$$: 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.a 2
3.b odd 2 1 18.38.a.f 2
4.b odd 2 1 16.38.a.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 423071208 T_{3} -$$858983057411401584

'>$$85\!\cdots\!84$$ acting on $$S_{38}^{\mathrm{new}}(\Gamma_0(2))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 262144 T )^{2}$$
$3$ $$1 - 423071208 T + 41584754370593142 T^{2} -$$$$19\!\cdots\!04$$$$T^{3} +$$$$20\!\cdots\!69$$$$T^{4}$$
$5$ $$1 + 13507530555540 T +$$$$18\!\cdots\!50$$$$T^{2} +$$$$98\!\cdots\!00$$$$T^{3} +$$$$52\!\cdots\!25$$$$T^{4}$$
$7$ $$1 - 3106174162962256 T +$$$$15\!\cdots\!98$$$$T^{2} -$$$$57\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 4002878661541032504 T +$$$$10\!\cdots\!46$$$$T^{2} -$$$$13\!\cdots\!84$$$$T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$13$ $$1 +$$$$41\!\cdots\!72$$$$T +$$$$37\!\cdots\!62$$$$T^{2} +$$$$68\!\cdots\!76$$$$T^{3} +$$$$27\!\cdots\!89$$$$T^{4}$$
$17$ $$1 -$$$$92\!\cdots\!36$$$$T +$$$$18\!\cdots\!78$$$$T^{2} -$$$$31\!\cdots\!72$$$$T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$19$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$97\!\cdots\!78$$$$T^{2} +$$$$30\!\cdots\!00$$$$T^{3} +$$$$42\!\cdots\!21$$$$T^{4}$$
$23$ $$1 -$$$$49\!\cdots\!08$$$$T +$$$$32\!\cdots\!22$$$$T^{2} -$$$$12\!\cdots\!24$$$$T^{3} +$$$$58\!\cdots\!09$$$$T^{4}$$
$29$ $$1 -$$$$11\!\cdots\!80$$$$T +$$$$22\!\cdots\!18$$$$T^{2} -$$$$15\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!81$$$$T^{4}$$
$31$ $$1 +$$$$55\!\cdots\!56$$$$T +$$$$37\!\cdots\!06$$$$T^{2} +$$$$83\!\cdots\!16$$$$T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$37$ $$1 +$$$$11\!\cdots\!44$$$$T +$$$$23\!\cdots\!18$$$$T^{2} +$$$$12\!\cdots\!48$$$$T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$41$ $$1 -$$$$47\!\cdots\!44$$$$T +$$$$89\!\cdots\!46$$$$T^{2} -$$$$22\!\cdots\!64$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4}$$
$43$ $$1 +$$$$31\!\cdots\!52$$$$T +$$$$63\!\cdots\!62$$$$T^{2} +$$$$87\!\cdots\!36$$$$T^{3} +$$$$75\!\cdots\!49$$$$T^{4}$$
$47$ $$1 +$$$$11\!\cdots\!04$$$$T +$$$$12\!\cdots\!78$$$$T^{2} +$$$$82\!\cdots\!48$$$$T^{3} +$$$$54\!\cdots\!69$$$$T^{4}$$
$53$ $$1 -$$$$53\!\cdots\!28$$$$T +$$$$13\!\cdots\!22$$$$T^{2} -$$$$33\!\cdots\!64$$$$T^{3} +$$$$39\!\cdots\!69$$$$T^{4}$$
$59$ $$1 -$$$$58\!\cdots\!60$$$$T +$$$$64\!\cdots\!38$$$$T^{2} -$$$$19\!\cdots\!40$$$$T^{3} +$$$$11\!\cdots\!61$$$$T^{4}$$
$61$ $$1 -$$$$94\!\cdots\!84$$$$T +$$$$19\!\cdots\!06$$$$T^{2} -$$$$10\!\cdots\!64$$$$T^{3} +$$$$13\!\cdots\!41$$$$T^{4}$$
$67$ $$1 -$$$$32\!\cdots\!96$$$$T +$$$$76\!\cdots\!58$$$$T^{2} -$$$$12\!\cdots\!92$$$$T^{3} +$$$$13\!\cdots\!29$$$$T^{4}$$
$71$ $$1 +$$$$36\!\cdots\!76$$$$T +$$$$84\!\cdots\!26$$$$T^{2} +$$$$11\!\cdots\!16$$$$T^{3} +$$$$98\!\cdots\!81$$$$T^{4}$$
$73$ $$1 +$$$$10\!\cdots\!72$$$$T +$$$$44\!\cdots\!02$$$$T^{2} +$$$$94\!\cdots\!16$$$$T^{3} +$$$$76\!\cdots\!09$$$$T^{4}$$
$79$ $$1 -$$$$41\!\cdots\!20$$$$T +$$$$22\!\cdots\!18$$$$T^{2} -$$$$68\!\cdots\!80$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4}$$
$83$ $$1 -$$$$17\!\cdots\!88$$$$T +$$$$18\!\cdots\!82$$$$T^{2} -$$$$17\!\cdots\!24$$$$T^{3} +$$$$10\!\cdots\!29$$$$T^{4}$$
$89$ $$1 +$$$$77\!\cdots\!20$$$$T +$$$$24\!\cdots\!58$$$$T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$17\!\cdots\!41$$$$T^{4}$$
$97$ $$1 +$$$$65\!\cdots\!44$$$$T +$$$$45\!\cdots\!58$$$$T^{2} +$$$$21\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!69$$$$T^{4}$$