# Properties

 Label 1.34.a.a Level $1$ Weight $34$ Character orbit 1.a Self dual yes Analytic conductor $6.898$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.89828288810$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 589050$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 72\sqrt{2356201}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -60840 - \beta ) q^{2} + ( 18959940 + 312 \beta ) q^{3} + ( 7326116992 + 121680 \beta ) q^{4} + ( -90530768250 - 1004000 \beta ) q^{5} + ( -4964461096608 - 37942020 \beta ) q^{6} + ( -33576540033400 + 801594864 \beta ) q^{7} + ( -1409375292549120 - 6139193600 \beta ) q^{8} + ( -4010568477485427 + 11831002560 \beta ) q^{9} +O(q^{10})$$ $$q +(-60840 - \beta) q^{2} +(18959940 + 312 \beta) q^{3} +(7326116992 + 121680 \beta) q^{4} +(-90530768250 - 1004000 \beta) q^{5} +(-4964461096608 - 37942020 \beta) q^{6} +(-33576540033400 + 801594864 \beta) q^{7} +(-1409375292549120 - 6139193600 \beta) q^{8} +(-4010568477485427 + 11831002560 \beta) q^{9} +(17771296108266000 + 151614128250 \beta) q^{10} +(66935907720957132 - 1346611783000 \beta) q^{11} +(602617716665233920 + 4592794000704 \beta) q^{12} +(-1490805239129721970 - 3738595861728 \beta) q^{13} +(-7748320631234170176 - 15192491492360 \beta) q^{14} +(-5542640034569937000 - 47281379454000 \beta) q^{15} +(97802989555947175936 + 737660590018560 \beta) q^{16} +(-39680574630587762670 - 2662090686805056 \beta) q^{17} +(99492661364271659640 + 3290770281735027 \beta) q^{18} +(-$$$$68\!\cdots\!00$$$$+ 4884703150768920 \beta) q^{19} +(-$$$$21\!\cdots\!00$$$$- 18371205340628000 \beta) q^{20} +($$$$24\!\cdots\!12$$$$+ 4722310035327360 \beta) q^{21} +($$$$12\!\cdots\!20$$$$+ 14991953156762868 \beta) q^{22} +($$$$13\!\cdots\!20$$$$+ 164629195362887632 \beta) q^{23} +(-$$$$50\!\cdots\!00$$$$- 556123833579709440 \beta) q^{24} +(-$$$$95\!\cdots\!25$$$$+ 181785782646000000 \beta) q^{25} +($$$$13\!\cdots\!52$$$$+ 1718261411357253490 \beta) q^{26} +(-$$$$13\!\cdots\!20$$$$- 2761409163063330000 \beta) q^{27} +($$$$94\!\cdots\!80$$$$+ 1786984362586217088 \beta) q^{28} +(-$$$$83\!\cdots\!50$$$$- 6085145360184173920 \beta) q^{29} +($$$$91\!\cdots\!00$$$$+ 8419239160551297000 \beta) q^{30} +(-$$$$31\!\cdots\!08$$$$+ 32916497064471576000 \beta) q^{31} +(-$$$$28\!\cdots\!40$$$$- 89946988381051355136 \beta) q^{32} +(-$$$$38\!\cdots\!20$$$$- 4647675400034394816 \beta) q^{33} +($$$$34\!\cdots\!04$$$$+$$$$20\!\cdots\!10$$$$\beta) q^{34} +(-$$$$67\!\cdots\!00$$$$- 38858152669640668000 \beta) q^{35} +(-$$$$11\!\cdots\!84$$$$-$$$$40\!\cdots\!40$$$$\beta) q^{36} +(-$$$$52\!\cdots\!10$$$$+$$$$20\!\cdots\!04$$$$\beta) q^{37} +(-$$$$18\!\cdots\!80$$$$+$$$$38\!\cdots\!00$$$$\beta) q^{38} +(-$$$$42\!\cdots\!24$$$$-$$$$53\!\cdots\!60$$$$\beta) q^{39} +($$$$20\!\cdots\!00$$$$+$$$$19\!\cdots\!00$$$$\beta) q^{40} +($$$$13\!\cdots\!22$$$$-$$$$32\!\cdots\!00$$$$\beta) q^{41} +(-$$$$20\!\cdots\!20$$$$-$$$$27\!\cdots\!12$$$$\beta) q^{42} +($$$$78\!\cdots\!00$$$$+$$$$88\!\cdots\!52$$$$\beta) q^{43} +(-$$$$15\!\cdots\!56$$$$-$$$$17\!\cdots\!40$$$$\beta) q^{44} +($$$$21\!\cdots\!50$$$$+$$$$29\!\cdots\!00$$$$\beta) q^{45} +(-$$$$28\!\cdots\!88$$$$-$$$$23\!\cdots\!00$$$$\beta) q^{46} +($$$$27\!\cdots\!20$$$$+$$$$49\!\cdots\!84$$$$\beta) q^{47} +($$$$46\!\cdots\!20$$$$+$$$$44\!\cdots\!32$$$$\beta) q^{48} +($$$$12\!\cdots\!57$$$$-$$$$53\!\cdots\!00$$$$\beta) q^{49} +($$$$36\!\cdots\!00$$$$+$$$$84\!\cdots\!25$$$$\beta) q^{50} +(-$$$$10\!\cdots\!48$$$$-$$$$62\!\cdots\!80$$$$\beta) q^{51} +(-$$$$16\!\cdots\!00$$$$-$$$$20\!\cdots\!76$$$$\beta) q^{52} +(-$$$$13\!\cdots\!10$$$$+$$$$20\!\cdots\!12$$$$\beta) q^{53} +($$$$42\!\cdots\!00$$$$+$$$$30\!\cdots\!20$$$$\beta) q^{54} +($$$$10\!\cdots\!00$$$$+$$$$54\!\cdots\!00$$$$\beta) q^{55} +(-$$$$12\!\cdots\!00$$$$-$$$$92\!\cdots\!80$$$$\beta) q^{56} +($$$$57\!\cdots\!60$$$$-$$$$11\!\cdots\!00$$$$\beta) q^{57} +($$$$12\!\cdots\!80$$$$+$$$$12\!\cdots\!50$$$$\beta) q^{58} +(-$$$$15\!\cdots\!00$$$$+$$$$31\!\cdots\!60$$$$\beta) q^{59} +(-$$$$11\!\cdots\!00$$$$-$$$$10\!\cdots\!00$$$$\beta) q^{60} +(-$$$$28\!\cdots\!18$$$$-$$$$63\!\cdots\!00$$$$\beta) q^{61} +(-$$$$21\!\cdots\!80$$$$+$$$$11\!\cdots\!08$$$$\beta) q^{62} +($$$$25\!\cdots\!60$$$$-$$$$36\!\cdots\!28$$$$\beta) q^{63} +($$$$43\!\cdots\!12$$$$+$$$$19\!\cdots\!60$$$$\beta) q^{64} +($$$$18\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{65} +($$$$29\!\cdots\!44$$$$+$$$$41\!\cdots\!60$$$$\beta) q^{66} +($$$$79\!\cdots\!80$$$$+$$$$28\!\cdots\!44$$$$\beta) q^{67} +(-$$$$42\!\cdots\!60$$$$-$$$$24\!\cdots\!52$$$$\beta) q^{68} +($$$$87\!\cdots\!56$$$$+$$$$72\!\cdots\!20$$$$\beta) q^{69} +($$$$88\!\cdots\!00$$$$+$$$$91\!\cdots\!00$$$$\beta) q^{70} +(-$$$$13\!\cdots\!88$$$$+$$$$18\!\cdots\!00$$$$\beta) q^{71} +($$$$47\!\cdots\!40$$$$+$$$$79\!\cdots\!00$$$$\beta) q^{72} +($$$$47\!\cdots\!70$$$$-$$$$59\!\cdots\!68$$$$\beta) q^{73} +($$$$72\!\cdots\!64$$$$+$$$$40\!\cdots\!50$$$$\beta) q^{74} +(-$$$$11\!\cdots\!00$$$$-$$$$26\!\cdots\!00$$$$\beta) q^{75} +($$$$22\!\cdots\!00$$$$-$$$$46\!\cdots\!60$$$$\beta) q^{76} +(-$$$$15\!\cdots\!00$$$$+$$$$98\!\cdots\!48$$$$\beta) q^{77} +($$$$91\!\cdots\!00$$$$+$$$$75\!\cdots\!24$$$$\beta) q^{78} +(-$$$$42\!\cdots\!00$$$$-$$$$11\!\cdots\!20$$$$\beta) q^{79} +(-$$$$17\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$\beta) q^{80} +($$$$91\!\cdots\!21$$$$-$$$$16\!\cdots\!20$$$$\beta) q^{81} +($$$$31\!\cdots\!20$$$$+$$$$58\!\cdots\!78$$$$\beta) q^{82} +($$$$14\!\cdots\!60$$$$+$$$$24\!\cdots\!92$$$$\beta) q^{83} +($$$$24\!\cdots\!04$$$$+$$$$32\!\cdots\!80$$$$\beta) q^{84} +($$$$36\!\cdots\!00$$$$+$$$$28\!\cdots\!00$$$$\beta) q^{85} +(-$$$$15\!\cdots\!68$$$$-$$$$13\!\cdots\!80$$$$\beta) q^{86} +(-$$$$39\!\cdots\!60$$$$-$$$$37\!\cdots\!00$$$$\beta) q^{87} +($$$$66\!\cdots\!60$$$$+$$$$14\!\cdots\!00$$$$\beta) q^{88} +($$$$66\!\cdots\!50$$$$-$$$$38\!\cdots\!60$$$$\beta) q^{89} +(-$$$$49\!\cdots\!00$$$$-$$$$39\!\cdots\!50$$$$\beta) q^{90} +($$$$13\!\cdots\!72$$$$-$$$$10\!\cdots\!80$$$$\beta) q^{91} +($$$$34\!\cdots\!80$$$$+$$$$27\!\cdots\!44$$$$\beta) q^{92} +($$$$66\!\cdots\!80$$$$-$$$$34\!\cdots\!96$$$$\beta) q^{93} +(-$$$$22\!\cdots\!56$$$$-$$$$30\!\cdots\!80$$$$\beta) q^{94} +($$$$16\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$\beta) q^{95} +(-$$$$39\!\cdots\!88$$$$-$$$$25\!\cdots\!20$$$$\beta) q^{96} +(-$$$$18\!\cdots\!30$$$$+$$$$53\!\cdots\!84$$$$\beta) q^{97} +($$$$58\!\cdots\!20$$$$+$$$$20\!\cdots\!43$$$$\beta) q^{98} +(-$$$$46\!\cdots\!64$$$$+$$$$61\!\cdots\!20$$$$\beta) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 121680q^{2} + 37919880q^{3} + 14652233984q^{4} - 181061536500q^{5} - 9928922193216q^{6} - 67153080066800q^{7} - 2818750585098240q^{8} - 8021136954970854q^{9} + O(q^{10})$$ $$2q - 121680q^{2} + 37919880q^{3} + 14652233984q^{4} - 181061536500q^{5} - 9928922193216q^{6} - 67153080066800q^{7} - 2818750585098240q^{8} - 8021136954970854q^{9} + 35542592216532000q^{10} + 133871815441914264q^{11} + 1205235433330467840q^{12} - 2981610478259443940q^{13} - 15496641262468340352q^{14} - 11085280069139874000q^{15} +$$$$19\!\cdots\!72$$$$q^{16} - 79361149261175525340q^{17} +$$$$19\!\cdots\!80$$$$q^{18} -$$$$13\!\cdots\!00$$$$q^{19} -$$$$43\!\cdots\!00$$$$q^{20} +$$$$48\!\cdots\!24$$$$q^{21} +$$$$24\!\cdots\!40$$$$q^{22} +$$$$26\!\cdots\!40$$$$q^{23} -$$$$10\!\cdots\!00$$$$q^{24} -$$$$19\!\cdots\!50$$$$q^{25} +$$$$27\!\cdots\!04$$$$q^{26} -$$$$27\!\cdots\!40$$$$q^{27} +$$$$18\!\cdots\!60$$$$q^{28} -$$$$16\!\cdots\!00$$$$q^{29} +$$$$18\!\cdots\!00$$$$q^{30} -$$$$62\!\cdots\!16$$$$q^{31} -$$$$57\!\cdots\!80$$$$q^{32} -$$$$77\!\cdots\!40$$$$q^{33} +$$$$69\!\cdots\!08$$$$q^{34} -$$$$13\!\cdots\!00$$$$q^{35} -$$$$23\!\cdots\!68$$$$q^{36} -$$$$10\!\cdots\!20$$$$q^{37} -$$$$36\!\cdots\!60$$$$q^{38} -$$$$85\!\cdots\!48$$$$q^{39} +$$$$40\!\cdots\!00$$$$q^{40} +$$$$27\!\cdots\!44$$$$q^{41} -$$$$40\!\cdots\!40$$$$q^{42} +$$$$15\!\cdots\!00$$$$q^{43} -$$$$30\!\cdots\!12$$$$q^{44} +$$$$43\!\cdots\!00$$$$q^{45} -$$$$56\!\cdots\!76$$$$q^{46} +$$$$54\!\cdots\!40$$$$q^{47} +$$$$93\!\cdots\!40$$$$q^{48} +$$$$24\!\cdots\!14$$$$q^{49} +$$$$72\!\cdots\!00$$$$q^{50} -$$$$21\!\cdots\!96$$$$q^{51} -$$$$32\!\cdots\!00$$$$q^{52} -$$$$26\!\cdots\!20$$$$q^{53} +$$$$84\!\cdots\!00$$$$q^{54} +$$$$20\!\cdots\!00$$$$q^{55} -$$$$25\!\cdots\!00$$$$q^{56} +$$$$11\!\cdots\!20$$$$q^{57} +$$$$25\!\cdots\!60$$$$q^{58} -$$$$30\!\cdots\!00$$$$q^{59} -$$$$22\!\cdots\!00$$$$q^{60} -$$$$57\!\cdots\!36$$$$q^{61} -$$$$42\!\cdots\!60$$$$q^{62} +$$$$50\!\cdots\!20$$$$q^{63} +$$$$86\!\cdots\!24$$$$q^{64} +$$$$36\!\cdots\!00$$$$q^{65} +$$$$58\!\cdots\!88$$$$q^{66} +$$$$15\!\cdots\!60$$$$q^{67} -$$$$84\!\cdots\!20$$$$q^{68} +$$$$17\!\cdots\!12$$$$q^{69} +$$$$17\!\cdots\!00$$$$q^{70} -$$$$26\!\cdots\!76$$$$q^{71} +$$$$95\!\cdots\!80$$$$q^{72} +$$$$94\!\cdots\!40$$$$q^{73} +$$$$14\!\cdots\!28$$$$q^{74} -$$$$22\!\cdots\!00$$$$q^{75} +$$$$45\!\cdots\!00$$$$q^{76} -$$$$30\!\cdots\!00$$$$q^{77} +$$$$18\!\cdots\!00$$$$q^{78} -$$$$85\!\cdots\!00$$$$q^{79} -$$$$35\!\cdots\!00$$$$q^{80} +$$$$18\!\cdots\!42$$$$q^{81} +$$$$62\!\cdots\!40$$$$q^{82} +$$$$29\!\cdots\!20$$$$q^{83} +$$$$49\!\cdots\!08$$$$q^{84} +$$$$72\!\cdots\!00$$$$q^{85} -$$$$31\!\cdots\!36$$$$q^{86} -$$$$78\!\cdots\!20$$$$q^{87} +$$$$13\!\cdots\!20$$$$q^{88} +$$$$13\!\cdots\!00$$$$q^{89} -$$$$98\!\cdots\!00$$$$q^{90} +$$$$26\!\cdots\!44$$$$q^{91} +$$$$68\!\cdots\!60$$$$q^{92} +$$$$13\!\cdots\!60$$$$q^{93} -$$$$45\!\cdots\!12$$$$q^{94} +$$$$33\!\cdots\!00$$$$q^{95} -$$$$79\!\cdots\!76$$$$q^{96} -$$$$36\!\cdots\!60$$$$q^{97} +$$$$11\!\cdots\!40$$$$q^{98} -$$$$92\!\cdots\!28$$$$q^{99} + O(q^{100})$$

## 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
 767.996 −766.996
−171359. 5.34420e7 2.07741e10 −2.01492e11 −9.15779e12 5.50153e13 −2.08788e15 −2.70301e15 3.45276e16
1.2 49679.4 −1.55221e7 −6.12189e9 2.04307e10 −7.71130e11 −1.22168e14 −7.30875e14 −5.31812e15 1.01499e15
 $$n$$: 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 1.34.a.a 2
3.b odd 2 1 9.34.a.b 2
4.b odd 2 1 16.34.a.b 2
5.b even 2 1 25.34.a.a 2
5.c odd 4 2 25.34.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 1.a even 1 1 trivial
9.34.a.b 2 3.b odd 2 1
16.34.a.b 2 4.b odd 2 1
25.34.a.a 2 5.b even 2 1
25.34.b.a 4 5.c odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{34}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 121680 T + 8666828800 T^{2} + 1045223241154560 T^{3} + 73786976294838206464 T^{4}$$
$3$ $$1 - 37919880 T + 10288587693648150 T^{2} -$$$$21\!\cdots\!40$$$$T^{3} +$$$$30\!\cdots\!29$$$$T^{4}$$
$5$ $$1 + 181061536500 T +$$$$22\!\cdots\!50$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!25$$$$T^{4}$$
$7$ $$1 + 67153080066800 T +$$$$87\!\cdots\!50$$$$T^{2} +$$$$51\!\cdots\!00$$$$T^{3} +$$$$59\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 133871815441914264 T +$$$$28\!\cdots\!86$$$$T^{2} -$$$$31\!\cdots\!84$$$$T^{3} +$$$$53\!\cdots\!61$$$$T^{4}$$
$13$ $$1 + 2981610478259443940 T +$$$$13\!\cdots\!50$$$$T^{2} +$$$$17\!\cdots\!20$$$$T^{3} +$$$$33\!\cdots\!09$$$$T^{4}$$
$17$ $$1 + 79361149261175525340 T -$$$$44\!\cdots\!50$$$$T^{2} +$$$$31\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!69$$$$T^{4}$$
$19$ $$1 +$$$$13\!\cdots\!00$$$$T +$$$$33\!\cdots\!18$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$24\!\cdots\!81$$$$T^{4}$$
$23$ $$1 -$$$$26\!\cdots\!40$$$$T +$$$$15\!\cdots\!50$$$$T^{2} -$$$$22\!\cdots\!20$$$$T^{3} +$$$$74\!\cdots\!89$$$$T^{4}$$
$29$ $$1 +$$$$16\!\cdots\!00$$$$T +$$$$38\!\cdots\!78$$$$T^{2} +$$$$30\!\cdots\!00$$$$T^{3} +$$$$32\!\cdots\!21$$$$T^{4}$$
$31$ $$1 +$$$$62\!\cdots\!16$$$$T +$$$$29\!\cdots\!46$$$$T^{2} +$$$$10\!\cdots\!56$$$$T^{3} +$$$$26\!\cdots\!81$$$$T^{4}$$
$37$ $$1 +$$$$10\!\cdots\!20$$$$T +$$$$13\!\cdots\!50$$$$T^{2} +$$$$59\!\cdots\!40$$$$T^{3} +$$$$31\!\cdots\!09$$$$T^{4}$$
$41$ $$1 -$$$$27\!\cdots\!44$$$$T +$$$$22\!\cdots\!26$$$$T^{2} -$$$$46\!\cdots\!24$$$$T^{3} +$$$$27\!\cdots\!41$$$$T^{4}$$
$43$ $$1 -$$$$15\!\cdots\!00$$$$T +$$$$12\!\cdots\!50$$$$T^{2} -$$$$12\!\cdots\!00$$$$T^{3} +$$$$64\!\cdots\!49$$$$T^{4}$$
$47$ $$1 -$$$$54\!\cdots\!40$$$$T +$$$$37\!\cdots\!50$$$$T^{2} -$$$$81\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!29$$$$T^{4}$$
$53$ $$1 +$$$$26\!\cdots\!20$$$$T +$$$$12\!\cdots\!50$$$$T^{2} +$$$$21\!\cdots\!60$$$$T^{3} +$$$$63\!\cdots\!29$$$$T^{4}$$
$59$ $$1 +$$$$30\!\cdots\!00$$$$T +$$$$77\!\cdots\!58$$$$T^{2} +$$$$83\!\cdots\!00$$$$T^{3} +$$$$75\!\cdots\!41$$$$T^{4}$$
$61$ $$1 +$$$$57\!\cdots\!36$$$$T +$$$$16\!\cdots\!86$$$$T^{2} +$$$$47\!\cdots\!16$$$$T^{3} +$$$$67\!\cdots\!61$$$$T^{4}$$
$67$ $$1 -$$$$15\!\cdots\!60$$$$T +$$$$41\!\cdots\!50$$$$T^{2} -$$$$29\!\cdots\!20$$$$T^{3} +$$$$33\!\cdots\!69$$$$T^{4}$$
$71$ $$1 +$$$$26\!\cdots\!76$$$$T +$$$$22\!\cdots\!66$$$$T^{2} +$$$$32\!\cdots\!36$$$$T^{3} +$$$$15\!\cdots\!21$$$$T^{4}$$
$73$ $$1 -$$$$94\!\cdots\!40$$$$T +$$$$19\!\cdots\!50$$$$T^{2} -$$$$29\!\cdots\!20$$$$T^{3} +$$$$95\!\cdots\!89$$$$T^{4}$$
$79$ $$1 +$$$$85\!\cdots\!00$$$$T +$$$$85\!\cdots\!78$$$$T^{2} +$$$$35\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!21$$$$T^{4}$$
$83$ $$1 -$$$$29\!\cdots\!20$$$$T +$$$$37\!\cdots\!50$$$$T^{2} -$$$$62\!\cdots\!60$$$$T^{3} +$$$$45\!\cdots\!69$$$$T^{4}$$
$89$ $$1 -$$$$13\!\cdots\!00$$$$T +$$$$47\!\cdots\!38$$$$T^{2} -$$$$28\!\cdots\!00$$$$T^{3} +$$$$45\!\cdots\!61$$$$T^{4}$$
$97$ $$1 +$$$$36\!\cdots\!60$$$$T +$$$$42\!\cdots\!50$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!29$$$$T^{4}$$