# Properties

 Label 25.36.a.a Level $25$ Weight $36$ Character orbit 25.a Self dual yes Analytic conductor $193.988$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$36$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$193.987826584$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 12422194 x - 2645665785$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{12}\cdot 3^{3}\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 1) 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.

 $$f(q)$$ $$=$$ $$q + ( -46552 - \beta_{1} ) q^{2} + ( 34958436 - \beta_{1} + \beta_{2} ) q^{3} + ( 11613754048 + 194112 \beta_{1} + 72 \beta_{2} ) q^{4} + ( -1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2} ) q^{6} + ( -292807383115352 - 723239370 \beta_{1} + 852426 \beta_{2} ) q^{7} + ( -7445336641779200 - 17597768192 \beta_{1} - 10055232 \beta_{2} ) q^{8} + ( 50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2} ) q^{9} +O(q^{10})$$ $$q +(-46552 - \beta_{1}) q^{2} +(34958436 - \beta_{1} + \beta_{2}) q^{3} +(11613754048 + 194112 \beta_{1} + 72 \beta_{2}) q^{4} +(-1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2}) q^{6} +(-292807383115352 - 723239370 \beta_{1} + 852426 \beta_{2}) q^{7} +(-7445336641779200 - 17597768192 \beta_{1} - 10055232 \beta_{2}) q^{8} +(50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2}) q^{9} +(-385981809516662348 + 583640593195 \beta_{1} - 3858182955 \beta_{2}) q^{11} +(7516297175592162048 + 21885019697408 \beta_{1} - 17183392736 \beta_{2}) q^{12} +(20713203516999550186 - 54628153929972 \beta_{1} + 55424393460 \beta_{2}) q^{13} +(45303114418357146176 + 261002424214616 \beta_{1} + 98492944896 \beta_{2}) q^{14} +($$$$71\!\cdots\!16$$$$+ 5006482791469056 \beta_{1} - 1754429566464 \beta_{2}) q^{16} +($$$$13\!\cdots\!98$$$$+ 2825301075479112 \beta_{1} - 6543407393352 \beta_{2}) q^{17} +($$$$76\!\cdots\!36$$$$- 1342002862888821 \beta_{1} + 11399973267456 \beta_{2}) q^{18} +(-$$$$10\!\cdots\!80$$$$- 84000151153806507 \beta_{1} - 9872745919317 \beta_{2}) q^{19} +($$$$74\!\cdots\!92$$$$- 336783816338068624 \beta_{1} - 328342926322544 \beta_{2}) q^{21} +(-$$$$75\!\cdots\!04$$$$+ 926845923948830028 \beta_{1} - 252123333707520 \beta_{2}) q^{22} +($$$$17\!\cdots\!36$$$$- 2541836449593511358 \beta_{1} - 1160877406548546 \beta_{2}) q^{23} +(-$$$$12\!\cdots\!40$$$$- 1173379931106134016 \beta_{1} - 4385028066775296 \beta_{2}) q^{24} +($$$$14\!\cdots\!12$$$$- 21659187993208072426 \beta_{1} + 6951417853215744 \beta_{2}) q^{26} +(-$$$$91\!\cdots\!00$$$$- 23978122270590027258 \beta_{1} + 34841458485708282 \beta_{2}) q^{27} +(-$$$$34\!\cdots\!36$$$$- 74972221489028408832 \beta_{1} - 42717777074276544 \beta_{2}) q^{28} +(-$$$$12\!\cdots\!70$$$$-$$$$16\!\cdots\!08$$$$\beta_{1} + 30897215823693852 \beta_{2}) q^{29} +($$$$34\!\cdots\!52$$$$+$$$$18\!\cdots\!60$$$$\beta_{1} + 125813443508636760 \beta_{2}) q^{31} +($$$$30\!\cdots\!48$$$$-$$$$56\!\cdots\!16$$$$\beta_{1} - 110510836707594240 \beta_{2}) q^{32} +(-$$$$39\!\cdots\!28$$$$+$$$$99\!\cdots\!08$$$$\beta_{1} + 46744170434631912 \beta_{2}) q^{33} +(-$$$$18\!\cdots\!04$$$$-$$$$66\!\cdots\!46$$$$\beta_{1} - 559749470446872576 \beta_{2}) q^{34} +(-$$$$20\!\cdots\!84$$$$-$$$$14\!\cdots\!64$$$$\beta_{1} + 3891889065775683816 \beta_{2}) q^{36} +(-$$$$82\!\cdots\!02$$$$-$$$$15\!\cdots\!64$$$$\beta_{1} + 5309319421076092452 \beta_{2}) q^{37} +($$$$41\!\cdots\!00$$$$+$$$$24\!\cdots\!12$$$$\beta_{1} + 5510380631291741952 \beta_{2}) q^{38} +($$$$62\!\cdots\!04$$$$-$$$$23\!\cdots\!54$$$$\beta_{1} + 19171250864870765626 \beta_{2}) q^{39} +($$$$78\!\cdots\!02$$$$-$$$$39\!\cdots\!40$$$$\beta_{1} - 24089178969404126640 \beta_{2}) q^{41} +($$$$11\!\cdots\!96$$$$+$$$$29\!\cdots\!32$$$$\beta_{1} + 6368192380520484864 \beta_{2}) q^{42} +($$$$15\!\cdots\!36$$$$-$$$$17\!\cdots\!35$$$$\beta_{1} - 66376501078524728013 \beta_{2}) q^{43} +(-$$$$26\!\cdots\!04$$$$-$$$$10\!\cdots\!16$$$$\beta_{1} + 52103622124984646304 \beta_{2}) q^{44} +($$$$10\!\cdots\!52$$$$+$$$$39\!\cdots\!00$$$$\beta_{1} +$$$$11\!\cdots\!00$$$$\beta_{2}) q^{46} +(-$$$$54\!\cdots\!52$$$$-$$$$86\!\cdots\!72$$$$\beta_{1} -$$$$51\!\cdots\!16$$$$\beta_{2}) q^{47} +(-$$$$14\!\cdots\!64$$$$+$$$$13\!\cdots\!12$$$$\beta_{1} +$$$$43\!\cdots\!24$$$$\beta_{2}) q^{48} +(-$$$$19\!\cdots\!07$$$$+$$$$11\!\cdots\!80$$$$\beta_{1} -$$$$45\!\cdots\!20$$$$\beta_{2}) q^{49} +(-$$$$60\!\cdots\!68$$$$+$$$$20\!\cdots\!62$$$$\beta_{1} +$$$$18\!\cdots\!22$$$$\beta_{2}) q^{51} +($$$$17\!\cdots\!48$$$$+$$$$25\!\cdots\!60$$$$\beta_{1} + 33640287635007496656 \beta_{2}) q^{52} +($$$$55\!\cdots\!86$$$$+$$$$47\!\cdots\!24$$$$\beta_{1} +$$$$46\!\cdots\!76$$$$\beta_{2}) q^{53} +($$$$14\!\cdots\!20$$$$+$$$$70\!\cdots\!28$$$$\beta_{1} +$$$$36\!\cdots\!68$$$$\beta_{2}) q^{54} +($$$$18\!\cdots\!80$$$$+$$$$12\!\cdots\!52$$$$\beta_{1} -$$$$31\!\cdots\!88$$$$\beta_{2}) q^{56} +(-$$$$13\!\cdots\!00$$$$-$$$$14\!\cdots\!36$$$$\beta_{1} -$$$$10\!\cdots\!56$$$$\beta_{2}) q^{57} +($$$$79\!\cdots\!00$$$$+$$$$32\!\cdots\!78$$$$\beta_{1} +$$$$13\!\cdots\!88$$$$\beta_{2}) q^{58} +($$$$14\!\cdots\!60$$$$+$$$$83\!\cdots\!79$$$$\beta_{1} +$$$$13\!\cdots\!49$$$$\beta_{2}) q^{59} +($$$$78\!\cdots\!02$$$$+$$$$49\!\cdots\!00$$$$\beta_{1} +$$$$10\!\cdots\!00$$$$\beta_{2}) q^{61} +(-$$$$98\!\cdots\!04$$$$-$$$$82\!\cdots\!12$$$$\beta_{1} -$$$$66\!\cdots\!60$$$$\beta_{2}) q^{62} +(-$$$$15\!\cdots\!64$$$$+$$$$44\!\cdots\!86$$$$\beta_{1} +$$$$10\!\cdots\!78$$$$\beta_{2}) q^{63} +($$$$31\!\cdots\!28$$$$-$$$$73\!\cdots\!56$$$$\beta_{1} +$$$$95\!\cdots\!64$$$$\beta_{2}) q^{64} +(-$$$$25\!\cdots\!56$$$$+$$$$24\!\cdots\!16$$$$\beta_{1} -$$$$69\!\cdots\!04$$$$\beta_{2}) q^{66} +($$$$62\!\cdots\!48$$$$-$$$$32\!\cdots\!53$$$$\beta_{1} +$$$$10\!\cdots\!33$$$$\beta_{2}) q^{67} +(-$$$$73\!\cdots\!36$$$$+$$$$27\!\cdots\!84$$$$\beta_{1} +$$$$24\!\cdots\!92$$$$\beta_{2}) q^{68} +(-$$$$10\!\cdots\!16$$$$-$$$$23\!\cdots\!32$$$$\beta_{1} +$$$$57\!\cdots\!08$$$$\beta_{2}) q^{69} +($$$$11\!\cdots\!52$$$$-$$$$58\!\cdots\!50$$$$\beta_{1} -$$$$52\!\cdots\!50$$$$\beta_{2}) q^{71} +(-$$$$10\!\cdots\!00$$$$+$$$$16\!\cdots\!76$$$$\beta_{1} -$$$$73\!\cdots\!04$$$$\beta_{2}) q^{72} +($$$$95\!\cdots\!86$$$$-$$$$17\!\cdots\!28$$$$\beta_{1} +$$$$75\!\cdots\!84$$$$\beta_{2}) q^{73} +($$$$70\!\cdots\!36$$$$+$$$$21\!\cdots\!70$$$$\beta_{1} +$$$$13\!\cdots\!20$$$$\beta_{2}) q^{74} +(-$$$$90\!\cdots\!40$$$$-$$$$58\!\cdots\!16$$$$\beta_{1} -$$$$11\!\cdots\!96$$$$\beta_{2}) q^{76} +(-$$$$23\!\cdots\!04$$$$+$$$$13\!\cdots\!40$$$$\beta_{1} +$$$$97\!\cdots\!32$$$$\beta_{2}) q^{77} +($$$$73\!\cdots\!72$$$$-$$$$58\!\cdots\!00$$$$\beta_{1} +$$$$27\!\cdots\!44$$$$\beta_{2}) q^{78} +(-$$$$14\!\cdots\!20$$$$-$$$$26\!\cdots\!48$$$$\beta_{1} +$$$$62\!\cdots\!12$$$$\beta_{2}) q^{79} +($$$$62\!\cdots\!21$$$$-$$$$12\!\cdots\!12$$$$\beta_{1} -$$$$65\!\cdots\!72$$$$\beta_{2}) q^{81} +($$$$13\!\cdots\!96$$$$+$$$$18\!\cdots\!38$$$$\beta_{1} +$$$$15\!\cdots\!40$$$$\beta_{2}) q^{82} +(-$$$$49\!\cdots\!64$$$$+$$$$42\!\cdots\!51$$$$\beta_{1} +$$$$21\!\cdots\!93$$$$\beta_{2}) q^{83} +(-$$$$43\!\cdots\!84$$$$-$$$$50\!\cdots\!88$$$$\beta_{1} +$$$$95\!\cdots\!72$$$$\beta_{2}) q^{84} +($$$$49\!\cdots\!32$$$$-$$$$24\!\cdots\!68$$$$\beta_{1} -$$$$23\!\cdots\!08$$$$\beta_{2}) q^{86} +($$$$26\!\cdots\!00$$$$-$$$$40\!\cdots\!34$$$$\beta_{1} +$$$$13\!\cdots\!86$$$$\beta_{2}) q^{87} +($$$$62\!\cdots\!00$$$$+$$$$26\!\cdots\!16$$$$\beta_{1} +$$$$19\!\cdots\!36$$$$\beta_{2}) q^{88} +($$$$10\!\cdots\!90$$$$-$$$$67\!\cdots\!44$$$$\beta_{1} +$$$$91\!\cdots\!36$$$$\beta_{2}) q^{89} +($$$$33\!\cdots\!32$$$$-$$$$19\!\cdots\!48$$$$\beta_{1} +$$$$53\!\cdots\!12$$$$\beta_{2}) q^{91} +(-$$$$27\!\cdots\!52$$$$-$$$$93\!\cdots\!08$$$$\beta_{1} +$$$$18\!\cdots\!28$$$$\beta_{2}) q^{92} +($$$$13\!\cdots\!72$$$$+$$$$81\!\cdots\!28$$$$\beta_{1} -$$$$35\!\cdots\!68$$$$\beta_{2}) q^{93} +($$$$63\!\cdots\!56$$$$+$$$$15\!\cdots\!48$$$$\beta_{1} -$$$$21\!\cdots\!12$$$$\beta_{2}) q^{94} +(-$$$$10\!\cdots\!28$$$$-$$$$86\!\cdots\!32$$$$\beta_{1} +$$$$74\!\cdots\!08$$$$\beta_{2}) q^{96} +($$$$35\!\cdots\!98$$$$-$$$$99\!\cdots\!92$$$$\beta_{1} -$$$$89\!\cdots\!36$$$$\beta_{2}) q^{97} +($$$$44\!\cdots\!64$$$$+$$$$25\!\cdots\!27$$$$\beta_{1} -$$$$32\!\cdots\!80$$$$\beta_{2}) q^{98} +($$$$10\!\cdots\!84$$$$+$$$$15\!\cdots\!23$$$$\beta_{1} -$$$$30\!\cdots\!87$$$$\beta_{2}) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} + O(q^{10})$$ $$3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} - 1157945428549987044q^{11} + 22548891526776486144q^{12} + 62139610550998650558q^{13} +$$$$13\!\cdots\!28$$$$q^{14} +$$$$21\!\cdots\!48$$$$q^{16} +$$$$39\!\cdots\!94$$$$q^{17} +$$$$23\!\cdots\!08$$$$q^{18} -$$$$32\!\cdots\!40$$$$q^{19} +$$$$22\!\cdots\!76$$$$q^{21} -$$$$22\!\cdots\!12$$$$q^{22} +$$$$51\!\cdots\!08$$$$q^{23} -$$$$37\!\cdots\!20$$$$q^{24} +$$$$42\!\cdots\!36$$$$q^{26} -$$$$27\!\cdots\!00$$$$q^{27} -$$$$10\!\cdots\!08$$$$q^{28} -$$$$38\!\cdots\!10$$$$q^{29} +$$$$10\!\cdots\!56$$$$q^{31} +$$$$92\!\cdots\!44$$$$q^{32} -$$$$11\!\cdots\!84$$$$q^{33} -$$$$55\!\cdots\!12$$$$q^{34} -$$$$60\!\cdots\!52$$$$q^{36} -$$$$24\!\cdots\!06$$$$q^{37} +$$$$12\!\cdots\!00$$$$q^{38} +$$$$18\!\cdots\!12$$$$q^{39} +$$$$23\!\cdots\!06$$$$q^{41} +$$$$33\!\cdots\!88$$$$q^{42} +$$$$47\!\cdots\!08$$$$q^{43} -$$$$80\!\cdots\!12$$$$q^{44} +$$$$31\!\cdots\!56$$$$q^{46} -$$$$16\!\cdots\!56$$$$q^{47} -$$$$44\!\cdots\!92$$$$q^{48} -$$$$59\!\cdots\!21$$$$q^{49} -$$$$18\!\cdots\!04$$$$q^{51} +$$$$51\!\cdots\!44$$$$q^{52} +$$$$16\!\cdots\!58$$$$q^{53} +$$$$44\!\cdots\!60$$$$q^{54} +$$$$56\!\cdots\!40$$$$q^{56} -$$$$40\!\cdots\!00$$$$q^{57} +$$$$23\!\cdots\!00$$$$q^{58} +$$$$43\!\cdots\!80$$$$q^{59} +$$$$23\!\cdots\!06$$$$q^{61} -$$$$29\!\cdots\!12$$$$q^{62} -$$$$45\!\cdots\!92$$$$q^{63} +$$$$93\!\cdots\!84$$$$q^{64} -$$$$75\!\cdots\!68$$$$q^{66} +$$$$18\!\cdots\!44$$$$q^{67} -$$$$21\!\cdots\!08$$$$q^{68} -$$$$32\!\cdots\!48$$$$q^{69} +$$$$34\!\cdots\!56$$$$q^{71} -$$$$31\!\cdots\!00$$$$q^{72} +$$$$28\!\cdots\!58$$$$q^{73} +$$$$21\!\cdots\!08$$$$q^{74} -$$$$27\!\cdots\!20$$$$q^{76} -$$$$69\!\cdots\!12$$$$q^{77} +$$$$22\!\cdots\!16$$$$q^{78} -$$$$42\!\cdots\!60$$$$q^{79} +$$$$18\!\cdots\!63$$$$q^{81} +$$$$40\!\cdots\!88$$$$q^{82} -$$$$14\!\cdots\!92$$$$q^{83} -$$$$13\!\cdots\!52$$$$q^{84} +$$$$14\!\cdots\!96$$$$q^{86} +$$$$78\!\cdots\!00$$$$q^{87} +$$$$18\!\cdots\!00$$$$q^{88} +$$$$30\!\cdots\!70$$$$q^{89} +$$$$10\!\cdots\!96$$$$q^{91} -$$$$83\!\cdots\!56$$$$q^{92} +$$$$40\!\cdots\!16$$$$q^{93} +$$$$19\!\cdots\!68$$$$q^{94} -$$$$32\!\cdots\!84$$$$q^{96} +$$$$10\!\cdots\!94$$$$q^{97} +$$$$13\!\cdots\!92$$$$q^{98} +$$$$30\!\cdots\!52$$$$q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 12422194 x - 2645665785$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$24 \nu^{2} + 44712 \nu - 198755104$$$$)/979$$ $$\beta_{2}$$ $$=$$ $$($$$$-39144 \nu^{2} + 84967848 \nu + 324169574624$$$$)/979$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 1631 \beta_{1}$$$$)/161280$$ $$\nu^{2}$$ $$=$$ $$($$$$-621 \beta_{2} + 1180109 \beta_{1} + 445211432960$$$$)/53760$$

## 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
 3626.53 −3412.77 −213.765
−331573. 1.54691e8 7.55810e10 0 −5.12913e13 −3.96640e14 −1.36679e16 −2.61023e16 0
1.2 26808.0 −3.95729e8 −3.36411e10 0 −1.06087e13 −6.06942e14 −1.82297e15 1.06570e17 0
1.3 165109. 3.45913e8 −7.09870e9 0 5.71135e13 1.25160e14 −6.84517e15 6.96245e16 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## 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 25.36.a.a 3
5.b even 2 1 1.36.a.a 3
5.c odd 4 2 25.36.b.a 6
15.d odd 2 1 9.36.a.b 3
20.d odd 2 1 16.36.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.36.a.a 3 5.b even 2 1
9.36.a.b 3 15.d odd 2 1
16.36.a.d 3 20.d odd 2 1
25.36.a.a 3 1.a even 1 1 trivial
25.36.b.a 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 139656 T_{2}^{2} - 59208339456 T_{2} +$$$$14\!\cdots\!64$$ acting on $$S_{36}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1467625047588864 - 59208339456 T + 139656 T^{2} + T^{3}$$
$3$ $$21\!\cdots\!32$$$$- 144593891972573904 T - 104875308 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-$$$$30\!\cdots\!16$$$$+$$$$11\!\cdots\!64$$$$T + 878422149346056 T^{2} + T^{3}$$
$11$ $$-$$$$14\!\cdots\!08$$$$-$$$$17\!\cdots\!88$$$$T + 1157945428549987044 T^{2} + T^{3}$$
$13$ $$-$$$$49\!\cdots\!48$$$$+$$$$63\!\cdots\!56$$$$T - 62139610550998650558 T^{2} + T^{3}$$
$17$ $$68\!\cdots\!24$$$$-$$$$17\!\cdots\!96$$$$T -$$$$39\!\cdots\!94$$$$T^{2} + T^{3}$$
$19$ $$-$$$$49\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T +$$$$32\!\cdots\!40$$$$T^{2} + T^{3}$$
$23$ $$20\!\cdots\!72$$$$-$$$$53\!\cdots\!84$$$$T -$$$$51\!\cdots\!08$$$$T^{2} + T^{3}$$
$29$ $$-$$$$25\!\cdots\!00$$$$-$$$$14\!\cdots\!00$$$$T +$$$$38\!\cdots\!10$$$$T^{2} + T^{3}$$
$31$ $$11\!\cdots\!92$$$$-$$$$10\!\cdots\!88$$$$T -$$$$10\!\cdots\!56$$$$T^{2} + T^{3}$$
$37$ $$-$$$$36\!\cdots\!96$$$$-$$$$17\!\cdots\!16$$$$T +$$$$24\!\cdots\!06$$$$T^{2} + T^{3}$$
$41$ $$11\!\cdots\!92$$$$-$$$$27\!\cdots\!88$$$$T -$$$$23\!\cdots\!06$$$$T^{2} + T^{3}$$
$43$ $$12\!\cdots\!12$$$$+$$$$78\!\cdots\!36$$$$T -$$$$47\!\cdots\!08$$$$T^{2} + T^{3}$$
$47$ $$-$$$$47\!\cdots\!56$$$$-$$$$30\!\cdots\!76$$$$T +$$$$16\!\cdots\!56$$$$T^{2} + T^{3}$$
$53$ $$44\!\cdots\!32$$$$-$$$$38\!\cdots\!04$$$$T -$$$$16\!\cdots\!58$$$$T^{2} + T^{3}$$
$59$ $$10\!\cdots\!00$$$$-$$$$25\!\cdots\!00$$$$T -$$$$43\!\cdots\!80$$$$T^{2} + T^{3}$$
$61$ $$57\!\cdots\!92$$$$+$$$$98\!\cdots\!12$$$$T -$$$$23\!\cdots\!06$$$$T^{2} + T^{3}$$
$67$ $$10\!\cdots\!24$$$$+$$$$34\!\cdots\!04$$$$T -$$$$18\!\cdots\!44$$$$T^{2} + T^{3}$$
$71$ $$33\!\cdots\!92$$$$-$$$$22\!\cdots\!88$$$$T -$$$$34\!\cdots\!56$$$$T^{2} + T^{3}$$
$73$ $$-$$$$22\!\cdots\!28$$$$-$$$$26\!\cdots\!84$$$$T -$$$$28\!\cdots\!58$$$$T^{2} + T^{3}$$
$79$ $$-$$$$88\!\cdots\!00$$$$-$$$$61\!\cdots\!00$$$$T +$$$$42\!\cdots\!60$$$$T^{2} + T^{3}$$
$83$ $$11\!\cdots\!92$$$$+$$$$71\!\cdots\!76$$$$T +$$$$14\!\cdots\!92$$$$T^{2} + T^{3}$$
$89$ $$-$$$$91\!\cdots\!00$$$$+$$$$29\!\cdots\!00$$$$T -$$$$30\!\cdots\!70$$$$T^{2} + T^{3}$$
$97$ $$83\!\cdots\!44$$$$+$$$$19\!\cdots\!24$$$$T -$$$$10\!\cdots\!94$$$$T^{2} + T^{3}$$