# Properties

 Label 9.48.a.c Level $9$ Weight $48$ Character orbit 9.a Self dual yes Analytic conductor $125.917$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$48$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$125.916896390$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{20}\cdot 3^{11}\cdot 5^{3}\cdot 7^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1446390 - \beta_{1} ) q^{2} + ( 101201874790288 + 2732420 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{4} + ( 7778670060568050 - 214856224 \beta_{1} - 1796 \beta_{2} + 96 \beta_{3} ) q^{5} + ( -9792304681472105800 + 3698202764048 \beta_{1} - 4878534 \beta_{2} - 133760 \beta_{3} ) q^{7} + ( -598179247018446084480 - 185574830780256 \beta_{1} - 105238288 \beta_{2} - 2897880 \beta_{3} ) q^{8} +O(q^{10})$$ $$q +(-1446390 - \beta_{1}) q^{2} +(101201874790288 + 2732420 \beta_{1} + 6 \beta_{2} + \beta_{3}) q^{4} +(7778670060568050 - 214856224 \beta_{1} - 1796 \beta_{2} + 96 \beta_{3}) q^{5} +(-9792304681472105800 + 3698202764048 \beta_{1} - 4878534 \beta_{2} - 133760 \beta_{3}) q^{7} +(-$$$$59\!\cdots\!80$$$$- 185574830780256 \beta_{1} - 105238288 \beta_{2} - 2897880 \beta_{3}) q^{8} +($$$$40\!\cdots\!00$$$$- 20577222944857586 \beta_{1} + 2263126656 \beta_{2} + 641207744 \beta_{3}) q^{10} +($$$$47\!\cdots\!28$$$$+ 80173046708155720 \beta_{1} - 65412162475 \beta_{2} + 6954435840 \beta_{3}) q^{11} +($$$$31\!\cdots\!30$$$$+ 2187545461397076512 \beta_{1} + 4897389979140 \beta_{2} - 467536231520 \beta_{3}) q^{13} +(-$$$$87\!\cdots\!76$$$$+ 48764929041038400840 \beta_{1} + 6024181979072 \beta_{2} - 2916146241888 \beta_{3}) q^{14} +($$$$31\!\cdots\!96$$$$+$$$$13\!\cdots\!20$$$$\beta_{1} + 878441883331968 \beta_{2} + 61695767581248 \beta_{3}) q^{16} +(-$$$$52\!\cdots\!70$$$$+$$$$16\!\cdots\!88$$$$\beta_{1} + 2513535368752552 \beta_{2} + 187599812159040 \beta_{3}) q^{17} +(-$$$$26\!\cdots\!60$$$$+$$$$32\!\cdots\!20$$$$\beta_{1} + 73337396648714469 \beta_{2} + 676931170946304 \beta_{3}) q^{19} +($$$$37\!\cdots\!00$$$$-$$$$12\!\cdots\!32$$$$\beta_{1} + 325607045576761772 \beta_{2} + 7255673126427378 \beta_{3}) q^{20} +(-$$$$19\!\cdots\!20$$$$-$$$$17\!\cdots\!68$$$$\beta_{1} - 669664337454884640 \beta_{2} - 61348773982345520 \beta_{3}) q^{22} +(-$$$$34\!\cdots\!20$$$$-$$$$10\!\cdots\!40$$$$\beta_{1} + 824862716416275506 \beta_{2} + 4835096198025600 \beta_{3}) q^{23} +($$$$25\!\cdots\!75$$$$-$$$$21\!\cdots\!00$$$$\beta_{1} + 9827455518879730800 \beta_{2} - 2144521474142940800 \beta_{3}) q^{25} +(-$$$$57\!\cdots\!72$$$$+$$$$42\!\cdots\!30$$$$\beta_{1} - 2446702235232694912 \beta_{2} - 3546261034268171712 \beta_{3}) q^{26} +(-$$$$90\!\cdots\!20$$$$+$$$$85\!\cdots\!80$$$$\beta_{1} +$$$$55\!\cdots\!96$$$$\beta_{2} - 33842643361650715080 \beta_{3}) q^{28} +($$$$56\!\cdots\!90$$$$+$$$$72\!\cdots\!00$$$$\beta_{1} +$$$$37\!\cdots\!64$$$$\beta_{2} - 85281536978719677216 \beta_{3}) q^{29} +($$$$18\!\cdots\!12$$$$-$$$$72\!\cdots\!60$$$$\beta_{1} +$$$$36\!\cdots\!00$$$$\beta_{2} -$$$$31\!\cdots\!20$$$$\beta_{3}) q^{31} +(-$$$$29\!\cdots\!40$$$$-$$$$22\!\cdots\!36$$$$\beta_{1} -$$$$10\!\cdots\!20$$$$\beta_{2} -$$$$10\!\cdots\!60$$$$\beta_{3}) q^{32} +(-$$$$31\!\cdots\!04$$$$+$$$$41\!\cdots\!30$$$$\beta_{1} -$$$$32\!\cdots\!88$$$$\beta_{2} -$$$$19\!\cdots\!68$$$$\beta_{3}) q^{34} +($$$$32\!\cdots\!00$$$$+$$$$99\!\cdots\!56$$$$\beta_{1} -$$$$91\!\cdots\!76$$$$\beta_{2} -$$$$89\!\cdots\!24$$$$\beta_{3}) q^{35} +(-$$$$28\!\cdots\!90$$$$+$$$$21\!\cdots\!80$$$$\beta_{1} -$$$$50\!\cdots\!28$$$$\beta_{2} -$$$$78\!\cdots\!60$$$$\beta_{3}) q^{37} +(-$$$$74\!\cdots\!20$$$$-$$$$16\!\cdots\!24$$$$\beta_{1} -$$$$53\!\cdots\!92$$$$\beta_{2} -$$$$45\!\cdots\!60$$$$\beta_{3}) q^{38} +($$$$17\!\cdots\!00$$$$-$$$$33\!\cdots\!40$$$$\beta_{1} -$$$$13\!\cdots\!60$$$$\beta_{2} -$$$$23\!\cdots\!40$$$$\beta_{3}) q^{40} +(-$$$$32\!\cdots\!82$$$$-$$$$37\!\cdots\!40$$$$\beta_{1} +$$$$21\!\cdots\!00$$$$\beta_{2} -$$$$77\!\cdots\!80$$$$\beta_{3}) q^{41} +(-$$$$11\!\cdots\!00$$$$+$$$$10\!\cdots\!96$$$$\beta_{1} +$$$$73\!\cdots\!87$$$$\beta_{2} -$$$$19\!\cdots\!00$$$$\beta_{3}) q^{43} +($$$$38\!\cdots\!64$$$$+$$$$25\!\cdots\!20$$$$\beta_{1} +$$$$26\!\cdots\!68$$$$\beta_{2} +$$$$83\!\cdots\!48$$$$\beta_{3}) q^{44} +($$$$30\!\cdots\!92$$$$+$$$$31\!\cdots\!40$$$$\beta_{1} +$$$$28\!\cdots\!20$$$$\beta_{2} +$$$$92\!\cdots\!00$$$$\beta_{3}) q^{46} +(-$$$$50\!\cdots\!80$$$$-$$$$55\!\cdots\!84$$$$\beta_{1} +$$$$72\!\cdots\!36$$$$\beta_{2} +$$$$73\!\cdots\!00$$$$\beta_{3}) q^{47} +($$$$22\!\cdots\!93$$$$-$$$$32\!\cdots\!60$$$$\beta_{1} -$$$$35\!\cdots\!40$$$$\beta_{2} +$$$$89\!\cdots\!40$$$$\beta_{3}) q^{49} +($$$$48\!\cdots\!50$$$$+$$$$17\!\cdots\!25$$$$\beta_{1} +$$$$22\!\cdots\!00$$$$\beta_{2} +$$$$17\!\cdots\!00$$$$\beta_{3}) q^{50} +(-$$$$13\!\cdots\!00$$$$+$$$$92\!\cdots\!88$$$$\beta_{1} -$$$$70\!\cdots\!84$$$$\beta_{2} +$$$$20\!\cdots\!50$$$$\beta_{3}) q^{52} +(-$$$$72\!\cdots\!90$$$$-$$$$11\!\cdots\!24$$$$\beta_{1} +$$$$33\!\cdots\!24$$$$\beta_{2} -$$$$80\!\cdots\!80$$$$\beta_{3}) q^{53} +($$$$48\!\cdots\!00$$$$+$$$$46\!\cdots\!28$$$$\beta_{1} +$$$$15\!\cdots\!62$$$$\beta_{2} -$$$$15\!\cdots\!12$$$$\beta_{3}) q^{55} +(-$$$$69\!\cdots\!60$$$$+$$$$59\!\cdots\!00$$$$\beta_{1} -$$$$60\!\cdots\!76$$$$\beta_{2} -$$$$58\!\cdots\!56$$$$\beta_{3}) q^{56} +(-$$$$18\!\cdots\!20$$$$+$$$$91\!\cdots\!46$$$$\beta_{1} -$$$$36\!\cdots\!12$$$$\beta_{2} -$$$$88\!\cdots\!40$$$$\beta_{3}) q^{58} +(-$$$$11\!\cdots\!20$$$$-$$$$91\!\cdots\!40$$$$\beta_{1} -$$$$23\!\cdots\!67$$$$\beta_{2} -$$$$27\!\cdots\!32$$$$\beta_{3}) q^{59} +($$$$15\!\cdots\!22$$$$-$$$$41\!\cdots\!00$$$$\beta_{1} +$$$$35\!\cdots\!00$$$$\beta_{2} +$$$$82\!\cdots\!00$$$$\beta_{3}) q^{61} +($$$$17\!\cdots\!20$$$$+$$$$41\!\cdots\!08$$$$\beta_{1} +$$$$49\!\cdots\!20$$$$\beta_{2} +$$$$62\!\cdots\!60$$$$\beta_{3}) q^{62} +($$$$13\!\cdots\!68$$$$+$$$$35\!\cdots\!20$$$$\beta_{1} +$$$$85\!\cdots\!92$$$$\beta_{2} +$$$$12\!\cdots\!92$$$$\beta_{3}) q^{64} +(-$$$$32\!\cdots\!00$$$$+$$$$11\!\cdots\!12$$$$\beta_{1} -$$$$21\!\cdots\!52$$$$\beta_{2} +$$$$11\!\cdots\!52$$$$\beta_{3}) q^{65} +($$$$46\!\cdots\!20$$$$+$$$$71\!\cdots\!32$$$$\beta_{1} -$$$$38\!\cdots\!67$$$$\beta_{2} +$$$$34\!\cdots\!60$$$$\beta_{3}) q^{67} +($$$$68\!\cdots\!60$$$$+$$$$57\!\cdots\!68$$$$\beta_{1} -$$$$12\!\cdots\!12$$$$\beta_{2} -$$$$26\!\cdots\!70$$$$\beta_{3}) q^{68} +(-$$$$24\!\cdots\!00$$$$+$$$$16\!\cdots\!84$$$$\beta_{1} +$$$$35\!\cdots\!36$$$$\beta_{2} -$$$$90\!\cdots\!36$$$$\beta_{3}) q^{70} +(-$$$$55\!\cdots\!92$$$$-$$$$41\!\cdots\!00$$$$\beta_{1} +$$$$14\!\cdots\!50$$$$\beta_{2} -$$$$65\!\cdots\!00$$$$\beta_{3}) q^{71} +($$$$26\!\cdots\!70$$$$+$$$$30\!\cdots\!00$$$$\beta_{1} +$$$$16\!\cdots\!84$$$$\beta_{2} -$$$$55\!\cdots\!40$$$$\beta_{3}) q^{73} +(-$$$$46\!\cdots\!36$$$$+$$$$58\!\cdots\!90$$$$\beta_{1} +$$$$12\!\cdots\!60$$$$\beta_{2} -$$$$12\!\cdots\!60$$$$\beta_{3}) q^{74} +($$$$87\!\cdots\!20$$$$+$$$$14\!\cdots\!00$$$$\beta_{1} -$$$$42\!\cdots\!68$$$$\beta_{2} +$$$$12\!\cdots\!92$$$$\beta_{3}) q^{76} +($$$$65\!\cdots\!00$$$$+$$$$17\!\cdots\!84$$$$\beta_{1} -$$$$18\!\cdots\!12$$$$\beta_{2} -$$$$21\!\cdots\!60$$$$\beta_{3}) q^{77} +(-$$$$33\!\cdots\!40$$$$-$$$$91\!\cdots\!40$$$$\beta_{1} -$$$$14\!\cdots\!04$$$$\beta_{2} +$$$$34\!\cdots\!96$$$$\beta_{3}) q^{79} +($$$$23\!\cdots\!00$$$$+$$$$13\!\cdots\!36$$$$\beta_{1} -$$$$18\!\cdots\!56$$$$\beta_{2} +$$$$25\!\cdots\!56$$$$\beta_{3}) q^{80} +($$$$95\!\cdots\!80$$$$+$$$$45\!\cdots\!62$$$$\beta_{1} +$$$$19\!\cdots\!80$$$$\beta_{2} +$$$$33\!\cdots\!40$$$$\beta_{3}) q^{82} +($$$$35\!\cdots\!40$$$$-$$$$22\!\cdots\!28$$$$\beta_{1} +$$$$80\!\cdots\!67$$$$\beta_{2} +$$$$34\!\cdots\!00$$$$\beta_{3}) q^{83} +(-$$$$25\!\cdots\!00$$$$+$$$$16\!\cdots\!04$$$$\beta_{1} +$$$$20\!\cdots\!16$$$$\beta_{2} -$$$$46\!\cdots\!16$$$$\beta_{3}) q^{85} +(-$$$$23\!\cdots\!32$$$$+$$$$76\!\cdots\!60$$$$\beta_{1} -$$$$91\!\cdots\!16$$$$\beta_{2} -$$$$12\!\cdots\!76$$$$\beta_{3}) q^{86} +(-$$$$38\!\cdots\!40$$$$-$$$$42\!\cdots\!68$$$$\beta_{1} -$$$$21\!\cdots\!64$$$$\beta_{2} -$$$$21\!\cdots\!40$$$$\beta_{3}) q^{88} +($$$$19\!\cdots\!70$$$$-$$$$45\!\cdots\!00$$$$\beta_{1} -$$$$23\!\cdots\!68$$$$\beta_{2} -$$$$79\!\cdots\!08$$$$\beta_{3}) q^{89} +($$$$13\!\cdots\!92$$$$-$$$$29\!\cdots\!00$$$$\beta_{1} -$$$$14\!\cdots\!64$$$$\beta_{2} +$$$$30\!\cdots\!16$$$$\beta_{3}) q^{91} +(-$$$$32\!\cdots\!20$$$$-$$$$39\!\cdots\!72$$$$\beta_{1} -$$$$37\!\cdots\!08$$$$\beta_{2} -$$$$32\!\cdots\!40$$$$\beta_{3}) q^{92} +($$$$86\!\cdots\!56$$$$-$$$$12\!\cdots\!80$$$$\beta_{1} -$$$$76\!\cdots\!16$$$$\beta_{2} -$$$$58\!\cdots\!56$$$$\beta_{3}) q^{94} +(-$$$$18\!\cdots\!00$$$$+$$$$64\!\cdots\!60$$$$\beta_{1} +$$$$15\!\cdots\!90$$$$\beta_{2} +$$$$37\!\cdots\!60$$$$\beta_{3}) q^{95} +($$$$23\!\cdots\!30$$$$-$$$$49\!\cdots\!56$$$$\beta_{1} +$$$$16\!\cdots\!64$$$$\beta_{2} +$$$$75\!\cdots\!20$$$$\beta_{3}) q^{97} +($$$$74\!\cdots\!30$$$$-$$$$24\!\cdots\!33$$$$\beta_{1} +$$$$27\!\cdots\!40$$$$\beta_{2} +$$$$39\!\cdots\!60$$$$\beta_{3}) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} - 2392716988073784337920q^{8} + O(q^{10})$$ $$4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} -$$$$23\!\cdots\!20$$$$q^{8} +$$$$16\!\cdots\!00$$$$q^{10} +$$$$19\!\cdots\!12$$$$q^{11} +$$$$12\!\cdots\!20$$$$q^{13} -$$$$34\!\cdots\!04$$$$q^{14} +$$$$12\!\cdots\!84$$$$q^{16} -$$$$21\!\cdots\!80$$$$q^{17} -$$$$10\!\cdots\!40$$$$q^{19} +$$$$15\!\cdots\!00$$$$q^{20} -$$$$79\!\cdots\!80$$$$q^{22} -$$$$13\!\cdots\!80$$$$q^{23} +$$$$10\!\cdots\!00$$$$q^{25} -$$$$22\!\cdots\!88$$$$q^{26} -$$$$36\!\cdots\!80$$$$q^{28} +$$$$22\!\cdots\!60$$$$q^{29} +$$$$75\!\cdots\!48$$$$q^{31} -$$$$11\!\cdots\!60$$$$q^{32} -$$$$12\!\cdots\!16$$$$q^{34} +$$$$13\!\cdots\!00$$$$q^{35} -$$$$11\!\cdots\!60$$$$q^{37} -$$$$29\!\cdots\!80$$$$q^{38} +$$$$70\!\cdots\!00$$$$q^{40} -$$$$13\!\cdots\!28$$$$q^{41} -$$$$44\!\cdots\!00$$$$q^{43} +$$$$15\!\cdots\!56$$$$q^{44} +$$$$12\!\cdots\!68$$$$q^{46} -$$$$20\!\cdots\!20$$$$q^{47} +$$$$91\!\cdots\!72$$$$q^{49} +$$$$19\!\cdots\!00$$$$q^{50} -$$$$55\!\cdots\!00$$$$q^{52} -$$$$29\!\cdots\!60$$$$q^{53} +$$$$19\!\cdots\!00$$$$q^{55} -$$$$27\!\cdots\!40$$$$q^{56} -$$$$73\!\cdots\!80$$$$q^{58} -$$$$47\!\cdots\!80$$$$q^{59} +$$$$62\!\cdots\!88$$$$q^{61} +$$$$68\!\cdots\!80$$$$q^{62} +$$$$55\!\cdots\!72$$$$q^{64} -$$$$12\!\cdots\!00$$$$q^{65} +$$$$18\!\cdots\!80$$$$q^{67} +$$$$27\!\cdots\!40$$$$q^{68} -$$$$97\!\cdots\!00$$$$q^{70} -$$$$22\!\cdots\!68$$$$q^{71} +$$$$10\!\cdots\!80$$$$q^{73} -$$$$18\!\cdots\!44$$$$q^{74} +$$$$35\!\cdots\!80$$$$q^{76} +$$$$26\!\cdots\!00$$$$q^{77} -$$$$13\!\cdots\!60$$$$q^{79} +$$$$94\!\cdots\!00$$$$q^{80} +$$$$38\!\cdots\!20$$$$q^{82} +$$$$14\!\cdots\!60$$$$q^{83} -$$$$10\!\cdots\!00$$$$q^{85} -$$$$95\!\cdots\!28$$$$q^{86} -$$$$15\!\cdots\!60$$$$q^{88} +$$$$79\!\cdots\!80$$$$q^{89} +$$$$53\!\cdots\!68$$$$q^{91} -$$$$13\!\cdots\!80$$$$q^{92} +$$$$34\!\cdots\!24$$$$q^{94} -$$$$75\!\cdots\!00$$$$q^{95} +$$$$95\!\cdots\!20$$$$q^{97} +$$$$29\!\cdots\!20$$$$q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$24 \nu - 6$$ $$\beta_{2}$$ $$=$$ $$($$$$27 \nu^{3} + 1621431 \nu^{2} - 21620549846568 \nu - 600047676703864212$$$$)/171584$$ $$\beta_{3}$$ $$=$$ $$($$$$-81 \nu^{3} + 44551899 \nu^{2} + 65191807354488 \nu - 18776838253817714244$$$$)/85792$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 6$$$$)/24$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 6 \beta_{2} - 160348 \beta_{1} + 239847319113552$$$$)/576$$ $$\nu^{3}$$ $$=$$ $$($$$$-180159 \beta_{3} + 9900422 \beta_{2} + 57683687726180 \beta_{1} - 4807255926333560112$$$$)/1728$$

## 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
 901372. 129356. −124721. −906006.
−2.30793e7 0 3.91917e14 1.15086e16 0 1.54211e19 −5.79706e21 0 −2.65611e23
1.2 −4.55092e6 0 −1.20027e14 3.08341e16 0 1.11073e20 1.18672e21 0 −1.40324e23
1.3 1.54692e6 0 −1.38345e14 −4.23962e16 0 −3.90714e19 −4.31719e20 0 −6.55837e22
1.4 2.02978e7 0 2.71261e14 3.11682e16 0 −1.26592e20 2.64934e21 0 6.32644e23
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## 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 9.48.a.c 4
3.b odd 2 1 1.48.a.a 4
12.b even 2 1 16.48.a.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.48.a.a 4 3.b odd 2 1
9.48.a.c 4 1.a even 1 1 trivial
16.48.a.d 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5785560 T_{2}^{3} -$$$$46\!\cdots\!32$$$$T_{2}^{2} -$$$$14\!\cdots\!60$$$$T_{2} +$$$$32\!\cdots\!56$$ acting on $$S_{48}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$32\!\cdots\!56$$$$-$$$$14\!\cdots\!60$$$$T - 467142374034432 T^{2} + 5785560 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$-$$$$46\!\cdots\!00$$$$+$$$$59\!\cdots\!00$$$$T -$$$$14\!\cdots\!00$$$$T^{2} - 31114680242272200 T^{3} + T^{4}$$
$7$ $$84\!\cdots\!96$$$$-$$$$34\!\cdots\!00$$$$T -$$$$14\!\cdots\!72$$$$T^{2} + 39169218725888423200 T^{3} + T^{4}$$
$11$ $$-$$$$18\!\cdots\!44$$$$+$$$$11\!\cdots\!92$$$$T -$$$$75\!\cdots\!96$$$$T^{2} -$$$$19\!\cdots\!12$$$$T^{3} + T^{4}$$
$13$ $$10\!\cdots\!76$$$$+$$$$33\!\cdots\!80$$$$T -$$$$23\!\cdots\!48$$$$T^{2} -$$$$12\!\cdots\!20$$$$T^{3} + T^{4}$$
$17$ $$-$$$$14\!\cdots\!24$$$$-$$$$21\!\cdots\!80$$$$T +$$$$10\!\cdots\!48$$$$T^{2} +$$$$21\!\cdots\!80$$$$T^{3} + T^{4}$$
$19$ $$37\!\cdots\!00$$$$-$$$$81\!\cdots\!00$$$$T -$$$$16\!\cdots\!00$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{3} + T^{4}$$
$23$ $$56\!\cdots\!16$$$$+$$$$10\!\cdots\!80$$$$T +$$$$63\!\cdots\!92$$$$T^{2} +$$$$13\!\cdots\!80$$$$T^{3} + T^{4}$$
$29$ $$-$$$$69\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$T -$$$$80\!\cdots\!00$$$$T^{2} -$$$$22\!\cdots\!60$$$$T^{3} + T^{4}$$
$31$ $$13\!\cdots\!36$$$$+$$$$24\!\cdots\!88$$$$T -$$$$36\!\cdots\!36$$$$T^{2} -$$$$75\!\cdots\!48$$$$T^{3} + T^{4}$$
$37$ $$-$$$$15\!\cdots\!64$$$$-$$$$78\!\cdots\!60$$$$T -$$$$49\!\cdots\!12$$$$T^{2} +$$$$11\!\cdots\!60$$$$T^{3} + T^{4}$$
$41$ $$89\!\cdots\!76$$$$-$$$$37\!\cdots\!28$$$$T -$$$$23\!\cdots\!56$$$$T^{2} +$$$$13\!\cdots\!28$$$$T^{3} + T^{4}$$
$43$ $$-$$$$57\!\cdots\!04$$$$-$$$$12\!\cdots\!00$$$$T +$$$$50\!\cdots\!72$$$$T^{2} +$$$$44\!\cdots\!00$$$$T^{3} + T^{4}$$
$47$ $$84\!\cdots\!16$$$$-$$$$55\!\cdots\!20$$$$T -$$$$50\!\cdots\!92$$$$T^{2} +$$$$20\!\cdots\!20$$$$T^{3} + T^{4}$$
$53$ $$10\!\cdots\!36$$$$-$$$$86\!\cdots\!40$$$$T -$$$$10\!\cdots\!88$$$$T^{2} +$$$$29\!\cdots\!60$$$$T^{3} + T^{4}$$
$59$ $$-$$$$14\!\cdots\!00$$$$+$$$$36\!\cdots\!00$$$$T +$$$$36\!\cdots\!00$$$$T^{2} +$$$$47\!\cdots\!80$$$$T^{3} + T^{4}$$
$61$ $$-$$$$47\!\cdots\!44$$$$+$$$$26\!\cdots\!08$$$$T -$$$$28\!\cdots\!96$$$$T^{2} -$$$$62\!\cdots\!88$$$$T^{3} + T^{4}$$
$67$ $$-$$$$23\!\cdots\!24$$$$-$$$$46\!\cdots\!20$$$$T +$$$$86\!\cdots\!48$$$$T^{2} -$$$$18\!\cdots\!80$$$$T^{3} + T^{4}$$
$71$ $$-$$$$15\!\cdots\!04$$$$-$$$$25\!\cdots\!48$$$$T -$$$$86\!\cdots\!16$$$$T^{2} +$$$$22\!\cdots\!68$$$$T^{3} + T^{4}$$
$73$ $$-$$$$60\!\cdots\!84$$$$+$$$$28\!\cdots\!20$$$$T -$$$$66\!\cdots\!08$$$$T^{2} -$$$$10\!\cdots\!80$$$$T^{3} + T^{4}$$
$79$ $$29\!\cdots\!00$$$$+$$$$74\!\cdots\!00$$$$T +$$$$55\!\cdots\!00$$$$T^{2} +$$$$13\!\cdots\!60$$$$T^{3} + T^{4}$$
$83$ $$-$$$$10\!\cdots\!44$$$$+$$$$38\!\cdots\!40$$$$T -$$$$23\!\cdots\!68$$$$T^{2} -$$$$14\!\cdots\!60$$$$T^{3} + T^{4}$$
$89$ $$-$$$$18\!\cdots\!00$$$$+$$$$94\!\cdots\!00$$$$T -$$$$94\!\cdots\!00$$$$T^{2} -$$$$79\!\cdots\!80$$$$T^{3} + T^{4}$$
$97$ $$-$$$$75\!\cdots\!84$$$$+$$$$26\!\cdots\!20$$$$T +$$$$19\!\cdots\!08$$$$T^{2} -$$$$95\!\cdots\!20$$$$T^{3} + T^{4}$$