# Properties

 Label 6422.2.a.bi Level $6422$ Weight $2$ Character orbit 6422.a Self dual yes Analytic conductor $51.280$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.2799281781$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 2 x^{8} - 12 x^{7} + 14 x^{6} + 54 x^{5} - 11 x^{4} - 84 x^{3} - 48 x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 + \beta_{7} ) q^{3} + q^{4} + \beta_{4} q^{5} + ( 1 - \beta_{7} ) q^{6} + ( 2 - \beta_{1} - \beta_{3} ) q^{7} - q^{8} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 + \beta_{7} ) q^{3} + q^{4} + \beta_{4} q^{5} + ( 1 - \beta_{7} ) q^{6} + ( 2 - \beta_{1} - \beta_{3} ) q^{7} - q^{8} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{9} -\beta_{4} q^{10} + ( 2 - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -1 + \beta_{7} ) q^{12} + ( -2 + \beta_{1} + \beta_{3} ) q^{14} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} ) q^{15} + q^{16} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{17} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{18} - q^{19} + \beta_{4} q^{20} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + \beta_{8} ) q^{21} + ( -2 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{22} + ( -1 - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{23} + ( 1 - \beta_{7} ) q^{24} + ( 3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 5 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{25} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + \beta_{7} + \beta_{8} ) q^{27} + ( 2 - \beta_{1} - \beta_{3} ) q^{28} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{29} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{30} + ( -1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{31} - q^{32} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{33} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{34} + ( -2 - \beta_{1} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{8} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{36} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{37} + q^{38} -\beta_{4} q^{40} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{8} ) q^{41} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - \beta_{8} ) q^{42} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{43} + ( 2 - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + ( 1 - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{45} + ( 1 + 2 \beta_{2} + \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{46} + ( 2 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} ) q^{47} + ( -1 + \beta_{7} ) q^{48} + ( 1 - 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{49} + ( -3 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 5 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{50} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{51} + ( 2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - \beta_{7} - \beta_{8} ) q^{54} + ( 2 + 2 \beta_{1} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{55} + ( -2 + \beta_{1} + \beta_{3} ) q^{56} + ( 1 - \beta_{7} ) q^{57} + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{58} + ( -2 - 3 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{59} + ( -1 + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} ) q^{60} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{61} + ( 1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{62} + ( 7 - 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{63} + q^{64} + ( 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} ) q^{66} + ( 7 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{7} + \beta_{8} ) q^{67} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{68} + ( \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{69} + ( 2 + \beta_{1} - \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{8} ) q^{70} + ( 8 + 4 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{71} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{8} ) q^{72} + ( 4 + 3 \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{73} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{74} + ( -8 - 2 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 9 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} ) q^{75} - q^{76} + ( 4 - 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{77} + ( -3 - \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + \beta_{7} - 3 \beta_{8} ) q^{79} + \beta_{4} q^{80} + ( 2 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{81} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{8} ) q^{82} + ( -3 \beta_{1} + \beta_{2} + 7 \beta_{3} + 2 \beta_{4} + \beta_{7} + 2 \beta_{8} ) q^{83} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + \beta_{8} ) q^{84} + ( 6 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} ) q^{85} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{86} + ( -7 - 4 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} + \beta_{4} + 7 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} + \beta_{8} ) q^{87} + ( -2 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{88} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} + 4 \beta_{7} + \beta_{8} ) q^{89} + ( -1 + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{90} + ( -1 - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{92} + ( 6 - 5 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{93} + ( -2 - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} ) q^{94} -\beta_{4} q^{95} + ( 1 - \beta_{7} ) q^{96} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{97} + ( -1 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{98} + ( -1 + 2 \beta_{1} - 6 \beta_{2} - 9 \beta_{3} - \beta_{5} + 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + O(q^{10})$$ $$9 q - 9 q^{2} - 5 q^{3} + 9 q^{4} - q^{5} + 5 q^{6} + 13 q^{7} - 9 q^{8} + 10 q^{9} + q^{10} + 13 q^{11} - 5 q^{12} - 13 q^{14} + q^{15} + 9 q^{16} - 12 q^{17} - 10 q^{18} - 9 q^{19} - q^{20} - 18 q^{21} - 13 q^{22} - 22 q^{23} + 5 q^{24} + 4 q^{25} - 26 q^{27} + 13 q^{28} + 12 q^{29} - q^{30} - q^{31} - 9 q^{32} + 28 q^{33} + 12 q^{34} - 18 q^{35} + 10 q^{36} + 25 q^{37} + 9 q^{38} + q^{40} - 11 q^{41} + 18 q^{42} - 10 q^{43} + 13 q^{44} + q^{45} + 22 q^{46} + 12 q^{47} - 5 q^{48} + 2 q^{49} - 4 q^{50} + 35 q^{51} + 9 q^{53} + 26 q^{54} + 18 q^{55} - 13 q^{56} + 5 q^{57} - 12 q^{58} - 10 q^{59} + q^{60} + 32 q^{61} + q^{62} + 63 q^{63} + 9 q^{64} - 28 q^{66} + 73 q^{67} - 12 q^{68} + 2 q^{69} + 18 q^{70} + 51 q^{71} - 10 q^{72} + 14 q^{73} - 25 q^{74} - 49 q^{75} - 9 q^{76} + 18 q^{77} - 28 q^{79} - q^{80} + 29 q^{81} + 11 q^{82} + 22 q^{83} - 18 q^{84} + 51 q^{85} + 10 q^{86} - 20 q^{87} - 13 q^{88} - 3 q^{89} - q^{90} - 22 q^{92} + 59 q^{93} - 12 q^{94} + q^{95} + 5 q^{96} - 2 q^{98} - 25 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 2 x^{8} - 12 x^{7} + 14 x^{6} + 54 x^{5} - 11 x^{4} - 84 x^{3} - 48 x^{2} - 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{8} - 3 \nu^{7} - 9 \nu^{6} + 23 \nu^{5} + 31 \nu^{4} - 42 \nu^{3} - 41 \nu^{2} - 7 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{8} - 6 \nu^{7} - 18 \nu^{6} + 46 \nu^{5} + 63 \nu^{4} - 87 \nu^{3} - 87 \nu^{2} - 2 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$2 \nu^{8} - 5 \nu^{7} - 22 \nu^{6} + 42 \nu^{5} + 86 \nu^{4} - 81 \nu^{3} - 118 \nu^{2} - 16 \nu + 3$$ $$\beta_{5}$$ $$=$$ $$3 \nu^{8} - 8 \nu^{7} - 31 \nu^{6} + 64 \nu^{5} + 120 \nu^{4} - 119 \nu^{3} - 171 \nu^{2} - 26 \nu + 5$$ $$\beta_{6}$$ $$=$$ $$-4 \nu^{8} + 11 \nu^{7} + 40 \nu^{6} - 87 \nu^{5} - 151 \nu^{4} + 162 \nu^{3} + 212 \nu^{2} + 27 \nu - 5$$ $$\beta_{7}$$ $$=$$ $$-6 \nu^{8} + 17 \nu^{7} + 58 \nu^{6} - 133 \nu^{5} - 214 \nu^{4} + 249 \nu^{3} + 298 \nu^{2} + 31 \nu - 5$$ $$\beta_{8}$$ $$=$$ $$10 \nu^{8} - 28 \nu^{7} - 97 \nu^{6} + 215 \nu^{5} + 366 \nu^{4} - 388 \nu^{3} - 527 \nu^{2} - 78 \nu + 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{6} - \beta_{3} + 2 \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{2} + 6 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{7} + 8 \beta_{6} + 3 \beta_{5} - 4 \beta_{3} + \beta_{2} + 16 \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{7} + 16 \beta_{6} + 12 \beta_{5} + \beta_{4} + 8 \beta_{2} + 45 \beta_{1} + 18$$ $$\nu^{6}$$ $$=$$ $$\beta_{8} - 26 \beta_{7} + 67 \beta_{6} + 34 \beta_{5} + 5 \beta_{4} - 13 \beta_{3} + 16 \beta_{2} + 125 \beta_{1} + 78$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{8} - 32 \beta_{7} + 173 \beta_{6} + 109 \beta_{5} + 25 \beta_{4} + 8 \beta_{3} + 67 \beta_{2} + 352 \beta_{1} + 143$$ $$\nu^{8}$$ $$=$$ $$21 \beta_{8} - 147 \beta_{7} + 589 \beta_{6} + 306 \beta_{5} + 97 \beta_{4} - 10 \beta_{3} + 173 \beta_{2} + 991 \beta_{1} + 492$$

## 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
 −0.816316 −1.85435 −1.89727 1.88062 2.69813 0.102176 −0.785289 2.96112 −0.288819
−1.00000 −3.11887 1.00000 −0.453021 3.11887 1.01438 −1.00000 6.72735 0.453021
1.2 −1.00000 −3.07508 1.00000 1.48707 3.07508 5.10133 −1.00000 6.45613 −1.48707
1.3 −1.00000 −2.42044 1.00000 −4.26312 2.42044 3.45222 −1.00000 2.85851 4.26312
1.4 −1.00000 −1.60929 1.00000 4.22571 1.60929 −0.325660 −1.00000 −0.410201 −4.22571
1.5 −1.00000 0.162557 1.00000 −2.16373 −0.162557 0.548851 −1.00000 −2.97358 2.16373
1.6 −1.00000 0.519384 1.00000 0.0567031 −0.519384 0.0958867 −1.00000 −2.73024 −0.0567031
1.7 −1.00000 0.782742 1.00000 −1.76453 −0.782742 2.34025 −1.00000 −2.38731 1.76453
1.8 −1.00000 1.04453 1.00000 1.64330 −1.04453 −2.76306 −1.00000 −1.90896 −1.64330
1.9 −1.00000 2.71446 1.00000 0.231615 −2.71446 3.53580 −1.00000 4.36830 −0.231615
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$13$$ $$-1$$
$$19$$ $$1$$

## 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 6422.2.a.bi 9
13.b even 2 1 6422.2.a.bk yes 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.bi 9 1.a even 1 1 trivial
6422.2.a.bk yes 9 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6422))$$:

 $$T_{3}^{9} + \cdots$$ $$T_{5}^{9} + \cdots$$ $$T_{7}^{9} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{9}$$
$3$ $$-7 + 63 T - 126 T^{2} - T^{3} + 136 T^{4} - 11 T^{5} - 53 T^{6} - 6 T^{7} + 5 T^{8} + T^{9}$$
$5$ $$-1 + 20 T - 32 T^{2} - 175 T^{3} + 64 T^{4} + 118 T^{5} - 21 T^{6} - 24 T^{7} + T^{8} + T^{9}$$
$7$ $$-7 + 77 T + 7 T^{2} - 587 T^{3} + 863 T^{4} - 380 T^{5} - 14 T^{6} + 52 T^{7} - 13 T^{8} + T^{9}$$
$11$ $$4459 - 1372 T - 8134 T^{2} + 4501 T^{3} + 2534 T^{4} - 2191 T^{5} + 419 T^{6} + 19 T^{7} - 13 T^{8} + T^{9}$$
$13$ $$T^{9}$$
$17$ $$47783 - 100909 T - 74654 T^{2} + 20615 T^{3} + 15753 T^{4} - 352 T^{5} - 812 T^{6} - 40 T^{7} + 12 T^{8} + T^{9}$$
$19$ $$( 1 + T )^{9}$$
$23$ $$796109 - 604142 T - 66155 T^{2} + 100540 T^{3} + 6707 T^{4} - 6322 T^{5} - 776 T^{6} + 103 T^{7} + 22 T^{8} + T^{9}$$
$29$ $$628069 - 967091 T + 333307 T^{2} + 50536 T^{3} - 36573 T^{4} + 993 T^{5} + 1190 T^{6} - 81 T^{7} - 12 T^{8} + T^{9}$$
$31$ $$34307 + 44681 T - 47992 T^{2} - 63405 T^{3} + 3915 T^{4} + 5724 T^{5} - 147 T^{6} - 144 T^{7} + T^{8} + T^{9}$$
$37$ $$-2899 + 9853 T - 2188 T^{2} - 11477 T^{3} + 6670 T^{4} + 68 T^{5} - 833 T^{6} + 229 T^{7} - 25 T^{8} + T^{9}$$
$41$ $$-3295669 - 4035408 T - 1259151 T^{2} + 132776 T^{3} + 96691 T^{4} + 3780 T^{5} - 1974 T^{6} - 150 T^{7} + 11 T^{8} + T^{9}$$
$43$ $$1362073 - 1060328 T - 694320 T^{2} + 68106 T^{3} + 64847 T^{4} + 2994 T^{5} - 1526 T^{6} - 128 T^{7} + 10 T^{8} + T^{9}$$
$47$ $$419 - 17651 T + 104810 T^{2} - 18039 T^{3} - 38678 T^{4} + 2182 T^{5} + 1477 T^{6} - 110 T^{7} - 12 T^{8} + T^{9}$$
$53$ $$-451543 - 1105334 T + 1328265 T^{2} - 31962 T^{3} - 112563 T^{4} + 10350 T^{5} + 2209 T^{6} - 237 T^{7} - 9 T^{8} + T^{9}$$
$59$ $$-1221571 - 1013470 T - 92748 T^{2} + 121975 T^{3} + 31621 T^{4} - 2176 T^{5} - 1417 T^{6} - 94 T^{7} + 10 T^{8} + T^{9}$$
$61$ $$2927897 - 3577245 T - 2779546 T^{2} + 739955 T^{3} + 161495 T^{4} - 59340 T^{5} + 4441 T^{6} + 165 T^{7} - 32 T^{8} + T^{9}$$
$67$ $$-35377399 + 57842050 T - 38795987 T^{2} + 14224958 T^{3} - 3173980 T^{4} + 450334 T^{5} - 40872 T^{6} + 2299 T^{7} - 73 T^{8} + T^{9}$$
$71$ $$48682591 - 89129235 T + 56604882 T^{2} - 17344947 T^{3} + 2779873 T^{4} - 213431 T^{5} + 2303 T^{6} + 787 T^{7} - 51 T^{8} + T^{9}$$
$73$ $$-219983 + 782432 T + 39438 T^{2} - 283287 T^{3} - 52059 T^{4} + 12460 T^{5} + 2062 T^{6} - 189 T^{7} - 14 T^{8} + T^{9}$$
$79$ $$-11213 - 329793 T + 243375 T^{2} + 580929 T^{3} + 145500 T^{4} - 17315 T^{5} - 4187 T^{6} + 24 T^{7} + 28 T^{8} + T^{9}$$
$83$ $$510299 - 9938640 T - 737842 T^{2} + 1609826 T^{3} - 170069 T^{4} - 37924 T^{5} + 7139 T^{6} - 189 T^{7} - 22 T^{8} + T^{9}$$
$89$ $$-28903 + 179865 T + 656271 T^{2} + 505189 T^{3} + 145528 T^{4} + 11248 T^{5} - 1859 T^{6} - 276 T^{7} + 3 T^{8} + T^{9}$$
$97$ $$-1628677 + 645659 T + 549227 T^{2} - 253887 T^{3} - 13069 T^{4} + 11970 T^{5} + 39 T^{6} - 196 T^{7} + T^{9}$$