Properties

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

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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: \(1\)
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:

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

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

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