Properties

Label 381.2.a.e
Level $381$
Weight $2$
Character orbit 381.a
Self dual yes
Analytic conductor $3.042$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.04230031701\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 2 x^{8} - 14 x^{7} + 26 x^{6} + 59 x^{5} - 99 x^{4} - 66 x^{3} + 102 x^{2} - 24 x + 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{5} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{8} ) q^{7} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{5} q^{5} -\beta_{1} q^{6} + ( 1 - \beta_{8} ) q^{7} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{10} + ( 1 + \beta_{1} - \beta_{6} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{14} -\beta_{5} q^{15} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{16} + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} -\beta_{1} q^{18} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{19} + ( -2 + 2 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{20} + ( 1 - \beta_{8} ) q^{21} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{22} -2 \beta_{7} q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{24} + ( 2 + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{25} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{26} + q^{27} + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{28} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{29} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{30} + ( -\beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{31} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{32} + ( 1 + \beta_{1} - \beta_{6} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{34} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{7} ) q^{37} + ( 1 + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{38} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{39} + ( -4 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} ) q^{40} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{42} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{43} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{44} -\beta_{5} q^{45} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{46} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{47} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{48} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{49} + ( -1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{50} + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{51} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{52} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{53} -\beta_{1} q^{54} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{55} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + \beta_{7} + 3 \beta_{8} ) q^{56} + ( -\beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{57} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{7} - \beta_{8} ) q^{58} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{60} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{62} + ( 1 - \beta_{8} ) q^{63} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{64} + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{65} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{66} + ( 2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{67} + ( -8 - \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{68} -2 \beta_{7} q^{69} + ( -9 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{70} + ( 5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{71} + ( -2 \beta_{1} - \beta_{3} ) q^{72} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{73} + ( 5 - 6 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{74} + ( 2 + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{75} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{76} + ( 2 \beta_{2} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{77} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{78} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{79} + ( -6 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{80} + q^{81} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{82} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{83} + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{84} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{85} + ( 6 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{86} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{87} + ( -6 - 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{88} + ( -4 + 2 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{90} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{91} + ( -6 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} ) q^{92} + ( -\beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{93} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} ) q^{94} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} ) q^{96} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{97} + ( -10 - \beta_{1} - 4 \beta_{2} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{98} + ( 1 + \beta_{1} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 2q^{2} + 9q^{3} + 14q^{4} - 4q^{5} - 2q^{6} + 10q^{7} - 6q^{8} + 9q^{9} + O(q^{10}) \) \( 9q - 2q^{2} + 9q^{3} + 14q^{4} - 4q^{5} - 2q^{6} + 10q^{7} - 6q^{8} + 9q^{9} - 4q^{10} + 8q^{11} + 14q^{12} + 14q^{13} + 4q^{14} - 4q^{15} + 32q^{16} - 6q^{17} - 2q^{18} + 12q^{19} - 28q^{20} + 10q^{21} - 18q^{22} - 4q^{23} - 6q^{24} + 21q^{25} - 14q^{26} + 9q^{27} - 8q^{29} - 4q^{30} + 4q^{31} - 29q^{32} + 8q^{33} - 3q^{34} + 6q^{35} + 14q^{36} + 22q^{37} - 7q^{38} + 14q^{39} - 2q^{41} + 4q^{42} + 6q^{43} + 17q^{44} - 4q^{45} - 10q^{46} - 2q^{47} + 32q^{48} + 23q^{49} - 20q^{50} - 6q^{51} - 9q^{52} - 12q^{53} - 2q^{54} - 22q^{55} + 18q^{56} + 12q^{57} - 28q^{58} - 6q^{59} - 28q^{60} + 2q^{61} - 15q^{62} + 10q^{63} + 24q^{64} + 4q^{65} - 18q^{66} + 18q^{67} - 24q^{68} - 4q^{69} - 72q^{70} + 24q^{71} - 6q^{72} + 14q^{73} + 3q^{74} + 21q^{75} + 4q^{76} - 18q^{77} - 14q^{78} + 12q^{79} - 86q^{80} + 9q^{81} + 4q^{82} - 20q^{83} - 24q^{85} + 16q^{86} - 8q^{87} - 55q^{88} - 30q^{89} - 4q^{90} + 14q^{91} - 46q^{92} + 4q^{93} - 66q^{94} - 32q^{95} - 29q^{96} + 12q^{97} - 62q^{98} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 2 x^{8} - 14 x^{7} + 26 x^{6} + 59 x^{5} - 99 x^{4} - 66 x^{3} + 102 x^{2} - 24 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{8} + \nu^{7} + 15 \nu^{6} - 11 \nu^{5} - 70 \nu^{4} + 29 \nu^{3} + 99 \nu^{2} - 3 \nu - 7 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{8} + 3 \nu^{7} + 14 \nu^{6} - 40 \nu^{5} - 58 \nu^{4} + 157 \nu^{3} + 59 \nu^{2} - 170 \nu + 32 \)\()/6\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{8} + 9 \nu^{7} + 73 \nu^{6} - 113 \nu^{5} - 338 \nu^{4} + 401 \nu^{3} + 499 \nu^{2} - 337 \nu - 5 \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{8} - 2 \nu^{7} - 14 \nu^{6} + 25 \nu^{5} + 60 \nu^{4} - 89 \nu^{3} - 74 \nu^{2} + 80 \nu - 11 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{8} - 5 \nu^{7} - 43 \nu^{6} + 65 \nu^{5} + 190 \nu^{4} - 247 \nu^{3} - 247 \nu^{2} + 251 \nu - 27 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(\beta_{8} - \beta_{7} + \beta_{4} + 10 \beta_{3} + 38 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(\beta_{8} + 3 \beta_{7} - 8 \beta_{6} + 14 \beta_{5} + 13 \beta_{4} + 2 \beta_{3} + 46 \beta_{2} + 11 \beta_{1} + 152\)
\(\nu^{7}\)\(=\)\(15 \beta_{8} - 13 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 13 \beta_{4} + 82 \beta_{3} + \beta_{2} + 250 \beta_{1} + 36\)
\(\nu^{8}\)\(=\)\(19 \beta_{8} + 43 \beta_{7} - 48 \beta_{6} + 144 \beta_{5} + 123 \beta_{4} + 31 \beta_{3} + 300 \beta_{2} + 98 \beta_{1} + 992\)

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
2.75841
2.55353
1.92789
0.682747
0.247918
0.0532857
−1.44437
−2.14900
−2.63041
−2.75841 1.00000 5.60882 −3.68284 −2.75841 −3.98729 −9.95461 1.00000 10.1588
1.2 −2.55353 1.00000 4.52051 2.44899 −2.55353 3.89705 −6.43620 1.00000 −6.25358
1.3 −1.92789 1.00000 1.71676 −1.15841 −1.92789 2.35752 0.546061 1.00000 2.23329
1.4 −0.682747 1.00000 −1.53386 3.92611 −0.682747 1.75230 2.41273 1.00000 −2.68054
1.5 −0.247918 1.00000 −1.93854 0.730098 −0.247918 −3.26267 0.976436 1.00000 −0.181005
1.6 −0.0532857 1.00000 −1.99716 −3.85537 −0.0532857 4.59060 0.212992 1.00000 0.205436
1.7 1.44437 1.00000 0.0862103 0.766977 1.44437 3.56571 −2.76422 1.00000 1.10780
1.8 2.14900 1.00000 2.61819 0.492551 2.14900 −0.251160 1.32848 1.00000 1.05849
1.9 2.63041 1.00000 4.91907 −3.66812 2.63041 1.33794 7.67834 1.00000 −9.64866
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(127\) \(1\)

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 381.2.a.e 9
3.b odd 2 1 1143.2.a.j 9
4.b odd 2 1 6096.2.a.bk 9
5.b even 2 1 9525.2.a.p 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.e 9 1.a even 1 1 trivial
1143.2.a.j 9 3.b odd 2 1
6096.2.a.bk 9 4.b odd 2 1
9525.2.a.p 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(381))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 24 T - 102 T^{2} - 66 T^{3} + 99 T^{4} + 59 T^{5} - 26 T^{6} - 14 T^{7} + 2 T^{8} + T^{9} \)
$3$ \( ( -1 + T )^{9} \)
$5$ \( -160 + 592 T - 384 T^{2} - 612 T^{3} + 524 T^{4} + 185 T^{5} - 94 T^{6} - 25 T^{7} + 4 T^{8} + T^{9} \)
$7$ \( 1152 + 2352 T - 7544 T^{2} + 5304 T^{3} - 532 T^{4} - 707 T^{5} + 222 T^{6} + 7 T^{7} - 10 T^{8} + T^{9} \)
$11$ \( -5644 - 3671 T + 7234 T^{2} + 1474 T^{3} - 2258 T^{4} - 29 T^{5} + 238 T^{6} - 18 T^{7} - 8 T^{8} + T^{9} \)
$13$ \( 418 - 4575 T - 5728 T^{2} + 10198 T^{3} - 894 T^{4} - 1521 T^{5} + 344 T^{6} + 30 T^{7} - 14 T^{8} + T^{9} \)
$17$ \( 4768 + 4912 T - 4960 T^{2} - 3248 T^{3} + 1630 T^{4} + 601 T^{5} - 182 T^{6} - 43 T^{7} + 6 T^{8} + T^{9} \)
$19$ \( 2816 - 6464 T - 3072 T^{2} + 7416 T^{3} - 572 T^{4} - 1499 T^{5} + 420 T^{6} + 5 T^{7} - 12 T^{8} + T^{9} \)
$23$ \( -169984 + 365824 T - 125824 T^{2} - 99520 T^{3} + 15712 T^{4} + 6784 T^{5} - 472 T^{6} - 148 T^{7} + 4 T^{8} + T^{9} \)
$29$ \( 288 - 10032 T - 43448 T^{2} - 28620 T^{3} + 7370 T^{4} + 3485 T^{5} - 530 T^{6} - 97 T^{7} + 8 T^{8} + T^{9} \)
$31$ \( 123392 + 262128 T + 106624 T^{2} - 53604 T^{3} - 16152 T^{4} + 4589 T^{5} + 504 T^{6} - 127 T^{7} - 4 T^{8} + T^{9} \)
$37$ \( 136082 + 399893 T - 556200 T^{2} + 183010 T^{3} + 8018 T^{4} - 12845 T^{5} + 1648 T^{6} + 62 T^{7} - 22 T^{8} + T^{9} \)
$41$ \( -30880 + 200624 T + 11768 T^{2} - 92500 T^{3} - 1890 T^{4} + 8613 T^{5} - 154 T^{6} - 175 T^{7} + 2 T^{8} + T^{9} \)
$43$ \( 292288 - 375536 T + 74552 T^{2} + 72736 T^{3} - 32672 T^{4} + 1189 T^{5} + 1222 T^{6} - 145 T^{7} - 6 T^{8} + T^{9} \)
$47$ \( 1356304 + 1092589 T - 570452 T^{2} - 407462 T^{3} + 36662 T^{4} + 18583 T^{5} - 614 T^{6} - 262 T^{7} + 2 T^{8} + T^{9} \)
$53$ \( 633760 - 101824 T - 782712 T^{2} - 31448 T^{3} + 82170 T^{4} + 5497 T^{5} - 2258 T^{6} - 181 T^{7} + 12 T^{8} + T^{9} \)
$59$ \( 3599360 - 6511872 T + 3304704 T^{2} - 502976 T^{3} - 57632 T^{4} + 20320 T^{5} - 264 T^{6} - 236 T^{7} + 6 T^{8} + T^{9} \)
$61$ \( -6116062 + 19037393 T + 1699844 T^{2} - 1587054 T^{3} - 72782 T^{4} + 40731 T^{5} + 696 T^{6} - 358 T^{7} - 2 T^{8} + T^{9} \)
$67$ \( -1696960 + 5063952 T - 4424696 T^{2} + 1573888 T^{3} - 199976 T^{4} - 10699 T^{5} + 4126 T^{6} - 145 T^{7} - 18 T^{8} + T^{9} \)
$71$ \( -165399656 - 20296923 T + 13163138 T^{2} + 898990 T^{3} - 405162 T^{4} - 6769 T^{5} + 5266 T^{6} - 98 T^{7} - 24 T^{8} + T^{9} \)
$73$ \( 271190 + 103241 T - 808152 T^{2} + 612086 T^{3} - 139862 T^{4} + 55 T^{5} + 2996 T^{6} - 178 T^{7} - 14 T^{8} + T^{9} \)
$79$ \( -104192 - 581712 T + 759392 T^{2} - 87808 T^{3} - 83000 T^{4} + 8349 T^{5} + 1996 T^{6} - 175 T^{7} - 12 T^{8} + T^{9} \)
$83$ \( 226643968 - 14961664 T - 18408448 T^{2} + 404864 T^{3} + 519104 T^{4} + 8336 T^{5} - 5568 T^{6} - 208 T^{7} + 20 T^{8} + T^{9} \)
$89$ \( -825248 - 57840 T + 479512 T^{2} + 124388 T^{3} - 45818 T^{4} - 19323 T^{5} - 1348 T^{6} + 207 T^{7} + 30 T^{8} + T^{9} \)
$97$ \( -13712896 + 197632 T + 3520896 T^{2} - 180480 T^{3} - 207712 T^{4} + 14896 T^{5} + 3616 T^{6} - 300 T^{7} - 12 T^{8} + T^{9} \)
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