Properties

Label 1911.2.a.x
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 4 x^{9} - 10 x^{8} + 52 x^{7} + 16 x^{6} - 212 x^{5} + 64 x^{4} + 300 x^{3} - 159 x^{2} - 80 x + 46\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 10 x^{8} + 52 x^{7} + 16 x^{6} - 212 x^{5} + 64 x^{4} + 300 x^{3} - 159 x^{2} - 80 x + 46\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{9} + 16 \nu^{8} + 149 \nu^{7} - 301 \nu^{6} - 766 \nu^{5} + 1800 \nu^{4} + 1314 \nu^{3} - 3578 \nu^{2} - 870 \nu + 1178 \)\()/211\)
\(\beta_{4}\)\(=\)\((\)\( 22 \nu^{9} - 86 \nu^{8} - 247 \nu^{7} + 1064 \nu^{6} + 794 \nu^{5} - 3978 \nu^{4} - 891 \nu^{3} + 5042 \nu^{2} + 298 \nu - 1426 \)\()/211\)
\(\beta_{5}\)\(=\)\((\)\( -10 \nu^{9} + 135 \nu^{8} - 22 \nu^{7} - 1788 \nu^{6} + 1001 \nu^{5} + 7486 \nu^{4} - 2971 \nu^{3} - 10962 \nu^{2} + 1284 \nu + 2950 \)\()/211\)
\(\beta_{6}\)\(=\)\((\)\( -31 \nu^{9} + 102 \nu^{8} + 396 \nu^{7} - 1365 \nu^{6} - 1560 \nu^{5} + 5778 \nu^{4} + 1994 \nu^{3} - 8409 \nu^{2} - 113 \nu + 2182 \)\()/211\)
\(\beta_{7}\)\(=\)\((\)\( -72 \nu^{9} + 128 \nu^{8} + 981 \nu^{7} - 1564 \nu^{6} - 4229 \nu^{5} + 5960 \nu^{4} + 6292 \nu^{3} - 8368 \nu^{2} - 2107 \nu + 2672 \)\()/211\)
\(\beta_{8}\)\(=\)\((\)\( 78 \nu^{9} - 209 \nu^{8} - 1010 \nu^{7} + 2679 \nu^{6} + 4177 \nu^{5} - 10747 \nu^{4} - 6535 \nu^{3} + 15114 \nu^{2} + 2898 \nu - 4442 \)\()/211\)
\(\beta_{9}\)\(=\)\((\)\( 113 \nu^{9} - 365 \nu^{8} - 1355 \nu^{7} + 4717 \nu^{6} + 4788 \nu^{5} - 19013 \nu^{4} - 5315 \nu^{3} + 26262 \nu^{2} + 514 \nu - 6538 \)\()/211\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + 7 \beta_{2} + 2 \beta_{1} + 22\)
\(\nu^{5}\)\(=\)\(2 \beta_{8} + \beta_{7} - 7 \beta_{6} - 10 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 30 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(11 \beta_{9} + \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 10 \beta_{5} - 4 \beta_{4} + 49 \beta_{2} + 20 \beta_{1} + 138\)
\(\nu^{7}\)\(=\)\(4 \beta_{9} + 22 \beta_{8} + 12 \beta_{7} - 39 \beta_{6} - 86 \beta_{4} + 69 \beta_{3} + 73 \beta_{2} + 193 \beta_{1} + 164\)
\(\nu^{8}\)\(=\)\(96 \beta_{9} + 16 \beta_{8} + 93 \beta_{7} + 100 \beta_{6} + 79 \beta_{5} - 67 \beta_{4} + 4 \beta_{3} + 352 \beta_{2} + 159 \beta_{1} + 928\)
\(\nu^{9}\)\(=\)\(69 \beta_{9} + 189 \beta_{8} + 111 \beta_{7} - 186 \beta_{6} + 6 \beta_{5} - 704 \beta_{4} + 506 \beta_{3} + 578 \beta_{2} + 1289 \beta_{1} + 1280\)

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
−2.58460
−1.83272
−1.56844
−0.583659
0.556045
0.691507
1.67947
2.36375
2.51105
2.76760
−2.58460 −1.00000 4.68016 3.63737 2.58460 0 −6.92714 1.00000 −9.40115
1.2 −1.83272 −1.00000 1.35888 −3.20067 1.83272 0 1.17500 1.00000 5.86594
1.3 −1.56844 −1.00000 0.460010 4.07203 1.56844 0 2.41539 1.00000 −6.38674
1.4 −0.583659 −1.00000 −1.65934 0.00312810 0.583659 0 2.13581 1.00000 −0.00182574
1.5 0.556045 −1.00000 −1.69081 3.27818 −0.556045 0 −2.05226 1.00000 1.82282
1.6 0.691507 −1.00000 −1.52182 −2.34977 −0.691507 0 −2.43536 1.00000 −1.62488
1.7 1.67947 −1.00000 0.820610 −1.01562 −1.67947 0 −1.98075 1.00000 −1.70571
1.8 2.36375 −1.00000 3.58734 4.08271 −2.36375 0 3.75207 1.00000 9.65053
1.9 2.51105 −1.00000 4.30539 −2.80810 −2.51105 0 5.78895 1.00000 −7.05129
1.10 2.76760 −1.00000 5.65960 0.300737 −2.76760 0 10.1283 1.00000 0.832320
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 1911.2.a.x 10
3.b odd 2 1 5733.2.a.bw 10
7.b odd 2 1 1911.2.a.y yes 10
21.c even 2 1 5733.2.a.bx 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.a.x 10 1.a even 1 1 trivial
1911.2.a.y yes 10 7.b odd 2 1
5733.2.a.bw 10 3.b odd 2 1
5733.2.a.bx 10 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 46 - 80 T - 159 T^{2} + 300 T^{3} + 64 T^{4} - 212 T^{5} + 16 T^{6} + 52 T^{7} - 10 T^{8} - 4 T^{9} + T^{10} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( 4 - 1288 T + 2948 T^{2} + 4704 T^{3} - 956 T^{4} - 1456 T^{5} + 196 T^{6} + 160 T^{7} - 23 T^{8} - 6 T^{9} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( 101776 - 16240 T - 112412 T^{2} + 38512 T^{3} + 21200 T^{4} - 8432 T^{5} - 956 T^{6} + 568 T^{7} - 12 T^{8} - 12 T^{9} + T^{10} \)
$13$ \( ( -1 + T )^{10} \)
$17$ \( 60928 - 418816 T + 128768 T^{2} + 108800 T^{3} - 41472 T^{4} - 6592 T^{5} + 3296 T^{6} + 112 T^{7} - 98 T^{8} + T^{10} \)
$19$ \( 2243584 + 2002944 T - 222208 T^{2} - 456192 T^{3} - 21184 T^{4} + 33344 T^{5} + 2624 T^{6} - 976 T^{7} - 89 T^{8} + 10 T^{9} + T^{10} \)
$23$ \( -21787136 - 9213952 T + 2626816 T^{2} + 1295872 T^{3} - 124448 T^{4} - 67072 T^{5} + 3824 T^{6} + 1560 T^{7} - 89 T^{8} - 14 T^{9} + T^{10} \)
$29$ \( 15341312 + 1526272 T - 4551168 T^{2} + 221952 T^{3} + 425312 T^{4} - 66688 T^{5} - 8304 T^{6} + 2224 T^{7} - 33 T^{8} - 18 T^{9} + T^{10} \)
$31$ \( -38642688 - 40519680 T - 13553152 T^{2} - 328704 T^{3} + 734848 T^{4} + 126720 T^{5} - 4496 T^{6} - 2592 T^{7} - 113 T^{8} + 14 T^{9} + T^{10} \)
$37$ \( 33918976 + 7268352 T - 12916480 T^{2} + 1073152 T^{3} + 844288 T^{4} - 122624 T^{5} - 15648 T^{6} + 3200 T^{7} + 16 T^{8} - 24 T^{9} + T^{10} \)
$41$ \( 150752 - 761792 T + 1149988 T^{2} - 689104 T^{3} + 122208 T^{4} + 35600 T^{5} - 15964 T^{6} + 1528 T^{7} + 104 T^{8} - 24 T^{9} + T^{10} \)
$43$ \( -868096 - 888320 T + 5022720 T^{2} - 830208 T^{3} - 902432 T^{4} + 14592 T^{5} + 28368 T^{6} + 176 T^{7} - 295 T^{8} - 2 T^{9} + T^{10} \)
$47$ \( -2080316 - 22016824 T - 3257148 T^{2} + 5034704 T^{3} + 172196 T^{4} - 236672 T^{5} + 3908 T^{6} + 3672 T^{7} - 151 T^{8} - 18 T^{9} + T^{10} \)
$53$ \( -702861568 - 113874432 T + 74401792 T^{2} + 10466560 T^{3} - 2541536 T^{4} - 277504 T^{5} + 41360 T^{6} + 2864 T^{7} - 329 T^{8} - 10 T^{9} + T^{10} \)
$59$ \( 30888592 - 44500752 T + 72292 T^{2} + 4458576 T^{3} - 227664 T^{4} - 159488 T^{5} + 11076 T^{6} + 2376 T^{7} - 188 T^{8} - 12 T^{9} + T^{10} \)
$61$ \( 131072 + 786432 T + 950272 T^{2} - 393216 T^{3} - 627712 T^{4} - 25600 T^{5} + 25728 T^{6} + 896 T^{7} - 316 T^{8} - 4 T^{9} + T^{10} \)
$67$ \( 224530432 + 149127168 T + 1769472 T^{2} - 12683264 T^{3} - 978816 T^{4} + 345216 T^{5} + 29728 T^{6} - 3520 T^{7} - 302 T^{8} + 12 T^{9} + T^{10} \)
$71$ \( -152918752 + 164551904 T - 32975132 T^{2} - 8105296 T^{3} + 2905792 T^{4} - 51504 T^{5} - 57340 T^{6} + 5176 T^{7} + 136 T^{8} - 32 T^{9} + T^{10} \)
$73$ \( -1751296 - 4267520 T - 2393856 T^{2} + 665856 T^{3} + 443168 T^{4} - 87168 T^{5} - 16096 T^{6} + 4240 T^{7} - 145 T^{8} - 18 T^{9} + T^{10} \)
$79$ \( 8925328 - 20965408 T + 17545136 T^{2} - 6157248 T^{3} + 609208 T^{4} + 154576 T^{5} - 41152 T^{6} + 1936 T^{7} + 289 T^{8} - 34 T^{9} + T^{10} \)
$83$ \( -225377852 + 389734072 T - 125884316 T^{2} - 459296 T^{3} + 4860196 T^{4} - 415616 T^{5} - 48972 T^{6} + 7040 T^{7} + 9 T^{8} - 30 T^{9} + T^{10} \)
$89$ \( 221391652 + 73029064 T - 92331100 T^{2} + 17032752 T^{3} + 1298692 T^{4} - 589488 T^{5} + 26420 T^{6} + 4584 T^{7} - 351 T^{8} - 10 T^{9} + T^{10} \)
$97$ \( 867344128 - 724011520 T + 89410560 T^{2} + 25195520 T^{3} - 4413280 T^{4} - 300096 T^{5} + 67600 T^{6} + 1408 T^{7} - 433 T^{8} - 2 T^{9} + T^{10} \)
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