Properties

Label 3381.2.a.bg
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 4 x^{9} - 6 x^{8} + 36 x^{7} + x^{6} - 100 x^{5} + 26 x^{4} + 100 x^{3} - 17 x^{2} - 28 x - 2\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 6 x^{8} + 36 x^{7} + x^{6} - 100 x^{5} + 26 x^{4} + 100 x^{3} - 17 x^{2} - 28 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{9} - 4 \nu^{8} + 35 \nu^{7} + 44 \nu^{6} - 161 \nu^{5} - 202 \nu^{4} + 193 \nu^{3} + 441 \nu^{2} + \nu - 220 \)\()/47\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{9} + 6 \nu^{8} - 76 \nu^{7} - 19 \nu^{6} + 453 \nu^{5} - 73 \nu^{4} - 783 \nu^{3} + 114 \nu^{2} + 163 \nu + 48 \)\()/47\)
\(\beta_{5}\)\(=\)\((\)\( -10 \nu^{9} + 27 \nu^{8} + 81 \nu^{7} - 203 \nu^{6} - 194 \nu^{5} + 353 \nu^{4} + 166 \nu^{3} - 4 \nu^{2} + 5 \nu - 66 \)\()/47\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{9} + 25 \nu^{8} + 122 \nu^{7} - 228 \nu^{6} - 486 \nu^{5} + 581 \nu^{4} + 850 \nu^{3} - 324 \nu^{2} - 535 \nu - 35 \)\()/47\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{9} + 21 \nu^{8} + 157 \nu^{7} - 184 \nu^{6} - 647 \nu^{5} + 426 \nu^{4} + 996 \nu^{3} - 165 \nu^{2} - 346 \nu - 20 \)\()/47\)
\(\beta_{8}\)\(=\)\((\)\( 15 \nu^{9} - 17 \nu^{8} - 192 \nu^{7} + 140 \nu^{6} + 855 \nu^{5} - 318 \nu^{4} - 1471 \nu^{3} + 194 \nu^{2} + 627 \nu - 42 \)\()/47\)
\(\beta_{9}\)\(=\)\((\)\( 26 \nu^{9} - 89 \nu^{8} - 173 \nu^{7} + 744 \nu^{6} + 166 \nu^{5} - 1745 \nu^{4} + 358 \nu^{3} + 1129 \nu^{2} - 295 \nu - 54 \)\()/47\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 7 \beta_{2} + 7 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{8} + 11 \beta_{7} - 2 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} - \beta_{3} + 10 \beta_{2} + 28 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{9} + 2 \beta_{8} + 24 \beta_{7} - 10 \beta_{6} - 9 \beta_{5} + 12 \beta_{4} - 10 \beta_{3} + 47 \beta_{2} + 49 \beta_{1} + 28\)
\(\nu^{7}\)\(=\)\(2 \beta_{9} + 13 \beta_{8} + 94 \beta_{7} - 22 \beta_{6} - 53 \beta_{5} + 57 \beta_{4} - 16 \beta_{3} + 84 \beta_{2} + 173 \beta_{1} - 2\)
\(\nu^{8}\)\(=\)\(13 \beta_{9} + 31 \beta_{8} + 220 \beta_{7} - 79 \beta_{6} - 70 \beta_{5} + 106 \beta_{4} - 85 \beta_{3} + 322 \beta_{2} + 352 \beta_{1} + 117\)
\(\nu^{9}\)\(=\)\(31 \beta_{9} + 129 \beta_{8} + 742 \beta_{7} - 185 \beta_{6} - 337 \beta_{5} + 401 \beta_{4} - 172 \beta_{3} + 665 \beta_{2} + 1144 \beta_{1} - 29\)

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
2.72964
2.24407
1.86909
1.69362
0.795395
−0.0765467
−0.488009
−1.01915
−1.54861
−2.19951
−2.72964 −1.00000 5.45092 1.75794 2.72964 0 −9.41975 1.00000 −4.79853
1.2 −2.24407 −1.00000 3.03587 −4.44184 2.24407 0 −2.32456 1.00000 9.96781
1.3 −1.86909 −1.00000 1.49350 3.83063 1.86909 0 0.946688 1.00000 −7.15979
1.4 −1.69362 −1.00000 0.868357 −1.90171 1.69362 0 1.91658 1.00000 3.22078
1.5 −0.795395 −1.00000 −1.36735 −1.31903 0.795395 0 2.67837 1.00000 1.04915
1.6 0.0765467 −1.00000 −1.99414 2.75264 −0.0765467 0 −0.305738 1.00000 0.210706
1.7 0.488009 −1.00000 −1.76185 −0.529828 −0.488009 0 −1.83582 1.00000 −0.258561
1.8 1.01915 −1.00000 −0.961343 2.85504 −1.01915 0 −3.01804 1.00000 2.90970
1.9 1.54861 −1.00000 0.398191 −1.02582 −1.54861 0 −2.48058 1.00000 −1.58859
1.10 2.19951 −1.00000 2.83784 2.02197 −2.19951 0 1.84284 1.00000 4.44733
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 3381.2.a.bg 10
7.b odd 2 1 3381.2.a.bh yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.bg 10 1.a even 1 1 trivial
3381.2.a.bh yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\(T_{2}^{10} + \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} + \cdots\)
\(T_{13}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + 28 T - 17 T^{2} - 100 T^{3} + 26 T^{4} + 100 T^{5} + T^{6} - 36 T^{7} - 6 T^{8} + 4 T^{9} + T^{10} \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( -648 - 1512 T + 279 T^{2} + 1704 T^{3} - 104 T^{4} - 708 T^{5} + 95 T^{6} + 112 T^{7} - 24 T^{8} - 4 T^{9} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( -2441 + 2654 T + 9373 T^{2} - 3220 T^{3} - 7618 T^{4} + 528 T^{5} + 1302 T^{6} - 68 T^{7} - 65 T^{8} + 2 T^{9} + T^{10} \)
$13$ \( 1358 + 9704 T - 46991 T^{2} - 19552 T^{3} + 26186 T^{4} - 1616 T^{5} - 2541 T^{6} + 464 T^{7} + 44 T^{8} - 16 T^{9} + T^{10} \)
$17$ \( 815806 - 343332 T - 316929 T^{2} + 130452 T^{3} + 34576 T^{4} - 16472 T^{5} - 804 T^{6} + 772 T^{7} - 30 T^{8} - 12 T^{9} + T^{10} \)
$19$ \( 22783 - 82098 T + 81377 T^{2} - 6024 T^{3} - 25710 T^{4} + 10316 T^{5} + 398 T^{6} - 1048 T^{7} + 255 T^{8} - 26 T^{9} + T^{10} \)
$23$ \( ( -1 + T )^{10} \)
$29$ \( -453058 + 909148 T - 322545 T^{2} - 200400 T^{3} + 64128 T^{4} + 22324 T^{5} - 2860 T^{6} - 1104 T^{7} - 6 T^{8} + 16 T^{9} + T^{10} \)
$31$ \( 1345636 - 135868 T - 1144607 T^{2} - 43384 T^{3} + 210118 T^{4} - 11120 T^{5} - 13354 T^{6} + 2232 T^{7} + 2 T^{8} - 20 T^{9} + T^{10} \)
$37$ \( 1480658 - 2116956 T + 7765 T^{2} + 831148 T^{3} - 174918 T^{4} - 53932 T^{5} + 10434 T^{6} + 1236 T^{7} - 184 T^{8} - 8 T^{9} + T^{10} \)
$41$ \( 116354273 + 14497398 T - 22852587 T^{2} + 265388 T^{3} + 1446698 T^{4} - 150880 T^{5} - 23282 T^{6} + 4236 T^{7} - 47 T^{8} - 22 T^{9} + T^{10} \)
$43$ \( 805678 - 2123960 T - 2073375 T^{2} + 2045512 T^{3} - 390998 T^{4} - 43604 T^{5} + 16635 T^{6} - 84 T^{7} - 216 T^{8} + 4 T^{9} + T^{10} \)
$47$ \( -11309552 + 6275328 T + 8613648 T^{2} - 321216 T^{3} - 744500 T^{4} - 13336 T^{5} + 22332 T^{6} + 764 T^{7} - 249 T^{8} - 6 T^{9} + T^{10} \)
$53$ \( -238691081 - 159472194 T - 14129065 T^{2} + 11621772 T^{3} + 2999591 T^{4} + 68930 T^{5} - 49523 T^{6} - 4700 T^{7} + 111 T^{8} + 30 T^{9} + T^{10} \)
$59$ \( -1101511 - 5072062 T + 2198403 T^{2} + 2914900 T^{3} - 2170327 T^{4} + 548170 T^{5} - 54055 T^{6} - 704 T^{7} + 569 T^{8} - 42 T^{9} + T^{10} \)
$61$ \( 22502303 + 180527758 T + 102542187 T^{2} + 9594252 T^{3} - 3395481 T^{4} - 415610 T^{5} + 50177 T^{6} + 4392 T^{7} - 363 T^{8} - 14 T^{9} + T^{10} \)
$67$ \( -2613735112 + 58420640 T + 263241297 T^{2} - 5263620 T^{3} - 8025086 T^{4} + 89100 T^{5} + 100299 T^{6} - 340 T^{7} - 532 T^{8} + T^{10} \)
$71$ \( -1340942 - 2574496 T + 2647311 T^{2} + 190240 T^{3} - 354274 T^{4} - 13156 T^{5} + 16021 T^{6} + 832 T^{7} - 218 T^{8} - 8 T^{9} + T^{10} \)
$73$ \( 497014664 + 9547608 T - 109200135 T^{2} + 16373800 T^{3} + 2728262 T^{4} - 586640 T^{5} - 12194 T^{6} + 6616 T^{7} - 146 T^{8} - 24 T^{9} + T^{10} \)
$79$ \( -1835233214 + 1214013780 T - 206239835 T^{2} - 20507332 T^{3} + 8835302 T^{4} - 456532 T^{5} - 79926 T^{6} + 8724 T^{7} + 28 T^{8} - 32 T^{9} + T^{10} \)
$83$ \( 1949250692 - 529693132 T - 252683521 T^{2} + 46217292 T^{3} + 5606352 T^{4} - 1128808 T^{5} - 24804 T^{6} + 9844 T^{7} - 164 T^{8} - 28 T^{9} + T^{10} \)
$89$ \( -4423288444 - 1013646052 T + 433872295 T^{2} + 37288428 T^{3} - 12554880 T^{4} - 396568 T^{5} + 143437 T^{6} + 1040 T^{7} - 648 T^{8} + T^{10} \)
$97$ \( -43708 + 245628 T - 511 T^{2} - 349080 T^{3} + 60200 T^{4} + 76600 T^{5} + 6622 T^{6} - 1832 T^{7} - 188 T^{8} + 12 T^{9} + T^{10} \)
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