Properties

Label 4235.2.a.bl
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 14 x^{8} + 26 x^{7} + 63 x^{6} - 106 x^{5} - 96 x^{4} + 140 x^{3} + 38 x^{2} - 38 x - 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 14 x^{8} + 26 x^{7} + 63 x^{6} - 106 x^{5} - 96 x^{4} + 140 x^{3} + 38 x^{2} - 38 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{9} + \nu^{8} + 17 \nu^{7} - 9 \nu^{6} - 95 \nu^{5} + 6 \nu^{4} + 188 \nu^{3} + 67 \nu^{2} - 97 \nu - 44 \)\()/11\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{9} + 7 \nu^{8} + 23 \nu^{7} - 92 \nu^{6} - 65 \nu^{5} + 381 \nu^{4} - 33 \nu^{3} - 511 \nu^{2} + 157 \nu + 121 \)\()/11\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{9} + 7 \nu^{8} + 57 \nu^{7} - 87 \nu^{6} - 261 \nu^{5} + 329 \nu^{4} + 390 \nu^{3} - 372 \nu^{2} - 85 \nu + 44 \)\()/11\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{9} - 8 \nu^{8} - 23 \nu^{7} + 109 \nu^{6} + 62 \nu^{5} - 468 \nu^{4} + 62 \nu^{3} + 640 \nu^{2} - 225 \nu - 143 \)\()/11\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{9} + 8 \nu^{8} + 57 \nu^{7} - 104 \nu^{6} - 258 \nu^{5} + 416 \nu^{4} + 372 \nu^{3} - 501 \nu^{2} - 72 \nu + 66 \)\()/11\)
\(\beta_{8}\)\(=\)\((\)\( 9 \nu^{9} - 20 \nu^{8} - 120 \nu^{7} + 257 \nu^{6} + 492 \nu^{5} - 1022 \nu^{4} - 570 \nu^{3} + 1245 \nu^{2} - 40 \nu - 198 \)\()/11\)
\(\beta_{9}\)\(=\)\((\)\( 12 \nu^{9} - 26 \nu^{8} - 160 \nu^{7} + 335 \nu^{6} + 647 \nu^{5} - 1334 \nu^{4} - 684 \nu^{3} + 1629 \nu^{2} - 184 \nu - 286 \)\()/11\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 7 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(-\beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 9 \beta_{6} - 8 \beta_{5} + 10 \beta_{4} + 2 \beta_{2} + 29 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(10 \beta_{8} + 9 \beta_{7} + 11 \beta_{6} + 10 \beta_{5} + 22 \beta_{4} - 8 \beta_{3} + 47 \beta_{2} + 13 \beta_{1} + 96\)
\(\nu^{7}\)\(=\)\(-10 \beta_{9} + 24 \beta_{8} + 69 \beta_{7} + 66 \beta_{6} - 52 \beta_{5} + 80 \beta_{4} + 25 \beta_{2} + 178 \beta_{1} + 19\)
\(\nu^{8}\)\(=\)\(3 \beta_{9} + 77 \beta_{8} + 68 \beta_{7} + 91 \beta_{6} + 78 \beta_{5} + 188 \beta_{4} - 49 \beta_{3} + 313 \beta_{2} + 124 \beta_{1} + 602\)
\(\nu^{9}\)\(=\)\(-72 \beta_{9} + 211 \beta_{8} + 499 \beta_{7} + 453 \beta_{6} - 318 \beta_{5} + 600 \beta_{4} + 6 \beta_{3} + 234 \beta_{2} + 1127 \beta_{1} + 219\)

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
−2.46970
−2.23949
−1.40864
−0.416850
−0.330750
0.799913
1.04772
1.78795
2.55273
2.67713
−2.46970 −0.581915 4.09943 −1.00000 1.43716 1.00000 −5.18496 −2.66137 2.46970
1.2 −2.23949 1.46605 3.01534 −1.00000 −3.28322 1.00000 −2.27384 −0.850686 2.23949
1.3 −1.40864 2.97205 −0.0157228 −1.00000 −4.18655 1.00000 2.83944 5.83306 1.40864
1.4 −0.416850 0.0737733 −1.82624 −1.00000 −0.0307524 1.00000 1.59497 −2.99456 0.416850
1.5 −0.330750 −2.01275 −1.89060 −1.00000 0.665719 1.00000 1.28682 1.05118 0.330750
1.6 0.799913 2.12674 −1.36014 −1.00000 1.70120 1.00000 −2.68782 1.52301 −0.799913
1.7 1.04772 −2.73241 −0.902293 −1.00000 −2.86278 1.00000 −3.04078 4.46605 −1.04772
1.8 1.78795 0.315097 1.19678 −1.00000 0.563380 1.00000 −1.43612 −2.90071 −1.78795
1.9 2.55273 2.87748 4.51641 −1.00000 7.34543 1.00000 6.42372 5.27991 −2.55273
1.10 2.67713 −0.504113 5.16704 −1.00000 −1.34958 1.00000 8.47858 −2.74587 −2.67713
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 4235.2.a.bl yes 10
11.b odd 2 1 4235.2.a.bj 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.bj 10 11.b odd 2 1
4235.2.a.bl yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\(T_{2}^{10} - \cdots\)
\(T_{3}^{10} - \cdots\)
\(T_{13}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -11 - 38 T + 38 T^{2} + 140 T^{3} - 96 T^{4} - 106 T^{5} + 63 T^{6} + 26 T^{7} - 14 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( 1 - 14 T - 3 T^{2} + 120 T^{3} + 45 T^{4} - 166 T^{5} + 13 T^{6} + 50 T^{7} - 10 T^{8} - 4 T^{9} + T^{10} \)
$5$ \( ( 1 + T )^{10} \)
$7$ \( ( -1 + T )^{10} \)
$11$ \( T^{10} \)
$13$ \( 14641 + 31944 T + 2904 T^{2} - 26840 T^{3} - 5473 T^{4} + 7894 T^{5} - 206 T^{6} - 896 T^{7} + 246 T^{8} - 26 T^{9} + T^{10} \)
$17$ \( -138923 - 79246 T + 79160 T^{2} + 37656 T^{3} - 17939 T^{4} - 5580 T^{5} + 1960 T^{6} + 270 T^{7} - 82 T^{8} - 4 T^{9} + T^{10} \)
$19$ \( -78107 + 22794 T + 84825 T^{2} - 922 T^{3} - 28038 T^{4} - 4360 T^{5} + 2269 T^{6} + 336 T^{7} - 79 T^{8} - 6 T^{9} + T^{10} \)
$23$ \( 18117 + 288 T - 71543 T^{2} - 47048 T^{3} + 13921 T^{4} + 11666 T^{5} + 300 T^{6} - 644 T^{7} - 65 T^{8} + 8 T^{9} + T^{10} \)
$29$ \( -4331 - 25220 T - 12973 T^{2} + 70764 T^{3} - 25385 T^{4} - 8436 T^{5} + 4276 T^{6} - 72 T^{7} - 115 T^{8} + 4 T^{9} + T^{10} \)
$31$ \( 2855497 - 4634094 T - 2196435 T^{2} + 1080792 T^{3} + 180159 T^{4} - 71810 T^{5} - 3555 T^{6} + 1914 T^{7} - 30 T^{8} - 18 T^{9} + T^{10} \)
$37$ \( 96009 + 409248 T - 836951 T^{2} - 162656 T^{3} + 145785 T^{4} + 34380 T^{5} - 4221 T^{6} - 1502 T^{7} - 28 T^{8} + 16 T^{9} + T^{10} \)
$41$ \( -912483 - 4435932 T - 403877 T^{2} + 906746 T^{3} + 174027 T^{4} - 47020 T^{5} - 14712 T^{6} - 286 T^{7} + 263 T^{8} + 30 T^{9} + T^{10} \)
$43$ \( -890747 - 3207838 T + 848875 T^{2} + 1979254 T^{3} + 239211 T^{4} - 125876 T^{5} - 6351 T^{6} + 2872 T^{7} - 22 T^{8} - 22 T^{9} + T^{10} \)
$47$ \( -20631 + 154974 T - 282476 T^{2} + 23876 T^{3} + 58910 T^{4} - 12358 T^{5} - 2969 T^{6} + 964 T^{7} - 16 T^{8} - 14 T^{9} + T^{10} \)
$53$ \( 13631577 + 7839114 T - 9411587 T^{2} - 5301906 T^{3} - 118111 T^{4} + 228610 T^{5} + 18027 T^{6} - 2864 T^{7} - 270 T^{8} + 10 T^{9} + T^{10} \)
$59$ \( -2783 - 9922 T + 1136 T^{2} + 29256 T^{3} + 14311 T^{4} - 11964 T^{5} - 2186 T^{6} + 978 T^{7} + 56 T^{8} - 22 T^{9} + T^{10} \)
$61$ \( -3681183 - 10051206 T - 3743252 T^{2} + 4833410 T^{3} + 256416 T^{4} - 319762 T^{5} + 17775 T^{6} + 3734 T^{7} - 280 T^{8} - 12 T^{9} + T^{10} \)
$67$ \( -685883 + 73096 T + 751521 T^{2} - 294368 T^{3} - 94810 T^{4} + 40710 T^{5} + 4791 T^{6} - 1418 T^{7} - 113 T^{8} + 14 T^{9} + T^{10} \)
$71$ \( 13539537 - 33795366 T + 19352971 T^{2} - 1819728 T^{3} - 1004677 T^{4} + 140300 T^{5} + 23751 T^{6} - 2362 T^{7} - 294 T^{8} + 8 T^{9} + T^{10} \)
$73$ \( -53820239 + 45089972 T - 693713 T^{2} - 8135860 T^{3} + 2131940 T^{4} - 14864 T^{5} - 51529 T^{6} + 5040 T^{7} + 91 T^{8} - 30 T^{9} + T^{10} \)
$79$ \( -27573743 - 1728644 T + 12553125 T^{2} + 1650788 T^{3} - 1128644 T^{4} - 109528 T^{5} + 33963 T^{6} + 2176 T^{7} - 349 T^{8} - 8 T^{9} + T^{10} \)
$83$ \( 4234914213 + 1051183524 T - 594992888 T^{2} - 154414242 T^{3} - 496045 T^{4} + 2094272 T^{5} + 91692 T^{6} - 9568 T^{7} - 576 T^{8} + 14 T^{9} + T^{10} \)
$89$ \( 721845289 + 70495518 T - 119678470 T^{2} - 25162038 T^{3} + 2736409 T^{4} + 1096042 T^{5} + 61779 T^{6} - 5516 T^{7} - 519 T^{8} + 6 T^{9} + T^{10} \)
$97$ \( 261366237 - 141536736 T - 18923384 T^{2} + 15959400 T^{3} + 146975 T^{4} - 518006 T^{5} + 15957 T^{6} + 5758 T^{7} - 261 T^{8} - 20 T^{9} + T^{10} \)
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