Properties

Label 4235.2.a.bk
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: \(x^{10} - 2 x^{9} - 14 x^{8} + 26 x^{7} + 67 x^{6} - 110 x^{5} - 132 x^{4} + 168 x^{3} + 94 x^{2} - 54 x - 11\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 14 x^{8} + 26 x^{7} + 67 x^{6} - 110 x^{5} - 132 x^{4} + 168 x^{3} + 94 x^{2} - 54 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - \nu^{7} - 12 \nu^{6} + 9 \nu^{5} + 43 \nu^{4} - 19 \nu^{3} - 49 \nu^{2} + 3 \nu + 7 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{9} + \nu^{8} + 12 \nu^{7} - 9 \nu^{6} - 43 \nu^{5} + 19 \nu^{4} + 49 \nu^{3} - 6 \nu^{2} - 7 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - 15 \nu^{7} + 73 \nu^{5} - 3 \nu^{4} - 124 \nu^{3} + 11 \nu^{2} + 36 \nu - 6 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{9} - 14 \nu^{7} - 3 \nu^{6} + 64 \nu^{5} + 27 \nu^{4} - 108 \nu^{3} - 61 \nu^{2} + 44 \nu + 14 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{9} + \nu^{8} + 57 \nu^{7} - 3 \nu^{6} - 265 \nu^{5} - 35 \nu^{4} + 442 \nu^{3} + 120 \nu^{2} - 145 \nu - 24 \)\()/3\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{9} + 58 \nu^{7} + 9 \nu^{6} - 274 \nu^{5} - 78 \nu^{4} + 458 \nu^{3} + 169 \nu^{2} - 136 \nu - 28 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 6 \nu^{9} - \nu^{8} - 85 \nu^{7} - 3 \nu^{6} + 390 \nu^{5} + 89 \nu^{4} - 628 \nu^{3} - 242 \nu^{2} + 167 \nu + 49 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + \beta_{7} - \beta_{3} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{9} - 10 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} - 10 \beta_{3} + 18 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} + 9 \beta_{8} + 12 \beta_{7} + 22 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} - 12 \beta_{3} + 47 \beta_{2} + 77\)
\(\nu^{7}\)\(=\)\(-9 \beta_{9} - 77 \beta_{8} + 71 \beta_{7} + 27 \beta_{6} + 3 \beta_{5} - 80 \beta_{3} + 3 \beta_{2} + 90 \beta_{1} + 72\)
\(\nu^{8}\)\(=\)\(77 \beta_{9} + 59 \beta_{8} + 110 \beta_{7} + 187 \beta_{6} + 104 \beta_{5} + 77 \beta_{4} - 107 \beta_{3} + 315 \beta_{2} + \beta_{1} + 472\)
\(\nu^{9}\)\(=\)\(-59 \beta_{9} - 546 \beta_{8} + 535 \beta_{7} + 265 \beta_{6} + 51 \beta_{5} + 3 \beta_{4} - 597 \beta_{3} + 55 \beta_{2} + 496 \beta_{1} + 562\)

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
−2.51238
−1.83371
−1.71475
−0.784066
−0.170316
0.547729
1.81029
1.85511
2.09068
2.71142
−2.51238 −2.89922 4.31207 1.00000 7.28396 −1.00000 −5.80880 5.40550 −2.51238
1.2 −1.83371 −1.24345 1.36250 1.00000 2.28012 −1.00000 1.16899 −1.45384 −1.83371
1.3 −1.71475 0.493317 0.940360 1.00000 −0.845915 −1.00000 1.81702 −2.75664 −1.71475
1.4 −0.784066 1.73981 −1.38524 1.00000 −1.36412 −1.00000 2.65425 0.0269309 −0.784066
1.5 −0.170316 0.667096 −1.97099 1.00000 −0.113617 −1.00000 0.676324 −2.55498 −0.170316
1.6 0.547729 −3.41697 −1.69999 1.00000 −1.87158 −1.00000 −2.02659 8.67570 0.547729
1.7 1.81029 2.40633 1.27715 1.00000 4.35616 −1.00000 −1.30857 2.79044 1.81029
1.8 1.85511 −2.66752 1.44142 1.00000 −4.94854 −1.00000 −1.03623 4.11568 1.85511
1.9 2.09068 −1.17179 2.37096 1.00000 −2.44983 −1.00000 0.775558 −1.62692 2.09068
1.10 2.71142 2.09240 5.35177 1.00000 5.67336 −1.00000 9.08805 1.37813 2.71142
\(n\): 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.bk yes 10
11.b odd 2 1 4235.2.a.bi 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.bi 10 11.b odd 2 1
4235.2.a.bk 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

$p$ $F_p(T)$
$2$ \( -11 - 54 T + 94 T^{2} + 168 T^{3} - 132 T^{4} - 110 T^{5} + 67 T^{6} + 26 T^{7} - 14 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( -111 + 258 T + 189 T^{2} - 504 T^{3} - 179 T^{4} + 282 T^{5} + 73 T^{6} - 58 T^{7} - 14 T^{8} + 4 T^{9} + T^{10} \)
$5$ \( ( -1 + T )^{10} \)
$7$ \( ( 1 + T )^{10} \)
$11$ \( T^{10} \)
$13$ \( 2497 - 10664 T + 15988 T^{2} - 8184 T^{3} - 2649 T^{4} + 4330 T^{5} - 1342 T^{6} - 60 T^{7} + 102 T^{8} - 18 T^{9} + T^{10} \)
$17$ \( -120531 - 224838 T + 108048 T^{2} + 88344 T^{3} - 30299 T^{4} - 8864 T^{5} + 2660 T^{6} + 330 T^{7} - 90 T^{8} - 4 T^{9} + T^{10} \)
$19$ \( -362363 + 756978 T - 432247 T^{2} - 27022 T^{3} + 80810 T^{4} - 13652 T^{5} - 3235 T^{6} + 984 T^{7} - 15 T^{8} - 14 T^{9} + T^{10} \)
$23$ \( -9867 + 67200 T + 122157 T^{2} - 76344 T^{3} - 82739 T^{4} + 11282 T^{5} + 6248 T^{6} - 408 T^{7} - 141 T^{8} + 4 T^{9} + T^{10} \)
$29$ \( 50990181 - 56834076 T + 23328843 T^{2} - 3272292 T^{3} - 491177 T^{4} + 228244 T^{5} - 23996 T^{6} - 1096 T^{7} + 445 T^{8} - 36 T^{9} + T^{10} \)
$31$ \( -443223 + 864666 T - 235059 T^{2} - 205464 T^{3} + 51895 T^{4} + 21498 T^{5} - 2787 T^{6} - 1046 T^{7} + 18 T^{8} + 18 T^{9} + T^{10} \)
$37$ \( 221353 + 2204544 T + 2182393 T^{2} + 34740 T^{3} - 258111 T^{4} - 15280 T^{5} + 10483 T^{6} + 490 T^{7} - 176 T^{8} - 4 T^{9} + T^{10} \)
$41$ \( 1406725 - 1763400 T + 390307 T^{2} + 349326 T^{3} - 203781 T^{4} + 22672 T^{5} + 10192 T^{6} - 3874 T^{7} + 559 T^{8} - 38 T^{9} + T^{10} \)
$43$ \( 80793693 + 93927438 T + 29563839 T^{2} - 1804626 T^{3} - 1837521 T^{4} - 49716 T^{5} + 38229 T^{6} + 1260 T^{7} - 330 T^{8} - 6 T^{9} + T^{10} \)
$47$ \( -2805575 - 5835370 T - 3115128 T^{2} + 503460 T^{3} + 709358 T^{4} + 112190 T^{5} - 11101 T^{6} - 3392 T^{7} - 100 T^{8} + 18 T^{9} + T^{10} \)
$53$ \( -4676351 - 5594486 T - 443407 T^{2} + 948890 T^{3} + 231833 T^{4} - 29790 T^{5} - 14945 T^{6} - 1000 T^{7} + 166 T^{8} + 26 T^{9} + T^{10} \)
$59$ \( 76565913 + 106780266 T + 19959000 T^{2} - 12454632 T^{3} - 1231553 T^{4} + 371400 T^{5} + 29014 T^{6} - 4070 T^{7} - 296 T^{8} + 14 T^{9} + T^{10} \)
$61$ \( -8879159 - 102690766 T + 30943852 T^{2} + 11795806 T^{3} - 5956280 T^{4} + 740894 T^{5} + 22479 T^{6} - 13926 T^{7} + 1384 T^{8} - 60 T^{9} + T^{10} \)
$67$ \( -43621883 - 9111104 T + 271768533 T^{2} - 29026500 T^{3} - 8714670 T^{4} + 617862 T^{5} + 102279 T^{6} - 4326 T^{7} - 525 T^{8} + 10 T^{9} + T^{10} \)
$71$ \( 9317697 - 5427870 T - 3668445 T^{2} + 2186256 T^{3} + 219235 T^{4} - 227748 T^{5} + 26799 T^{6} + 1406 T^{7} - 318 T^{8} + T^{10} \)
$73$ \( -2443752719 + 2199826592 T - 696488417 T^{2} + 71639664 T^{3} + 6706308 T^{4} - 1783384 T^{5} + 44207 T^{6} + 10356 T^{7} - 489 T^{8} - 18 T^{9} + T^{10} \)
$79$ \( -524411063 + 448747180 T - 62241187 T^{2} - 22049428 T^{3} + 5547956 T^{4} + 46128 T^{5} - 99653 T^{6} + 6296 T^{7} + 307 T^{8} - 40 T^{9} + T^{10} \)
$83$ \( -206723 - 4630660 T - 4130368 T^{2} + 2921566 T^{3} + 134111 T^{4} - 258276 T^{5} + 29512 T^{6} + 2068 T^{7} - 368 T^{8} - 2 T^{9} + T^{10} \)
$89$ \( 3891537 - 4630446 T + 453042 T^{2} + 788466 T^{3} - 155855 T^{4} - 37666 T^{5} + 8883 T^{6} + 612 T^{7} - 175 T^{8} - 2 T^{9} + T^{10} \)
$97$ \( -20760073331 - 4882034892 T + 959475444 T^{2} + 175046676 T^{3} - 22124473 T^{4} - 1689614 T^{5} + 199149 T^{6} + 6270 T^{7} - 749 T^{8} - 8 T^{9} + T^{10} \)
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