Properties

Label 847.2.a.o
Level $847$
Weight $2$
Character orbit 847.a
Self dual yes
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} - 5 \nu^{5} - 30 \nu^{4} - 30 \nu^{3} + 130 \nu^{2} + 135 \nu - 140 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} - 4 \nu^{6} + 35 \nu^{5} + 35 \nu^{4} - 165 \nu^{3} - 60 \nu^{2} + 180 \nu - 45 \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} + 6 \nu^{6} - 40 \nu^{5} - 65 \nu^{4} + 160 \nu^{3} + 165 \nu^{2} - 170 \nu - 20 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{7} + 7 \nu^{6} + 95 \nu^{5} - 55 \nu^{4} - 280 \nu^{3} + 105 \nu^{2} + 210 \nu - 90 \)\()/25\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 7 \nu^{6} - 95 \nu^{5} + 55 \nu^{4} + 280 \nu^{3} - 80 \nu^{2} - 210 \nu + 15 \)\()/25\)
\(\beta_{7}\)\(=\)\((\)\( 12 \nu^{7} - \nu^{6} - 135 \nu^{5} + 15 \nu^{4} + 440 \nu^{3} - 90 \nu^{2} - 405 \nu + 170 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 6 \beta_{6} + 7 \beta_{5} - \beta_{4} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 7 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(11 \beta_{7} + 35 \beta_{6} + 47 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 77\)
\(\nu^{7}\)\(=\)\(13 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} - 42 \beta_{2} + 165 \beta_{1} + 31\)

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
2.55194
1.98451
1.11447
0.669744
0.226211
−1.40927
−1.70716
−2.43045
−2.55194 −0.410598 4.51241 3.42898 1.04782 −1.00000 −6.41153 −2.83141 −8.75055
1.2 −1.98451 2.78639 1.93830 −0.0269243 −5.52964 −1.00000 0.122446 4.76399 0.0534317
1.3 −1.11447 −2.85882 −0.757964 3.45608 3.18606 −1.00000 3.07366 5.17284 −3.85168
1.4 −0.669744 3.13977 −1.55144 2.14378 −2.10284 −1.00000 2.37856 6.85818 −1.43578
1.5 −0.226211 −0.219130 −1.94883 −2.49552 0.0495696 −1.00000 0.893270 −2.95198 0.564516
1.6 1.40927 −2.16338 −0.0139645 −1.83139 −3.04878 −1.00000 −2.83822 1.68022 −2.58091
1.7 1.70716 2.29155 0.914391 4.06637 3.91205 −1.00000 −1.85331 2.25122 6.94194
1.8 2.43045 1.43421 3.90710 1.25863 3.48577 −1.00000 4.63512 −0.943053 3.05904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 847.2.a.o 8
3.b odd 2 1 7623.2.a.cw 8
7.b odd 2 1 5929.2.a.bs 8
11.b odd 2 1 847.2.a.p 8
11.c even 5 2 847.2.f.v 16
11.c even 5 2 847.2.f.x 16
11.d odd 10 2 77.2.f.b 16
11.d odd 10 2 847.2.f.w 16
33.d even 2 1 7623.2.a.ct 8
33.f even 10 2 693.2.m.i 16
77.b even 2 1 5929.2.a.bt 8
77.l even 10 2 539.2.f.e 16
77.n even 30 4 539.2.q.f 32
77.o odd 30 4 539.2.q.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.d odd 10 2
539.2.f.e 16 77.l even 10 2
539.2.q.f 32 77.n even 30 4
539.2.q.g 32 77.o odd 30 4
693.2.m.i 16 33.f even 10 2
847.2.a.o 8 1.a even 1 1 trivial
847.2.a.p 8 11.b odd 2 1
847.2.f.v 16 11.c even 5 2
847.2.f.w 16 11.d odd 10 2
847.2.f.x 16 11.c even 5 2
5929.2.a.bs 8 7.b odd 2 1
5929.2.a.bt 8 77.b even 2 1
7623.2.a.ct 8 33.d even 2 1
7623.2.a.cw 8 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\(T_{2}^{8} + \cdots\)
\(T_{3}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - 30 T - 30 T^{2} + 30 T^{3} + 35 T^{4} - 10 T^{5} - 11 T^{6} + T^{7} + T^{8} \)
$3$ \( 16 + 96 T + 64 T^{2} - 186 T^{3} + 15 T^{4} + 54 T^{5} - 11 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( -16 - 576 T + 680 T^{2} + 62 T^{3} - 285 T^{4} + 67 T^{5} + 22 T^{6} - 10 T^{7} + T^{8} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( T^{8} \)
$13$ \( -13744 + 10784 T + 2104 T^{2} - 3114 T^{3} + 195 T^{4} + 261 T^{5} - 36 T^{6} - 6 T^{7} + T^{8} \)
$17$ \( -1616 + 1832 T + 660 T^{2} - 1206 T^{3} + 65 T^{4} + 211 T^{5} - 37 T^{6} - 5 T^{7} + T^{8} \)
$19$ \( 7920 + 9960 T + 344 T^{2} - 3028 T^{3} - 415 T^{4} + 305 T^{5} + 15 T^{6} - 13 T^{7} + T^{8} \)
$23$ \( 859 + 1310 T - 1702 T^{2} - 3298 T^{3} - 765 T^{4} + 342 T^{5} + 40 T^{6} - 16 T^{7} + T^{8} \)
$29$ \( -495 + 3210 T - 5599 T^{2} + 3049 T^{3} + 310 T^{4} - 445 T^{5} - 45 T^{6} + 9 T^{7} + T^{8} \)
$31$ \( -51280 + 53800 T + 34100 T^{2} - 27410 T^{3} + 2135 T^{4} + 990 T^{5} - 106 T^{6} - 9 T^{7} + T^{8} \)
$37$ \( -461 + 2540 T - 2433 T^{2} - 2041 T^{3} + 1060 T^{4} + 261 T^{5} - 65 T^{6} - 7 T^{7} + T^{8} \)
$41$ \( 3664 - 18552 T + 25420 T^{2} - 11214 T^{3} + 305 T^{4} + 694 T^{5} - 67 T^{6} - 10 T^{7} + T^{8} \)
$43$ \( -971 + 8740 T - 10617 T^{2} - 812 T^{3} + 2780 T^{4} + 268 T^{5} - 105 T^{6} - 4 T^{7} + T^{8} \)
$47$ \( -2312080 + 522080 T + 221680 T^{2} - 59580 T^{3} - 4355 T^{4} + 1945 T^{5} - 56 T^{6} - 16 T^{7} + T^{8} \)
$53$ \( 557531 - 944112 T + 116873 T^{2} + 80581 T^{3} - 16950 T^{4} - 769 T^{5} + 433 T^{6} - 37 T^{7} + T^{8} \)
$59$ \( -13680 - 51000 T - 35684 T^{2} + 7274 T^{3} + 5385 T^{4} - 50 T^{5} - 140 T^{6} - T^{7} + T^{8} \)
$61$ \( 15920 - 27640 T + 1920 T^{2} + 12220 T^{3} - 1585 T^{4} - 1070 T^{5} + 4 T^{6} + 19 T^{7} + T^{8} \)
$67$ \( -27395 - 35255 T + 5380 T^{2} + 12660 T^{3} - 255 T^{4} - 1160 T^{5} - 16 T^{6} + 19 T^{7} + T^{8} \)
$71$ \( -19471 - 151555 T + 130762 T^{2} - 13964 T^{3} - 10945 T^{4} + 3004 T^{5} - 170 T^{6} - 13 T^{7} + T^{8} \)
$73$ \( 324144 + 744504 T + 303200 T^{2} - 89828 T^{3} - 11135 T^{4} + 2812 T^{5} + 32 T^{6} - 25 T^{7} + T^{8} \)
$79$ \( -3982275 - 2765925 T - 306995 T^{2} + 121275 T^{3} + 22330 T^{4} - 1005 T^{5} - 295 T^{6} + T^{8} \)
$83$ \( 27504 + 46944 T + 15080 T^{2} - 7078 T^{3} - 2785 T^{4} + 542 T^{5} + 122 T^{6} - 25 T^{7} + T^{8} \)
$89$ \( 952400 - 1415520 T + 512284 T^{2} + 45228 T^{3} - 34845 T^{4} + 2150 T^{5} + 320 T^{6} - 37 T^{7} + T^{8} \)
$97$ \( -2155696 + 977888 T + 283960 T^{2} - 165414 T^{3} + 7665 T^{4} + 3144 T^{5} - 222 T^{6} - 15 T^{7} + T^{8} \)
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