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

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:

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

Atkin-Lehner signs

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

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

$p$ $F_p(T)$
$2$ \( 1 + T + 5 T^{2} + 4 T^{3} + 15 T^{4} + 14 T^{5} + 38 T^{6} + 30 T^{7} + 75 T^{8} + 60 T^{9} + 152 T^{10} + 112 T^{11} + 240 T^{12} + 128 T^{13} + 320 T^{14} + 128 T^{15} + 256 T^{16} \)
$3$ \( 1 - 4 T + 13 T^{2} - 30 T^{3} + 69 T^{4} - 132 T^{5} + 271 T^{6} - 498 T^{7} + 940 T^{8} - 1494 T^{9} + 2439 T^{10} - 3564 T^{11} + 5589 T^{12} - 7290 T^{13} + 9477 T^{14} - 8748 T^{15} + 6561 T^{16} \)
$5$ \( 1 - 10 T + 62 T^{2} - 283 T^{3} + 1075 T^{4} - 3513 T^{5} + 10230 T^{6} - 26646 T^{7} + 62784 T^{8} - 133230 T^{9} + 255750 T^{10} - 439125 T^{11} + 671875 T^{12} - 884375 T^{13} + 968750 T^{14} - 781250 T^{15} + 390625 T^{16} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( \)
$13$ \( 1 - 6 T + 68 T^{2} - 285 T^{3} + 2119 T^{4} - 7443 T^{5} + 44016 T^{6} - 130942 T^{7} + 656120 T^{8} - 1702246 T^{9} + 7438704 T^{10} - 16352271 T^{11} + 60520759 T^{12} - 105818505 T^{13} + 328223012 T^{14} - 376491102 T^{15} + 815730721 T^{16} \)
$17$ \( 1 - 5 T + 99 T^{2} - 384 T^{3} + 4383 T^{4} - 13616 T^{5} + 119813 T^{6} - 309659 T^{7} + 2344384 T^{8} - 5264203 T^{9} + 34625957 T^{10} - 66895408 T^{11} + 366072543 T^{12} - 545225088 T^{13} + 2389619331 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 13 T + 167 T^{2} - 1424 T^{3} + 11403 T^{4} - 72606 T^{5} + 434133 T^{6} - 2182431 T^{7} + 10302272 T^{8} - 41466189 T^{9} + 156722013 T^{10} - 498004554 T^{11} + 1486050363 T^{12} - 3525964976 T^{13} + 7856662127 T^{14} - 11620332607 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 926670 T^{6} - 5230592 T^{7} + 26816927 T^{8} - 120303616 T^{9} + 490208430 T^{10} - 1724209904 T^{11} + 5475648847 T^{12} - 14378790262 T^{13} + 33160039136 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 9 T + 187 T^{2} + 1382 T^{3} + 16028 T^{4} + 97473 T^{5} + 828470 T^{6} + 4208558 T^{7} + 28798593 T^{8} + 122048182 T^{9} + 696743270 T^{10} + 2377268997 T^{11} + 11336299868 T^{12} + 28346407918 T^{13} + 111231961027 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 9 T + 142 T^{2} - 963 T^{3} + 9327 T^{4} - 55589 T^{5} + 439146 T^{6} - 2365595 T^{7} + 15862880 T^{8} - 73333445 T^{9} + 422019306 T^{10} - 1656051899 T^{11} + 8613680367 T^{12} - 27569872413 T^{13} + 126025522702 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 7 T + 231 T^{2} - 1552 T^{3} + 24962 T^{4} - 154999 T^{5} + 1656240 T^{6} - 9060906 T^{7} + 73868707 T^{8} - 335253522 T^{9} + 2267392560 T^{10} - 7851164347 T^{11} + 46782806882 T^{12} - 107621821264 T^{13} + 592682800479 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 10 T + 261 T^{2} - 2176 T^{3} + 30891 T^{4} - 221954 T^{5} + 2245611 T^{6} - 13854084 T^{7} + 110613464 T^{8} - 568017444 T^{9} + 3774872091 T^{10} - 15297291634 T^{11} + 87290583051 T^{12} - 252103093376 T^{13} + 1239777206901 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 4 T + 239 T^{2} - 936 T^{3} + 27462 T^{4} - 98508 T^{5} + 2007760 T^{6} - 6271668 T^{7} + 102278657 T^{8} - 269681724 T^{9} + 3712348240 T^{10} - 7832075556 T^{11} + 93887113062 T^{12} - 137599902648 T^{13} + 1510805768711 T^{14} - 1087274444428 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 16 T + 320 T^{2} - 3319 T^{3} + 41705 T^{4} - 344729 T^{5} + 3361468 T^{6} - 23054530 T^{7} + 186100580 T^{8} - 1083562910 T^{9} + 7425482812 T^{10} - 35790798967 T^{11} + 203507096105 T^{12} - 761196078233 T^{13} + 3449348905280 T^{14} - 8105969927408 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 37 T + 857 T^{2} - 14496 T^{3} + 199396 T^{4} - 2305797 T^{5} + 23105040 T^{6} - 202528658 T^{7} + 1568879259 T^{8} - 10734018874 T^{9} + 64902057360 T^{10} - 343280139969 T^{11} + 1573330349476 T^{12} - 6062161866528 T^{13} + 18994857487553 T^{14} - 43464312173969 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - T + 332 T^{2} - 463 T^{3} + 53293 T^{4} - 80577 T^{5} + 5426300 T^{6} - 7692267 T^{7} + 381400788 T^{8} - 453843753 T^{9} + 18888950300 T^{10} - 16548823683 T^{11} + 645770519773 T^{12} - 331009950437 T^{13} + 14003937168812 T^{14} - 2488651484819 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 19 T + 492 T^{2} + 7043 T^{3} + 104067 T^{4} + 1170549 T^{5} + 12549376 T^{6} + 113336285 T^{7} + 952230800 T^{8} + 6913513385 T^{9} + 46696228096 T^{10} + 265692382569 T^{11} + 1440895135347 T^{12} + 5948491747943 T^{13} + 25348024185612 T^{14} + 59712113884399 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 13 T + 398 T^{2} - 3457 T^{3} + 57783 T^{4} - 323737 T^{5} + 4210848 T^{6} - 14543752 T^{7} + 249426533 T^{8} - 1032606392 T^{9} + 21226884768 T^{10} - 115869033407 T^{11} + 1468363163223 T^{12} - 6237220866407 T^{13} + 50983913000558 T^{14} - 118236562059083 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 25 T + 616 T^{2} - 9963 T^{3} + 152093 T^{4} - 1861173 T^{5} + 21394652 T^{6} - 209466223 T^{7} + 1925408604 T^{8} - 15291034279 T^{9} + 114012100508 T^{10} - 724027936941 T^{11} + 4319173668413 T^{12} - 20654012281059 T^{13} + 93221883394024 T^{14} - 276184962977425 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 337 T^{2} - 1005 T^{3} + 57248 T^{4} - 275700 T^{5} + 6743044 T^{6} - 36745800 T^{7} + 601257265 T^{8} - 2902918200 T^{9} + 42083337604 T^{10} - 135930852300 T^{11} + 2229814237088 T^{12} - 3092441680995 T^{13} + 81920472510577 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 - 25 T + 786 T^{2} - 13983 T^{3} + 250863 T^{4} - 3398873 T^{5} + 43717402 T^{6} - 464690723 T^{7} + 4604658344 T^{8} - 38569330009 T^{9} + 301169182378 T^{10} - 1943431396051 T^{11} + 11905536781023 T^{12} - 55079605311069 T^{13} + 256975133468034 T^{14} - 678401274740675 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 65606528 T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 65145596911 T^{9} + 519669308288 T^{10} - 3632450763191 T^{11} + 22450616901343 T^{12} - 116712426543549 T^{13} + 512884692271752 T^{14} - 1636559391134573 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 15 T + 554 T^{2} - 7041 T^{3} + 141913 T^{4} - 1604409 T^{5} + 23035698 T^{6} - 230491951 T^{7} + 2630434004 T^{8} - 22357719247 T^{9} + 216742882482 T^{10} - 1464300775257 T^{11} + 12563455854553 T^{12} - 60463462749537 T^{13} + 461466490730666 T^{14} - 1211974267171695 T^{15} + 7837433594376961 T^{16} \)
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