Properties

Label 5054.2.a.bg
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \(x^{8} - 2 x^{7} - 12 x^{6} + 16 x^{5} + 50 x^{4} - 24 x^{3} - 72 x^{2} - 32 x - 4\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( 1 + \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{5} + ( -1 - \beta_{2} - \beta_{5} ) q^{6} + q^{7} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 + \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{5} + ( -1 - \beta_{2} - \beta_{5} ) q^{6} + q^{7} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} ) q^{10} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{2} + \beta_{5} ) q^{12} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} - q^{14} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{15} + q^{16} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{17} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{18} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{20} + ( 1 + \beta_{2} + \beta_{5} ) q^{21} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{22} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{23} + ( -1 - \beta_{2} - \beta_{5} ) q^{24} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 6 \beta_{6} + \beta_{7} ) q^{25} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{26} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{27} + q^{28} + ( 1 - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{30} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} - q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{33} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{34} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{35} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{36} + ( 5 + 3 \beta_{1} + \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{37} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{39} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} ) q^{40} + ( -1 + 3 \beta_{2} + \beta_{5} - 4 \beta_{6} ) q^{41} + ( -1 - \beta_{2} - \beta_{5} ) q^{42} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{43} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{44} + ( -3 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{45} + ( 3 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{46} + ( 2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 7 \beta_{6} ) q^{47} + ( 1 + \beta_{2} + \beta_{5} ) q^{48} + q^{49} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{6} - \beta_{7} ) q^{50} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{51} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{52} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - 4 \beta_{7} ) q^{54} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{55} - q^{56} + ( -1 + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{58} + ( -1 + \beta_{1} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{60} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{61} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{62} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{63} + q^{64} + ( 2 - 3 \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{65} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{66} + ( 7 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{68} + ( -1 - 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{69} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} ) q^{70} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{71} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{72} + ( 6 + 3 \beta_{1} - \beta_{2} + 4 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{73} + ( -5 - 3 \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{74} + ( -3 - 8 \beta_{1} + 6 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 7 \beta_{6} + 6 \beta_{7} ) q^{75} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{77} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{78} + ( 6 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{79} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} ) q^{80} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{81} + ( 1 - 3 \beta_{2} - \beta_{5} + 4 \beta_{6} ) q^{82} + ( -5 + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{83} + ( 1 + \beta_{2} + \beta_{5} ) q^{84} + ( -2 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{85} + ( -\beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{86} + ( 4 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{87} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{88} + ( 2 + 2 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 3 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{90} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{92} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{93} + ( -2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 7 \beta_{6} ) q^{94} + ( -1 - \beta_{2} - \beta_{5} ) q^{96} + ( -1 - \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{97} - q^{98} + ( -6 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + O(q^{10}) \) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 12 x^{6} + 16 x^{5} + 50 x^{4} - 24 x^{3} - 72 x^{2} - 32 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 8 \nu + 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 18 \nu - 4 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 10 \nu^{2} + 16 \nu - 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 8 \nu^{2} + 18 \nu + 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - 12 \nu^{5} + 17 \nu^{4} + 48 \nu^{3} - 32 \nu^{2} - 64 \nu - 16 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + 5 \nu^{6} + 22 \nu^{5} - 43 \nu^{4} - 84 \nu^{3} + 86 \nu^{2} + 116 \nu + 24 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(-14 \beta_{5} + 16 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} + 42 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} + 4 \beta_{6} - 66 \beta_{5} + 70 \beta_{4} + 30 \beta_{3} + 56 \beta_{2} + 92 \beta_{1} + 102\)
\(\nu^{7}\)\(=\)\(4 \beta_{7} + 10 \beta_{6} - 148 \beta_{5} + 180 \beta_{4} + 122 \beta_{3} + 188 \beta_{2} + 326 \beta_{1} + 188\)

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
2.35877
−0.761439
−2.15124
−1.91631
3.02989
−0.368905
−0.236286
2.04552
−1.00000 −2.35992 1.00000 −4.10066 2.35992 1.00000 −1.00000 2.56920 4.10066
1.2 −1.00000 −1.40760 1.00000 −2.26420 1.40760 1.00000 −1.00000 −1.01865 2.26420
1.3 −1.00000 −1.30521 1.00000 −0.772004 1.30521 1.00000 −1.00000 −1.29642 0.772004
1.4 −1.00000 0.282233 1.00000 2.81658 −0.282233 1.00000 −1.00000 −2.92034 −2.81658
1.5 −1.00000 1.02954 1.00000 −1.38232 −1.02954 1.00000 −1.00000 −1.94005 1.38232
1.6 −1.00000 1.57867 1.00000 0.329540 −1.57867 1.00000 −1.00000 −0.507804 −0.329540
1.7 −1.00000 3.04815 1.00000 3.90247 −3.04815 1.00000 −1.00000 6.29119 −3.90247
1.8 −1.00000 3.13415 1.00000 −0.529405 −3.13415 1.00000 −1.00000 6.82288 0.529405
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 5054.2.a.bg 8
19.b odd 2 1 5054.2.a.bh yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.bg 8 1.a even 1 1 trivial
5054.2.a.bh yes 8 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\(T_{3}^{8} - \cdots\)
\(T_{5}^{8} + \cdots\)
\(T_{13}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( -19 + 74 T + 2 T^{2} - 98 T^{3} + 15 T^{4} + 38 T^{5} - 8 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( -19 - 18 T + 138 T^{2} + 254 T^{3} + 90 T^{4} - 46 T^{5} - 22 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( -1 + T )^{8} \)
$11$ \( -779 + 3212 T + 3268 T^{2} + 4 T^{3} - 710 T^{4} - 156 T^{5} + 28 T^{6} + 12 T^{7} + T^{8} \)
$13$ \( 905 - 2960 T + 2600 T^{2} - 290 T^{3} - 505 T^{4} + 170 T^{5} + 10 T^{6} - 10 T^{7} + T^{8} \)
$17$ \( -24719 - 1556 T + 19052 T^{2} + 8822 T^{3} + 295 T^{4} - 462 T^{5} - 58 T^{6} + 6 T^{7} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( -5795 + 8380 T + 4320 T^{2} - 3940 T^{3} - 1910 T^{4} + 20 T^{5} + 120 T^{6} + 20 T^{7} + T^{8} \)
$29$ \( -2179 + 10792 T + 948 T^{2} - 5736 T^{3} + 690 T^{4} + 424 T^{5} - 52 T^{6} - 8 T^{7} + T^{8} \)
$31$ \( 4541 - 9258 T + 3238 T^{2} + 2394 T^{3} - 1630 T^{4} + 154 T^{5} + 78 T^{6} - 18 T^{7} + T^{8} \)
$37$ \( -396379 + 895876 T - 648248 T^{2} + 193748 T^{3} - 21470 T^{4} - 948 T^{5} + 432 T^{6} - 36 T^{7} + T^{8} \)
$41$ \( -9559 - 17886 T + 7402 T^{2} + 17662 T^{3} + 5855 T^{4} + 78 T^{5} - 148 T^{6} - 4 T^{7} + T^{8} \)
$43$ \( 2944801 - 2593096 T - 234638 T^{2} + 159352 T^{3} + 8455 T^{4} - 2872 T^{5} - 158 T^{6} + 16 T^{7} + T^{8} \)
$47$ \( 8597405 - 2495090 T - 820830 T^{2} + 152430 T^{3} + 23210 T^{4} - 2390 T^{5} - 270 T^{6} + 10 T^{7} + T^{8} \)
$53$ \( -48299 + 468648 T + 292448 T^{2} - 12624 T^{3} - 17470 T^{4} + 1256 T^{5} + 248 T^{6} - 32 T^{7} + T^{8} \)
$59$ \( 29921 - 22338 T - 27882 T^{2} + 4054 T^{3} + 3350 T^{4} - 206 T^{5} - 122 T^{6} + 2 T^{7} + T^{8} \)
$61$ \( 7421 + 9638 T - 2202 T^{2} - 3414 T^{3} + 610 T^{4} + 346 T^{5} - 82 T^{6} - 2 T^{7} + T^{8} \)
$67$ \( -51619 + 238284 T - 324498 T^{2} + 155152 T^{3} - 13985 T^{4} - 3072 T^{5} + 662 T^{6} - 44 T^{7} + T^{8} \)
$71$ \( 17936 + 24832 T - 40832 T^{2} - 8416 T^{3} + 3960 T^{4} + 544 T^{5} - 112 T^{6} - 8 T^{7} + T^{8} \)
$73$ \( 27334525 + 14084100 T + 564300 T^{2} - 461650 T^{3} - 17665 T^{4} + 6270 T^{5} - 10 T^{6} - 30 T^{7} + T^{8} \)
$79$ \( -6438695 + 3898520 T - 206650 T^{2} - 375860 T^{3} + 125995 T^{4} - 18660 T^{5} + 1470 T^{6} - 60 T^{7} + T^{8} \)
$83$ \( -23760304 - 3954912 T + 1543488 T^{2} + 284096 T^{3} - 22020 T^{4} - 5264 T^{5} - 12 T^{6} + 28 T^{7} + T^{8} \)
$89$ \( 580621 - 3478988 T + 1255068 T^{2} + 396734 T^{3} - 15385 T^{4} - 7246 T^{5} - 222 T^{6} + 22 T^{7} + T^{8} \)
$97$ \( -1404719 + 1126874 T + 1442622 T^{2} + 378162 T^{3} + 18750 T^{4} - 3962 T^{5} - 378 T^{6} + 6 T^{7} + T^{8} \)
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