Properties

Label 4598.2.a.cb
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 15 \nu^{5} + 11 \nu^{4} + 64 \nu^{3} - 24 \nu^{2} - 66 \nu - 2 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 20 \nu^{3} + 44 \nu^{2} - 40 \nu - 18 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 19 \nu^{5} - 15 \nu^{4} + 108 \nu^{3} + 56 \nu^{2} - 166 \nu - 18 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - 2 \nu^{6} - 45 \nu^{5} + 14 \nu^{4} + 196 \nu^{3} + 8 \nu^{2} - 218 \nu - 32 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 3 \nu^{6} - 87 \nu^{5} - 53 \nu^{4} + 448 \nu^{3} + 232 \nu^{2} - 610 \nu - 98 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 3 \nu^{6} - 109 \nu^{5} + 13 \nu^{4} + 496 \nu^{3} + 72 \nu^{2} - 582 \nu - 78 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-9 \beta_{7} + \beta_{6} + 8 \beta_{5} + 6 \beta_{3} + 10 \beta_{2} + 18\)
\(\nu^{5}\)\(=\)\(-25 \beta_{7} + 3 \beta_{6} + 24 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} + 30 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(-83 \beta_{7} + 19 \beta_{6} + 68 \beta_{5} - 4 \beta_{4} + 34 \beta_{3} + 86 \beta_{2} + 124\)
\(\nu^{7}\)\(=\)\(-255 \beta_{7} + 53 \beta_{6} + 236 \beta_{5} + 52 \beta_{4} - 38 \beta_{3} + 8 \beta_{2} + 196 \beta_{1} + 116\)

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
−1.93433
−0.115899
2.55131
3.02984
−1.84109
1.24609
−0.182716
−2.75320
1.00000 −2.48823 1.00000 3.25915 −2.48823 4.58972 1.00000 3.19127 3.25915
1.2 1.00000 −2.11129 1.00000 −2.75070 −2.11129 3.66381 1.00000 1.45756 −2.75070
1.3 1.00000 −0.103249 1.00000 −3.91798 −0.103249 −4.45623 1.00000 −2.98934 −3.91798
1.4 1.00000 1.55594 1.00000 2.24033 1.55594 −2.27752 1.00000 −0.579065 2.24033
1.5 1.00000 1.67352 1.00000 −0.566489 1.67352 3.40033 1.00000 −0.199325 −0.566489
1.6 1.00000 2.79123 1.00000 4.02325 2.79123 0.274917 1.00000 4.79095 4.02325
1.7 1.00000 3.30349 1.00000 1.88260 3.30349 2.41197 1.00000 7.91304 1.88260
1.8 1.00000 3.37860 1.00000 −4.17017 3.37860 −3.60700 1.00000 8.41492 −4.17017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 4598.2.a.cb yes 8
11.b odd 2 1 4598.2.a.by 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.by 8 11.b odd 2 1
4598.2.a.cb yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\(T_{3}^{8} - \cdots\)
\( T_{5}^{8} - 38 T_{5}^{6} + 12 T_{5}^{5} + 470 T_{5}^{4} - 288 T_{5}^{3} - 1984 T_{5}^{2} + 1536 T_{5} + 1408 \)
\(T_{7}^{8} - \cdots\)
\(T_{13}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( -44 - 368 T + 557 T^{2} - 80 T^{3} - 183 T^{4} + 72 T^{5} + 9 T^{6} - 8 T^{7} + T^{8} \)
$5$ \( 1408 + 1536 T - 1984 T^{2} - 288 T^{3} + 470 T^{4} + 12 T^{5} - 38 T^{6} + T^{8} \)
$7$ \( -1388 + 5408 T - 867 T^{2} - 1716 T^{3} + 387 T^{4} + 152 T^{5} - 37 T^{6} - 4 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( -368 + 96 T + 1088 T^{2} - 72 T^{3} - 451 T^{4} - 84 T^{5} + 34 T^{6} + 12 T^{7} + T^{8} \)
$17$ \( -92 - 1480 T - 1912 T^{2} + 1076 T^{3} + 613 T^{4} - 128 T^{5} - 46 T^{6} + 4 T^{7} + T^{8} \)
$19$ \( ( 1 + T )^{8} \)
$23$ \( 35296 + 67256 T + 15805 T^{2} - 13070 T^{3} - 1713 T^{4} + 980 T^{5} - 29 T^{6} - 14 T^{7} + T^{8} \)
$29$ \( 964 + 92436 T - 62863 T^{2} + 26 T^{3} + 4833 T^{4} - 148 T^{5} - 123 T^{6} + 2 T^{7} + T^{8} \)
$31$ \( -9344 + 11648 T + 37888 T^{2} + 24096 T^{3} + 4264 T^{4} - 368 T^{5} - 132 T^{6} + T^{8} \)
$37$ \( 29569 - 23888 T - 55602 T^{2} + 49120 T^{3} - 13230 T^{4} + 952 T^{5} + 134 T^{6} - 24 T^{7} + T^{8} \)
$41$ \( -3872 - 40480 T + 50000 T^{2} - 14680 T^{3} - 2162 T^{4} + 1444 T^{5} - 130 T^{6} - 8 T^{7} + T^{8} \)
$43$ \( -67328 + 55936 T + 17568 T^{2} - 20128 T^{3} + 1736 T^{4} + 800 T^{5} - 96 T^{6} - 8 T^{7} + T^{8} \)
$47$ \( -2659283 + 1731808 T + 1385868 T^{2} + 218304 T^{3} - 8910 T^{4} - 4208 T^{5} - 172 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 2770816 - 1436784 T + 23497 T^{2} + 103740 T^{3} - 17347 T^{4} - 744 T^{5} + 413 T^{6} - 36 T^{7} + T^{8} \)
$59$ \( -20032604 - 2459400 T + 1154669 T^{2} + 175128 T^{3} - 17311 T^{4} - 3672 T^{5} + T^{6} + 24 T^{7} + T^{8} \)
$61$ \( 16192 + 90912 T + 109160 T^{2} - 54264 T^{3} - 2338 T^{4} + 2772 T^{5} - 194 T^{6} - 12 T^{7} + T^{8} \)
$67$ \( 457168 - 830496 T + 431552 T^{2} - 44968 T^{3} - 15135 T^{4} + 3464 T^{5} - 126 T^{6} - 16 T^{7} + T^{8} \)
$71$ \( 3806464 + 1199424 T - 456232 T^{2} - 94072 T^{3} + 21990 T^{4} + 1244 T^{5} - 282 T^{6} - 4 T^{7} + T^{8} \)
$73$ \( -37382204 - 12732376 T + 1400232 T^{2} + 610900 T^{3} + 6341 T^{4} - 6848 T^{5} - 270 T^{6} + 20 T^{7} + T^{8} \)
$79$ \( 51808 + 32864 T - 27728 T^{2} - 33528 T^{3} - 12530 T^{4} - 1988 T^{5} - 78 T^{6} + 12 T^{7} + T^{8} \)
$83$ \( -251648 + 356416 T + 393656 T^{2} - 211784 T^{3} - 15090 T^{4} + 6444 T^{5} - 234 T^{6} - 20 T^{7} + T^{8} \)
$89$ \( -320672 - 4640 T + 262448 T^{2} - 81848 T^{3} - 594 T^{4} + 2532 T^{5} - 210 T^{6} - 8 T^{7} + T^{8} \)
$97$ \( -1921664 + 2470912 T + 117760 T^{2} - 325888 T^{3} + 38568 T^{4} + 2896 T^{5} - 452 T^{6} - 4 T^{7} + T^{8} \)
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