Properties

Label 546.2.s.e
Level $546$
Weight $2$
Character orbit 546.s
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.195105024.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 4 x^{5} - 20 x^{4} + 12 x^{3} + 45 x^{2} - 108 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{4} ) q^{3} -\beta_{4} q^{4} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} -\beta_{6} q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{6} ) q^{8} + \beta_{4} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 1 + \beta_{4} ) q^{3} -\beta_{4} q^{4} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} -\beta_{6} q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{6} ) q^{8} + \beta_{4} q^{9} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{10} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + q^{12} + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{13} + q^{14} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{15} + ( -1 - \beta_{4} ) q^{16} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{2} - \beta_{6} ) q^{18} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{19} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( -\beta_{2} - \beta_{6} ) q^{21} + ( \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{22} + ( 3 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{23} + \beta_{2} q^{24} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{26} - q^{27} + \beta_{2} q^{28} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{6} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} ) q^{30} + ( -3 - 2 \beta_{2} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{31} + \beta_{6} q^{32} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} ) q^{35} + ( 1 + \beta_{4} ) q^{36} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{38} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{39} + ( 2 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{40} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( 1 + \beta_{4} ) q^{42} + ( -5 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{43} + ( -1 - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{44} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{45} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{46} + ( 5 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} ) q^{47} -\beta_{4} q^{48} + ( 1 + \beta_{4} ) q^{49} + ( 4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{50} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{51} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{52} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{53} -\beta_{2} q^{54} + ( -5 + \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{55} -\beta_{4} q^{56} + ( 1 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{57} + ( -4 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{58} + ( 3 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{60} + ( 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{61} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{62} -\beta_{2} q^{63} - q^{64} + ( 7 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} ) q^{66} + ( 1 - \beta_{1} - 7 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{68} + ( 2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{69} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{70} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{71} -\beta_{6} q^{72} + ( 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} ) q^{73} + ( -1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{74} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{75} + ( -3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{76} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} ) q^{77} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{78} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{80} + ( -1 - \beta_{4} ) q^{81} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{82} + ( -4 + \beta_{1} - 5 \beta_{2} + \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{83} -\beta_{6} q^{84} + ( -9 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{85} + ( 3 - 2 \beta_{2} + 6 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{86} + ( -1 + \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{88} + ( -2 \beta_{1} - 9 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{89} + ( -2 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{90} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{92} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{93} + ( -2 + 5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( -7 + 7 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + 15 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} ) q^{95} + ( \beta_{2} + \beta_{6} ) q^{96} + ( -9 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{97} -\beta_{6} q^{98} + ( 1 + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 4q^{4} - 4q^{9} + O(q^{10}) \) \( 8q + 4q^{3} + 4q^{4} - 4q^{9} + 6q^{10} + 6q^{11} + 8q^{12} + 12q^{13} + 8q^{14} + 6q^{15} - 4q^{16} + 2q^{17} - 12q^{19} - 6q^{20} - 4q^{22} + 8q^{23} - 24q^{25} + 6q^{26} - 8q^{27} + 2q^{29} - 6q^{30} + 6q^{33} - 6q^{35} + 4q^{36} + 18q^{37} - 4q^{38} + 12q^{40} + 12q^{41} + 4q^{42} - 8q^{43} + 6q^{45} + 18q^{46} + 4q^{48} + 4q^{49} + 12q^{50} + 4q^{51} + 12q^{52} - 12q^{53} - 22q^{55} + 4q^{56} - 24q^{58} + 18q^{59} - 8q^{61} + 8q^{62} - 8q^{64} + 46q^{65} - 8q^{66} + 18q^{67} - 2q^{68} - 8q^{69} + 6q^{71} - 6q^{74} - 12q^{75} - 12q^{76} - 8q^{77} + 6q^{78} - 4q^{79} - 6q^{80} - 4q^{81} + 10q^{82} - 54q^{85} - 2q^{87} + 4q^{88} - 18q^{89} - 12q^{90} + 6q^{91} + 16q^{92} + 30q^{93} - 2q^{94} - 50q^{95} - 54q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 4 x^{5} - 20 x^{4} + 12 x^{3} + 45 x^{2} - 108 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 2 \nu^{5} + 10 \nu^{4} + 10 \nu^{3} + 42 \nu^{2} - 45 \nu - 27 \)\()/216\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 2 \nu^{5} - 10 \nu^{4} - 10 \nu^{3} + 30 \nu^{2} - 27 \nu + 27 \)\()/72\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 8 \nu^{6} - 4 \nu^{5} - 26 \nu^{4} + 52 \nu^{3} + 18 \nu^{2} - 153 \nu + 162 \)\()/108\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{7} + 13 \nu^{6} - 2 \nu^{5} - 22 \nu^{4} + 26 \nu^{3} + 54 \nu^{2} - 225 \nu + 243 \)\()/108\)
\(\beta_{6}\)\(=\)\((\)\( 19 \nu^{7} - 49 \nu^{6} + 14 \nu^{5} + 130 \nu^{4} - 218 \nu^{3} - 150 \nu^{2} + 801 \nu - 891 \)\()/216\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 14 \nu^{6} - 10 \nu^{5} - 32 \nu^{4} + 58 \nu^{3} + 24 \nu^{2} - 207 \nu + 270 \)\()/36\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + 6 \beta_{2} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(-5 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 3\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} - 24 \beta_{4} - 8 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} - 15\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} - 36 \beta_{4} + 6 \beta_{2} - 9 \beta_{1} + 6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.560908 + 1.63871i
1.30512 1.13871i
−1.58726 0.693255i
1.72124 + 0.193255i
1.30512 + 1.13871i
0.560908 1.63871i
1.72124 0.193255i
−1.58726 + 0.693255i
−0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0.332808i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i 0.166404 0.288220i
43.2 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 4.39924i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i 2.19962 3.80986i
43.3 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 3.05596i 0.866025 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i 1.52798 2.64654i
43.4 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.78801i 0.866025 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i −0.894007 + 1.54846i
127.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 4.39924i −0.866025 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i 2.19962 + 3.80986i
127.2 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0.332808i −0.866025 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i 0.166404 + 0.288220i
127.3 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.78801i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i −0.894007 1.54846i
127.4 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 3.05596i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i 1.52798 + 2.64654i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.s.e 8
3.b odd 2 1 1638.2.bj.f 8
13.e even 6 1 inner 546.2.s.e 8
13.f odd 12 1 7098.2.a.cn 4
13.f odd 12 1 7098.2.a.co 4
39.h odd 6 1 1638.2.bj.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.e 8 1.a even 1 1 trivial
546.2.s.e 8 13.e even 6 1 inner
1638.2.bj.f 8 3.b odd 2 1
1638.2.bj.f 8 39.h odd 6 1
7098.2.a.cn 4 13.f odd 12 1
7098.2.a.co 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{8} + 32 T_{5}^{6} + 276 T_{5}^{4} + 608 T_{5}^{2} + 64 \)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{4} \)
$5$ \( 64 + 608 T^{2} + 276 T^{4} + 32 T^{6} + T^{8} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$11$ \( 1024 - 2304 T + 2016 T^{2} - 648 T^{3} - 31 T^{4} + 54 T^{5} + 3 T^{6} - 6 T^{7} + T^{8} \)
$13$ \( 28561 - 26364 T + 11154 T^{2} - 3120 T^{3} + 815 T^{4} - 240 T^{5} + 66 T^{6} - 12 T^{7} + T^{8} \)
$17$ \( 37249 + 4246 T + 6274 T^{2} + 112 T^{3} + 751 T^{4} + 16 T^{5} + 34 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( 234256 - 191664 T + 30492 T^{2} + 17820 T^{3} - 43 T^{4} - 540 T^{5} + 3 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( 183184 - 99296 T + 47404 T^{2} - 10328 T^{3} + 2509 T^{4} - 344 T^{5} + 79 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( 4096 - 1024 T + 2240 T^{2} + 752 T^{3} + 865 T^{4} + 94 T^{5} + 35 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 47524 + 53616 T^{2} + 4253 T^{4} + 114 T^{6} + T^{8} \)
$37$ \( 5184 - 15552 T + 15120 T^{2} + 1296 T^{3} - 1188 T^{4} - 108 T^{5} + 114 T^{6} - 18 T^{7} + T^{8} \)
$41$ \( 256 - 384 T - 272 T^{2} + 696 T^{3} + 921 T^{4} + 348 T^{5} + 19 T^{6} - 12 T^{7} + T^{8} \)
$43$ \( 20164 - 10508 T + 9026 T^{2} - 422 T^{3} + 1075 T^{4} - 52 T^{5} + 89 T^{6} + 8 T^{7} + T^{8} \)
$47$ \( 59783824 + 2914872 T^{2} + 51041 T^{4} + 378 T^{6} + T^{8} \)
$53$ \( ( 2461 - 426 T - 106 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$59$ \( 4477456 + 2336064 T + 234876 T^{2} - 89424 T^{3} - 2179 T^{4} + 1458 T^{5} + 27 T^{6} - 18 T^{7} + T^{8} \)
$61$ \( 2283121 + 1680232 T + 1058246 T^{2} + 155392 T^{3} + 24331 T^{4} + 1280 T^{5} + 182 T^{6} + 8 T^{7} + T^{8} \)
$67$ \( 11916304 - 8201952 T + 2258060 T^{2} - 258984 T^{3} + 1077 T^{4} + 1962 T^{5} - T^{6} - 18 T^{7} + T^{8} \)
$71$ \( 324 + 1620 T + 3402 T^{2} + 3510 T^{3} + 1719 T^{4} + 234 T^{5} - 27 T^{6} - 6 T^{7} + T^{8} \)
$73$ \( 65536 + 368640 T^{2} + 22592 T^{4} + 288 T^{6} + T^{8} \)
$79$ \( ( -104 - 148 T - 39 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$83$ \( 22146436 + 1924816 T^{2} + 46077 T^{4} + 382 T^{6} + T^{8} \)
$89$ \( 20584369 - 7268274 T + 84178 T^{2} + 272340 T^{3} + 14751 T^{4} - 3060 T^{5} - 62 T^{6} + 18 T^{7} + T^{8} \)
$97$ \( 1290496 - 1063296 T - 19232 T^{2} + 256464 T^{3} + 93060 T^{4} + 14796 T^{5} + 1246 T^{6} + 54 T^{7} + T^{8} \)
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