Properties

Label 322.2.e.a
Level $322$
Weight $2$
Character orbit 322.e
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
Defining polynomial: \(x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -49 \nu^{7} + 64 \nu^{6} - 256 \nu^{5} + 338 \nu^{4} - 1088 \nu^{3} + 1216 \nu^{2} - 1385 \nu + 256 \)\()/289\)
\(\beta_{3}\)\(=\)\((\)\( 60 \nu^{7} + 16 \nu^{6} + 225 \nu^{5} - 60 \nu^{4} + 884 \nu^{3} + 15 \nu^{2} + 304 \nu + 64 \)\()/289\)
\(\beta_{4}\)\(=\)\((\)\( 63 \nu^{7} - 41 \nu^{6} + 164 \nu^{5} - 352 \nu^{4} + 697 \nu^{3} - 779 \nu^{2} - 490 \nu - 164 \)\()/289\)
\(\beta_{5}\)\(=\)\((\)\( -64 \nu^{7} + 60 \nu^{6} - 240 \nu^{5} + 353 \nu^{4} - 1020 \nu^{3} + 1140 \nu^{2} - 305 \nu + 240 \)\()/289\)
\(\beta_{6}\)\(=\)\((\)\( -164 \nu^{7} - 63 \nu^{6} - 615 \nu^{5} + 164 \nu^{4} - 2108 \nu^{3} - 41 \nu^{2} - 41 \nu + 37 \)\()/289\)
\(\beta_{7}\)\(=\)\((\)\( 180 \nu^{7} + 48 \nu^{6} + 675 \nu^{5} - 180 \nu^{4} + 2363 \nu^{3} + 45 \nu^{2} + 45 \nu + 481 \)\()/289\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} - \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 3 \beta_{3} - 3 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-7 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} + 5 \beta_{1}\)
\(\nu^{5}\)\(=\)\(5 \beta_{7} + \beta_{6} + 6 \beta_{5} + \beta_{4} - 11 \beta_{3} - 5 \beta_{2} - 5 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(-16 \beta_{7} - 15 \beta_{6} + 7 \beta_{3} - 7 \beta_{1} + 27\)
\(\nu^{7}\)\(=\)\(-30 \beta_{5} - 8 \beta_{4} + 23 \beta_{2} + 65 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−1.03075 1.78531i
0.346911 + 0.600868i
0.882007 + 1.52768i
−0.198169 0.343239i
−1.03075 + 1.78531i
0.346911 0.600868i
0.882007 1.52768i
−0.198169 + 0.343239i
−0.500000 + 0.866025i −1.28821 2.23124i −0.500000 0.866025i −0.663319 + 1.14890i 2.57641 −1.46157 + 2.20541i 1.00000 −1.81896 + 3.15053i −0.663319 1.14890i
93.2 −0.500000 + 0.866025i −0.873734 1.51335i −0.500000 0.866025i −2.13304 + 3.69453i 1.74747 1.89234 1.84906i 1.00000 −0.0268230 + 0.0464588i −2.13304 3.69453i
93.3 −0.500000 + 0.866025i 0.0985631 + 0.170716i −0.500000 0.866025i 0.154437 0.267494i −0.197126 −1.71031 2.01862i 1.00000 1.48057 2.56442i 0.154437 + 0.267494i
93.4 −0.500000 + 0.866025i 0.563379 + 0.975800i −0.500000 0.866025i −0.858079 + 1.48624i −1.12676 0.779537 + 2.52830i 1.00000 0.865209 1.49859i −0.858079 1.48624i
277.1 −0.500000 0.866025i −1.28821 + 2.23124i −0.500000 + 0.866025i −0.663319 1.14890i 2.57641 −1.46157 2.20541i 1.00000 −1.81896 3.15053i −0.663319 + 1.14890i
277.2 −0.500000 0.866025i −0.873734 + 1.51335i −0.500000 + 0.866025i −2.13304 3.69453i 1.74747 1.89234 + 1.84906i 1.00000 −0.0268230 0.0464588i −2.13304 + 3.69453i
277.3 −0.500000 0.866025i 0.0985631 0.170716i −0.500000 + 0.866025i 0.154437 + 0.267494i −0.197126 −1.71031 + 2.01862i 1.00000 1.48057 + 2.56442i 0.154437 0.267494i
277.4 −0.500000 0.866025i 0.563379 0.975800i −0.500000 + 0.866025i −0.858079 1.48624i −1.12676 0.779537 2.52830i 1.00000 0.865209 + 1.49859i −0.858079 + 1.48624i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.e.a 8
7.c even 3 1 inner 322.2.e.a 8
7.c even 3 1 2254.2.a.z 4
7.d odd 6 1 2254.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.a 8 1.a even 1 1 trivial
322.2.e.a 8 7.c even 3 1 inner
2254.2.a.x 4 7.d odd 6 1
2254.2.a.z 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( 1 - 5 T + 26 T^{2} - T^{3} + 15 T^{4} + 7 T^{5} + 10 T^{6} + 3 T^{7} + T^{8} \)
$5$ \( 9 - 15 T + 64 T^{2} + 107 T^{3} + 137 T^{4} + 81 T^{5} + 36 T^{6} + 7 T^{7} + T^{8} \)
$7$ \( 2401 + 343 T + 490 T^{2} + 35 T^{3} + 101 T^{4} + 5 T^{5} + 10 T^{6} + T^{7} + T^{8} \)
$11$ \( 81 + 261 T + 733 T^{2} + 384 T^{3} + 211 T^{4} + 34 T^{5} + 16 T^{6} + 2 T^{7} + T^{8} \)
$13$ \( ( 343 - 42 T^{2} - T^{3} + T^{4} )^{2} \)
$17$ \( 178929 + 112518 T + 55528 T^{2} + 13806 T^{3} + 3049 T^{4} + 352 T^{5} + 61 T^{6} + 5 T^{7} + T^{8} \)
$19$ \( 10201 - 28583 T + 81402 T^{2} + 1457 T^{3} + 3181 T^{4} + 423 T^{5} + 134 T^{6} + 11 T^{7} + T^{8} \)
$23$ \( ( 1 - T + T^{2} )^{4} \)
$29$ \( ( -21 + 32 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$31$ \( 499849 + 325927 T + 165859 T^{2} + 38910 T^{3} + 7829 T^{4} + 526 T^{5} + 102 T^{6} + 6 T^{7} + T^{8} \)
$37$ \( 1868689 - 906321 T + 357549 T^{2} - 61652 T^{3} + 10271 T^{4} - 846 T^{5} + 124 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( ( 417 + 194 T - 34 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$43$ \( ( 1021 + 285 T - 84 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$47$ \( 471969 - 172437 T + 87046 T^{2} - 6329 T^{3} + 3299 T^{4} + 117 T^{5} + 156 T^{6} + 11 T^{7} + T^{8} \)
$53$ \( 870489 + 566331 T + 270484 T^{2} + 65601 T^{3} + 12565 T^{4} + 1109 T^{5} + 106 T^{6} + T^{7} + T^{8} \)
$59$ \( 6561 + 6561 T + 8019 T^{2} + 486 T^{3} + 1377 T^{4} + 378 T^{5} + 126 T^{6} + 12 T^{7} + T^{8} \)
$61$ \( 155575729 + 33627208 T + 6894226 T^{2} + 604746 T^{3} + 69989 T^{4} + 4762 T^{5} + 471 T^{6} + 21 T^{7} + T^{8} \)
$67$ \( 33953929 + 2563880 T + 1370654 T^{2} - 53918 T^{3} + 36297 T^{4} - 274 T^{5} + 211 T^{6} - 3 T^{7} + T^{8} \)
$71$ \( ( -189 + 424 T - 76 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$73$ \( 2229049 - 828615 T + 311011 T^{2} - 46666 T^{3} + 10377 T^{4} - 1142 T^{5} + 254 T^{6} - 16 T^{7} + T^{8} \)
$79$ \( 2809 + 106 T + 5410 T^{2} - 2430 T^{3} + 10415 T^{4} - 2138 T^{5} + 339 T^{6} - 21 T^{7} + T^{8} \)
$83$ \( ( 5913 + 352 T - 155 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$89$ \( 15992001 + 3315171 T + 1235104 T^{2} + 102373 T^{3} + 45151 T^{4} + 5357 T^{5} + 592 T^{6} + 27 T^{7} + T^{8} \)
$97$ \( ( -23 + 194 T - 33 T^{2} - 6 T^{3} + T^{4} )^{2} \)
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