Properties

Label 385.2.i.a
Level $385$
Weight $2$
Character orbit 385.i
Analytic conductor $3.074$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.07424047782\)
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.

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

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 \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−0.198169 + 0.343239i
0.882007 1.52768i
0.346911 0.600868i
−1.03075 + 1.78531i
−0.198169 0.343239i
0.882007 + 1.52768i
0.346911 + 0.600868i
−1.03075 1.78531i
−0.563379 0.975800i −0.761548 + 1.31904i 0.365209 0.632561i −0.500000 0.866025i 1.71616 −0.779537 + 2.52830i −3.07652 0.340090 + 0.589053i −0.563379 + 0.975800i
221.2 −0.0985631 0.170716i 0.783444 1.35697i 0.980571 1.69840i −0.500000 0.866025i −0.308875 1.71031 2.01862i −0.780845 0.272430 + 0.471863i −0.0985631 + 0.170716i
221.3 0.873734 + 1.51335i 1.22065 2.11422i −0.526823 + 0.912484i −0.500000 0.866025i 4.26608 −1.89234 1.84906i 1.65372 −1.47995 2.56335i 0.873734 1.51335i
221.4 1.28821 + 2.23124i 0.257458 0.445930i −2.31896 + 4.01655i −0.500000 0.866025i 1.32664 1.46157 + 2.20541i −6.79636 1.36743 + 2.36846i 1.28821 2.23124i
331.1 −0.563379 + 0.975800i −0.761548 1.31904i 0.365209 + 0.632561i −0.500000 + 0.866025i 1.71616 −0.779537 2.52830i −3.07652 0.340090 0.589053i −0.563379 0.975800i
331.2 −0.0985631 + 0.170716i 0.783444 + 1.35697i 0.980571 + 1.69840i −0.500000 + 0.866025i −0.308875 1.71031 + 2.01862i −0.780845 0.272430 0.471863i −0.0985631 0.170716i
331.3 0.873734 1.51335i 1.22065 + 2.11422i −0.526823 0.912484i −0.500000 + 0.866025i 4.26608 −1.89234 + 1.84906i 1.65372 −1.47995 + 2.56335i 0.873734 + 1.51335i
331.4 1.28821 2.23124i 0.257458 + 0.445930i −2.31896 4.01655i −0.500000 + 0.866025i 1.32664 1.46157 2.20541i −6.79636 1.36743 2.36846i 1.28821 + 2.23124i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.4
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 385.2.i.a 8
7.c even 3 1 inner 385.2.i.a 8
7.c even 3 1 2695.2.a.j 4
7.d odd 6 1 2695.2.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.a 8 1.a even 1 1 trivial
385.2.i.a 8 7.c even 3 1 inner
2695.2.a.j 4 7.c even 3 1
2695.2.a.k 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 26 T^{2} + T^{3} + 15 T^{4} - 7 T^{5} + 10 T^{6} - 3 T^{7} + T^{8} \)
$3$ \( 9 - 21 T + 46 T^{2} - 25 T^{3} + 25 T^{4} - 11 T^{5} + 10 T^{6} - 3 T^{7} + T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{4} \)
$7$ \( 2401 - 343 T + 490 T^{2} - 35 T^{3} + 101 T^{4} - 5 T^{5} + 10 T^{6} - T^{7} + T^{8} \)
$11$ \( ( 1 + T + T^{2} )^{4} \)
$13$ \( ( 3 - 22 T - 4 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$17$ \( 441 - 609 T + 694 T^{2} - 329 T^{3} + 157 T^{4} - 37 T^{5} + 16 T^{6} - 3 T^{7} + T^{8} \)
$19$ \( 139129 + 20888 T + 18056 T^{2} - 2 T^{3} + 1395 T^{4} + 8 T^{5} + 49 T^{6} - 3 T^{7} + T^{8} \)
$23$ \( 16129 - 13843 T + 10103 T^{2} - 3050 T^{3} + 977 T^{4} - 134 T^{5} + 50 T^{6} - 6 T^{7} + T^{8} \)
$29$ \( ( 43 - 42 T - 9 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$31$ \( 1849 + 4859 T + 12167 T^{2} + 2442 T^{3} + 1369 T^{4} + 86 T^{5} + 114 T^{6} + 10 T^{7} + T^{8} \)
$37$ \( 385641 - 207414 T + 97894 T^{2} - 18526 T^{3} + 4111 T^{4} - 470 T^{5} + 103 T^{6} - 9 T^{7} + T^{8} \)
$41$ \( ( -557 + 353 T + 21 T^{2} - 15 T^{3} + T^{4} )^{2} \)
$43$ \( ( 2851 - 84 T - 127 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$47$ \( 90601 - 1505 T + 18386 T^{2} - 8725 T^{3} + 4097 T^{4} - 925 T^{5} + 164 T^{6} - 15 T^{7} + T^{8} \)
$53$ \( 1671849 - 1492122 T + 939937 T^{2} - 272082 T^{3} + 55896 T^{4} - 6782 T^{5} + 597 T^{6} - 30 T^{7} + T^{8} \)
$59$ \( 2712609 + 1609119 T + 919942 T^{2} + 76515 T^{3} + 18697 T^{4} + 1597 T^{5} + 310 T^{6} + 17 T^{7} + T^{8} \)
$61$ \( 3249 - 1083 T + 2983 T^{2} + 874 T^{3} + 2059 T^{4} + 38 T^{5} + 46 T^{6} + T^{8} \)
$67$ \( 1103369089 - 177378780 T + 26589014 T^{2} - 1970570 T^{3} + 170081 T^{4} - 9230 T^{5} + 683 T^{6} - 25 T^{7} + T^{8} \)
$71$ \( ( 717 - 385 T - 29 T^{2} + 13 T^{3} + T^{4} )^{2} \)
$73$ \( 9 + 66 T + 712 T^{2} - 1690 T^{3} + 5707 T^{4} - 272 T^{5} + 85 T^{6} + 3 T^{7} + T^{8} \)
$79$ \( 24649 - 12717 T + 27285 T^{2} + 11948 T^{3} + 16943 T^{4} + 690 T^{5} + 148 T^{6} - 4 T^{7} + T^{8} \)
$83$ \( ( -7629 + 1741 T - 18 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$89$ \( 856908529 + 140305489 T + 21831202 T^{2} + 1650577 T^{3} + 150619 T^{4} + 8611 T^{5} + 664 T^{6} + 25 T^{7} + T^{8} \)
$97$ \( ( -881 + 140 T + 150 T^{2} + 23 T^{3} + T^{4} )^{2} \)
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