Properties

Label 420.2.l.f
Level $420$
Weight $2$
Character orbit 420.l
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
Defining polynomial: \(x^{8} - x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} - 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - \nu^{5} + 2 \nu^{4} + 2 \nu^{3} - 4 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - \nu^{5} - 2 \nu^{4} + 2 \nu^{3} - 4 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} - 2 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 4 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - \nu^{4} + 2 \nu^{3} + 2 \nu^{2} - 4 \nu - 8 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} - \nu^{5} - 2 \nu^{3} - 4 \nu^{2} - 12 \nu \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{3} - 4 \nu^{2} - 12 \nu - 24 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + \beta_{5} - \beta_{4} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{4} - 2 \beta_{3} + \beta_{2}\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_{1} + 4\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} + \beta_{6} - 3 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
1.40961 0.114062i
1.40961 + 0.114062i
0.621372 1.27039i
0.621372 + 1.27039i
−0.835949 1.14070i
−0.835949 + 1.14070i
−1.19503 0.756243i
−1.19503 + 0.756243i
−1.40961 0.114062i −1.47398 + 0.909606i 1.97398 + 0.321565i 1.00000 2.00000i 2.18148 1.11406i 1.00000 −2.74586 0.678435i 1.34523 2.68148i −1.63773 + 2.70515i
239.2 −1.40961 + 0.114062i −1.47398 0.909606i 1.97398 0.321565i 1.00000 + 2.00000i 2.18148 + 1.11406i 1.00000 −2.74586 + 0.678435i 1.34523 + 2.68148i −1.63773 2.70515i
239.3 −0.621372 1.27039i 1.72779 + 0.121372i −1.22779 + 1.57877i 1.00000 2.00000i −0.919412 2.27039i 1.00000 2.76858 + 0.578773i 2.97054 + 0.419412i −3.16216 0.0276478i
239.4 −0.621372 + 1.27039i 1.72779 0.121372i −1.22779 1.57877i 1.00000 + 2.00000i −0.919412 + 2.27039i 1.00000 2.76858 0.578773i 2.97054 0.419412i −3.16216 + 0.0276478i
239.5 0.835949 1.14070i 1.10238 + 1.33595i −0.602380 1.90713i 1.00000 + 2.00000i 2.44545 0.140697i 1.00000 −2.67901 0.907128i −0.569517 + 2.94545i 3.11734 + 0.531200i
239.6 0.835949 + 1.14070i 1.10238 1.33595i −0.602380 + 1.90713i 1.00000 2.00000i 2.44545 + 0.140697i 1.00000 −2.67901 + 0.907128i −0.569517 2.94545i 3.11734 0.531200i
239.7 1.19503 0.756243i −0.356193 1.69503i 0.856193 1.80747i 1.00000 2.00000i −1.70752 1.75624i 1.00000 −0.343707 2.80747i −2.74625 + 1.20752i −0.317456 3.14630i
239.8 1.19503 + 0.756243i −0.356193 + 1.69503i 0.856193 + 1.80747i 1.00000 + 2.00000i −1.70752 + 1.75624i 1.00000 −0.343707 + 2.80747i −2.74625 1.20752i −0.317456 + 3.14630i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.l.f yes 8
3.b odd 2 1 420.2.l.e yes 8
4.b odd 2 1 420.2.l.d yes 8
5.b even 2 1 420.2.l.c 8
12.b even 2 1 420.2.l.c 8
15.d odd 2 1 420.2.l.d yes 8
20.d odd 2 1 420.2.l.e yes 8
60.h even 2 1 inner 420.2.l.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.c 8 5.b even 2 1
420.2.l.c 8 12.b even 2 1
420.2.l.d yes 8 4.b odd 2 1
420.2.l.d yes 8 15.d odd 2 1
420.2.l.e yes 8 3.b odd 2 1
420.2.l.e yes 8 20.d odd 2 1
420.2.l.f yes 8 1.a even 1 1 trivial
420.2.l.f yes 8 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{4} - 2 T_{11}^{3} - 11 T_{11}^{2} + 16 T_{11} + 16 \)
\( T_{17}^{4} + 2 T_{17}^{3} - 35 T_{17}^{2} - 52 T_{17} + 100 \)
\( T_{43}^{4} + 4 T_{43}^{3} - 56 T_{43}^{2} - 192 T_{43} - 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + 4 T^{3} + 2 T^{4} + 2 T^{5} - T^{6} + T^{8} \)
$3$ \( 81 - 54 T + 9 T^{2} - 6 T^{3} + 4 T^{4} - 2 T^{5} + T^{6} - 2 T^{7} + T^{8} \)
$5$ \( ( 5 - 2 T + T^{2} )^{4} \)
$7$ \( ( -1 + T )^{8} \)
$11$ \( ( 16 + 16 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 6400 + 5664 T^{2} + 1321 T^{4} + 70 T^{6} + T^{8} \)
$17$ \( ( 100 - 52 T - 35 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$19$ \( 1024 + 2048 T^{2} + 1104 T^{4} + 88 T^{6} + T^{8} \)
$23$ \( 6400 + 18944 T^{2} + 2656 T^{4} + 96 T^{6} + T^{8} \)
$29$ \( 6400 + 5664 T^{2} + 1321 T^{4} + 70 T^{6} + T^{8} \)
$31$ \( 12544 + 209920 T^{2} + 11488 T^{4} + 192 T^{6} + T^{8} \)
$37$ \( 4096 + 25088 T^{2} + 3984 T^{4} + 136 T^{6} + T^{8} \)
$41$ \( 65536 + 38912 T^{2} + 3472 T^{4} + 104 T^{6} + T^{8} \)
$43$ \( ( -128 - 192 T - 56 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$47$ \( 16 + 280 T^{2} + 145 T^{4} + 22 T^{6} + T^{8} \)
$53$ \( ( 5392 - 256 T - 200 T^{2} + T^{4} )^{2} \)
$59$ \( ( -2240 + 1152 T - 140 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( ( -1312 - 1232 T - 148 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( ( -6848 - 2112 T - 124 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$71$ \( ( 8 + T )^{8} \)
$73$ \( ( 64 + T^{2} )^{4} \)
$79$ \( 753424 + 895192 T^{2} + 66289 T^{4} + 502 T^{6} + T^{8} \)
$83$ \( 29073664 + 2034432 T^{2} + 48224 T^{4} + 432 T^{6} + T^{8} \)
$89$ \( 6553600 + 712704 T^{2} + 26176 T^{4} + 352 T^{6} + T^{8} \)
$97$ \( 16777216 + 2859008 T^{2} + 128601 T^{4} + 694 T^{6} + T^{8} \)
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