Properties

Label 507.2.f.f
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{2} + ( 1 - \beta_{2} - \beta_{4} ) q^{3} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( 3 - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{6} + ( 1 + \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( -2 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{2} + ( 1 - \beta_{2} - \beta_{4} ) q^{3} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{5} + ( 3 - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{6} + ( 1 + \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{8} + ( -2 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( -1 + \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{10} + ( 1 - \beta_{3} - 2 \beta_{7} ) q^{11} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{12} + ( -1 + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{14} + ( 3 - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{15} + ( 1 - 2 \beta_{2} + 2 \beta_{5} ) q^{16} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{18} + 2 \beta_{5} q^{19} + ( -4 - 3 \beta_{1} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{20} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{21} + ( -1 + 3 \beta_{2} - 3 \beta_{5} ) q^{22} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{24} + ( -1 + \beta_{2} + \beta_{5} ) q^{25} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -3 + 3 \beta_{3} + 2 \beta_{5} ) q^{28} + ( -1 + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( 4 + 5 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{30} + ( 3 - 3 \beta_{3} - 4 \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{32} + ( 2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{33} + ( -5 + \beta_{2} - 4 \beta_{3} ) q^{34} + ( -1 + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{35} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{36} + ( 2 - 5 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -3 - 4 \beta_{2} + 4 \beta_{5} ) q^{40} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{41} + ( 5 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{42} + ( 3 - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{5} ) q^{43} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{44} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{45} + ( -4 - 4 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{48} -5 \beta_{3} q^{49} + ( -1 + \beta_{3} + 2 \beta_{7} ) q^{50} + ( -\beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{51} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{53} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{54} + ( -1 + 3 \beta_{2} - 3 \beta_{5} ) q^{55} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{56} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{57} + ( -9 + 9 \beta_{3} + \beta_{5} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 4 + \beta_{1} + 6 \beta_{2} + 8 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{60} -7 q^{61} + ( 3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{62} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{63} + ( -5 + 5 \beta_{2} + 7 \beta_{3} + 5 \beta_{5} ) q^{64} + ( -1 + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{66} + ( 4 - 4 \beta_{3} - 6 \beta_{5} ) q^{67} + ( 3 - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{68} + ( -5 + 5 \beta_{3} + 2 \beta_{5} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{71} + ( 4 \beta_{1} + \beta_{2} - 5 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{72} + ( -2 - 3 \beta_{2} - 5 \beta_{3} ) q^{73} + ( 8 - 3 \beta_{2} - 5 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{74} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{75} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{77} + 2 q^{79} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{80} + ( 1 + 2 \beta_{2} - 4 \beta_{4} - 6 \beta_{5} ) q^{81} + ( -2 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{82} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{83} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{84} + ( -5 + \beta_{2} - 4 \beta_{3} ) q^{85} + ( 9 + 6 \beta_{1} - 3 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} ) q^{86} + ( 11 + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{87} + ( 1 - \beta_{2} + 8 \beta_{3} - \beta_{5} ) q^{88} + ( 5 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{7} ) q^{89} + ( 4 - 2 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} - 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{90} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{93} + ( -16 - 4 \beta_{2} + 4 \beta_{5} ) q^{94} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{95} + ( 3 - 3 \beta_{3} - \beta_{5} - \beta_{6} ) q^{96} + ( 7 - 7 \beta_{3} - 6 \beta_{5} ) q^{97} + ( 5 + 5 \beta_{1} - 5 \beta_{4} - 5 \beta_{5} ) q^{98} + ( 5 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 16q^{6} + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 4q^{3} + 16q^{6} + 8q^{7} - 8q^{9} + 16q^{15} + 8q^{16} + 4q^{18} + 8q^{19} + 4q^{21} - 8q^{22} + 12q^{24} + 4q^{27} - 16q^{28} + 8q^{31} + 4q^{33} - 36q^{34} - 4q^{37} - 24q^{40} + 32q^{42} + 4q^{45} + 28q^{48} - 8q^{54} - 8q^{55} + 16q^{57} - 68q^{58} + 44q^{60} - 56q^{61} - 8q^{63} - 40q^{66} + 8q^{67} - 32q^{70} - 36q^{72} - 28q^{73} - 8q^{76} + 16q^{79} - 8q^{81} + 4q^{84} - 36q^{85} + 68q^{87} - 20q^{93} - 128q^{94} + 16q^{96} + 32q^{97} + 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{7} - \nu^{6} - 25 \nu^{5} - 46 \nu^{4} - 5 \nu^{3} - 132 \nu^{2} + 28 \nu - 55 \)\()/37\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 41 \nu - 29 \)\()/37\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 7 \nu^{6} + 47 \nu^{5} - 100 \nu^{4} + 261 \nu^{3} - 295 \nu^{2} + 344 \nu - 126 \)\()/37\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 36 \nu^{6} - 136 \nu^{5} + 361 \nu^{4} - 634 \nu^{3} + 793 \nu^{2} - 601 \nu + 278 \)\()/37\)
\(\beta_{6}\)\(=\)\((\)\( -18 \nu^{7} + 63 \nu^{6} - 238 \nu^{5} + 419 \nu^{4} - 684 \nu^{3} + 546 \nu^{2} - 395 \nu + 61 \)\()/37\)
\(\beta_{7}\)\(=\)\((\)\( -18 \nu^{7} + 63 \nu^{6} - 238 \nu^{5} + 456 \nu^{4} - 758 \nu^{3} + 805 \nu^{2} - 617 \nu + 283 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + 3 \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{6} - 3 \beta_{5} - 12 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} - 8 \beta_{1} + 10\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{7} - \beta_{6} - 2 \beta_{5} + 11 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} - 25 \beta_{1} + 19\)\()/2\)
\(\nu^{6}\)\(=\)\(5 \beta_{7} + 12 \beta_{6} + 2 \beta_{5} + 32 \beta_{4} - 42 \beta_{3} + 7 \beta_{1} - 11\)
\(\nu^{7}\)\(=\)\((\)\(-46 \beta_{7} + 38 \beta_{6} + 17 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 140 \beta_{1} - 96\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\)

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
0.500000 2.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 0.564882i
0.500000 1.19293i
−1.69293 1.69293i −0.366025 + 1.69293i 3.73205i −1.69293 1.69293i 3.48568 2.24637i 1.00000 + 1.00000i 2.93225 2.93225i −2.73205 1.23931i 5.73205i
239.2 −1.06488 1.06488i 1.36603 + 1.06488i 0.267949i −1.06488 1.06488i −0.320682 2.58863i 1.00000 + 1.00000i −1.84443 + 1.84443i 0.732051 + 2.90931i 2.26795i
239.3 1.06488 + 1.06488i 1.36603 1.06488i 0.267949i 1.06488 + 1.06488i 2.58863 + 0.320682i 1.00000 + 1.00000i 1.84443 1.84443i 0.732051 2.90931i 2.26795i
239.4 1.69293 + 1.69293i −0.366025 1.69293i 3.73205i 1.69293 + 1.69293i 2.24637 3.48568i 1.00000 + 1.00000i −2.93225 + 2.93225i −2.73205 + 1.23931i 5.73205i
437.1 −1.69293 + 1.69293i −0.366025 1.69293i 3.73205i −1.69293 + 1.69293i 3.48568 + 2.24637i 1.00000 1.00000i 2.93225 + 2.93225i −2.73205 + 1.23931i 5.73205i
437.2 −1.06488 + 1.06488i 1.36603 1.06488i 0.267949i −1.06488 + 1.06488i −0.320682 + 2.58863i 1.00000 1.00000i −1.84443 1.84443i 0.732051 2.90931i 2.26795i
437.3 1.06488 1.06488i 1.36603 + 1.06488i 0.267949i 1.06488 1.06488i 2.58863 0.320682i 1.00000 1.00000i 1.84443 + 1.84443i 0.732051 + 2.90931i 2.26795i
437.4 1.69293 1.69293i −0.366025 + 1.69293i 3.73205i 1.69293 1.69293i 2.24637 + 3.48568i 1.00000 1.00000i −2.93225 2.93225i −2.73205 1.23931i 5.73205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 437.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.f 8
3.b odd 2 1 inner 507.2.f.f 8
13.b even 2 1 507.2.f.e 8
13.c even 3 1 39.2.k.b 8
13.c even 3 1 507.2.k.e 8
13.d odd 4 1 507.2.f.e 8
13.d odd 4 1 inner 507.2.f.f 8
13.e even 6 1 507.2.k.d 8
13.e even 6 1 507.2.k.f 8
13.f odd 12 1 39.2.k.b 8
13.f odd 12 1 507.2.k.d 8
13.f odd 12 1 507.2.k.e 8
13.f odd 12 1 507.2.k.f 8
39.d odd 2 1 507.2.f.e 8
39.f even 4 1 507.2.f.e 8
39.f even 4 1 inner 507.2.f.f 8
39.h odd 6 1 507.2.k.d 8
39.h odd 6 1 507.2.k.f 8
39.i odd 6 1 39.2.k.b 8
39.i odd 6 1 507.2.k.e 8
39.k even 12 1 39.2.k.b 8
39.k even 12 1 507.2.k.d 8
39.k even 12 1 507.2.k.e 8
39.k even 12 1 507.2.k.f 8
52.j odd 6 1 624.2.cn.c 8
52.l even 12 1 624.2.cn.c 8
65.n even 6 1 975.2.bo.d 8
65.o even 12 1 975.2.bp.f 8
65.q odd 12 1 975.2.bp.e 8
65.q odd 12 1 975.2.bp.f 8
65.s odd 12 1 975.2.bo.d 8
65.t even 12 1 975.2.bp.e 8
156.p even 6 1 624.2.cn.c 8
156.v odd 12 1 624.2.cn.c 8
195.x odd 6 1 975.2.bo.d 8
195.bc odd 12 1 975.2.bp.e 8
195.bh even 12 1 975.2.bo.d 8
195.bl even 12 1 975.2.bp.e 8
195.bl even 12 1 975.2.bp.f 8
195.bn odd 12 1 975.2.bp.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 13.c even 3 1
39.2.k.b 8 13.f odd 12 1
39.2.k.b 8 39.i odd 6 1
39.2.k.b 8 39.k even 12 1
507.2.f.e 8 13.b even 2 1
507.2.f.e 8 13.d odd 4 1
507.2.f.e 8 39.d odd 2 1
507.2.f.e 8 39.f even 4 1
507.2.f.f 8 1.a even 1 1 trivial
507.2.f.f 8 3.b odd 2 1 inner
507.2.f.f 8 13.d odd 4 1 inner
507.2.f.f 8 39.f even 4 1 inner
507.2.k.d 8 13.e even 6 1
507.2.k.d 8 13.f odd 12 1
507.2.k.d 8 39.h odd 6 1
507.2.k.d 8 39.k even 12 1
507.2.k.e 8 13.c even 3 1
507.2.k.e 8 13.f odd 12 1
507.2.k.e 8 39.i odd 6 1
507.2.k.e 8 39.k even 12 1
507.2.k.f 8 13.e even 6 1
507.2.k.f 8 13.f odd 12 1
507.2.k.f 8 39.h odd 6 1
507.2.k.f 8 39.k even 12 1
624.2.cn.c 8 52.j odd 6 1
624.2.cn.c 8 52.l even 12 1
624.2.cn.c 8 156.p even 6 1
624.2.cn.c 8 156.v odd 12 1
975.2.bo.d 8 65.n even 6 1
975.2.bo.d 8 65.s odd 12 1
975.2.bo.d 8 195.x odd 6 1
975.2.bo.d 8 195.bh even 12 1
975.2.bp.e 8 65.q odd 12 1
975.2.bp.e 8 65.t even 12 1
975.2.bp.e 8 195.bc odd 12 1
975.2.bp.e 8 195.bl even 12 1
975.2.bp.f 8 65.o even 12 1
975.2.bp.f 8 65.q odd 12 1
975.2.bp.f 8 195.bl even 12 1
975.2.bp.f 8 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} + 38 T_{2}^{4} + 169 \)
\( T_{5}^{8} + 38 T_{5}^{4} + 169 \)
\( T_{7}^{2} - 2 T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 169 + 38 T^{4} + T^{8} \)
$3$ \( ( 9 - 6 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$5$ \( 169 + 38 T^{4} + T^{8} \)
$7$ \( ( 2 - 2 T + T^{2} )^{4} \)
$11$ \( 2704 + 296 T^{4} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 117 - 30 T^{2} + T^{4} )^{2} \)
$19$ \( ( 16 + 16 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( ( 1573 + 82 T^{2} + T^{4} )^{2} \)
$31$ \( ( 484 + 88 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$37$ \( ( 1369 - 74 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$41$ \( 169 + 998 T^{4} + T^{8} \)
$43$ \( ( 324 + 72 T^{2} + T^{4} )^{2} \)
$47$ \( 11075584 + 9728 T^{4} + T^{8} \)
$53$ \( ( 13 + 22 T^{2} + T^{4} )^{2} \)
$59$ \( 43264 + 608 T^{4} + T^{8} \)
$61$ \( ( 7 + T )^{8} \)
$67$ \( ( 2704 + 208 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$71$ \( 43264 + 608 T^{4} + T^{8} \)
$73$ \( ( 121 + 154 T + 98 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$79$ \( ( -2 + T )^{8} \)
$83$ \( 2704 + 296 T^{4} + T^{8} \)
$89$ \( 77228944 + 17768 T^{4} + T^{8} \)
$97$ \( ( 484 + 352 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2} \)
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