Properties

Label 280.2.b.c
Level $280$
Weight $2$
Character orbit 280.b
Analytic conductor $2.236$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 42 \nu^{4} - 59 \nu^{3} + 48 \nu^{2} - 24 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11 \)
\(\beta_{4}\)\(=\)\( -7 \nu^{7} + 25 \nu^{6} - 87 \nu^{5} + 158 \nu^{4} - 231 \nu^{3} + 206 \nu^{2} - 118 \nu + 31 \)
\(\beta_{5}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(\beta_{6}\)\(=\)\( 8 \nu^{7} - 28 \nu^{6} + 98 \nu^{5} - 175 \nu^{4} + 257 \nu^{3} - 224 \nu^{2} + 130 \nu - 32 \)
\(\beta_{7}\)\(=\)\( 10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 166 \nu - 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 4 \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} - 5 \beta_{4} - \beta_{3} - 9 \beta_{2} + 2 \beta_{1} + 5\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{7} - 17 \beta_{6} - 13 \beta_{5} + 13 \beta_{4} - 23 \beta_{3} + 3 \beta_{2} - 18 \beta_{1} + 21\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-46 \beta_{7} + 29 \beta_{6} - 67 \beta_{5} + 37 \beta_{4} - 15 \beta_{3} + 47 \beta_{2} - 24 \beta_{1} - 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-76 \beta_{7} + 105 \beta_{6} - 7 \beta_{5} - 31 \beta_{4} + 91 \beta_{3} + 39 \beta_{2} + 68 \beta_{1} - 83\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1
0.500000 + 0.0297061i
0.500000 0.0297061i
0.500000 + 0.691860i
0.500000 0.691860i
0.500000 + 1.44392i
0.500000 1.44392i
0.500000 2.10607i
0.500000 + 2.10607i
−0.874559 1.11137i 2.41421i −0.470294 + 1.94392i 1.00000i 2.68309 2.11137i −1.00000 2.57172 1.17740i −2.82843 1.11137 0.874559i
141.2 −0.874559 + 1.11137i 2.41421i −0.470294 1.94392i 1.00000i 2.68309 + 2.11137i −1.00000 2.57172 + 1.17740i −2.82843 1.11137 + 0.874559i
141.3 −0.635665 1.26330i 0.414214i −1.19186 + 1.60607i 1.00000i 0.523276 0.263301i −1.00000 2.78658 + 0.484753i 2.82843 −1.26330 + 0.635665i
141.4 −0.635665 + 1.26330i 0.414214i −1.19186 1.60607i 1.00000i 0.523276 + 0.263301i −1.00000 2.78658 0.484753i 2.82843 −1.26330 0.635665i
141.5 0.167452 1.40426i 2.41421i −1.94392 0.470294i 1.00000i −3.39020 0.404265i −1.00000 −0.985930 + 2.65103i −2.82843 −1.40426 0.167452i
141.6 0.167452 + 1.40426i 2.41421i −1.94392 + 0.470294i 1.00000i −3.39020 + 0.404265i −1.00000 −0.985930 2.65103i −2.82843 −1.40426 + 0.167452i
141.7 1.34277 0.443806i 0.414214i 1.60607 1.19186i 1.00000i 0.183830 + 0.556194i −1.00000 1.62764 2.31318i 2.82843 −0.443806 1.34277i
141.8 1.34277 + 0.443806i 0.414214i 1.60607 + 1.19186i 1.00000i 0.183830 0.556194i −1.00000 1.62764 + 2.31318i 2.82843 −0.443806 + 1.34277i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.b.c 8
4.b odd 2 1 1120.2.b.c 8
8.b even 2 1 inner 280.2.b.c 8
8.d odd 2 1 1120.2.b.c 8
16.e even 4 1 8960.2.a.bt 4
16.e even 4 1 8960.2.a.bu 4
16.f odd 4 1 8960.2.a.bs 4
16.f odd 4 1 8960.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.c 8 1.a even 1 1 trivial
280.2.b.c 8 8.b even 2 1 inner
1120.2.b.c 8 4.b odd 2 1
1120.2.b.c 8 8.d odd 2 1
8960.2.a.bs 4 16.f odd 4 1
8960.2.a.bt 4 16.e even 4 1
8960.2.a.bu 4 16.e even 4 1
8960.2.a.bv 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\):

\( T_{3}^{4} + 6 T_{3}^{2} + 1 \)
\( T_{13}^{8} + 52 T_{13}^{6} + 166 T_{13}^{4} + 52 T_{13}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 8 T^{2} - 8 T^{3} + 2 T^{4} - 4 T^{5} + 2 T^{6} + T^{8} \)
$3$ \( ( 1 + 6 T^{2} + T^{4} )^{2} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( 1681 + 1620 T^{2} + 502 T^{4} + 52 T^{6} + T^{8} \)
$13$ \( 1 + 52 T^{2} + 166 T^{4} + 52 T^{6} + T^{8} \)
$17$ \( ( 41 - 52 T - 22 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$19$ \( 602176 + 93248 T^{2} + 5152 T^{4} + 120 T^{6} + T^{8} \)
$23$ \( ( -56 - 80 T - 20 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$29$ \( 160801 + 78940 T^{2} + 6886 T^{4} + 156 T^{6} + T^{8} \)
$31$ \( ( -56 - 192 T - 44 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$37$ \( 61504 + 20288 T^{2} + 2208 T^{4} + 88 T^{6} + T^{8} \)
$41$ \( ( 392 + 240 T - 4 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$43$ \( 18496 + 11968 T^{2} + 2208 T^{4} + 104 T^{6} + T^{8} \)
$47$ \( ( 721 + 64 T - 74 T^{2} + T^{4} )^{2} \)
$53$ \( 4096 + 40960 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8} \)
$59$ \( 66064384 + 3434496 T^{2} + 59776 T^{4} + 416 T^{6} + T^{8} \)
$61$ \( 47004736 + 3103424 T^{2} + 59424 T^{4} + 424 T^{6} + T^{8} \)
$67$ \( 4096 + 40960 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8} \)
$71$ \( ( 3728 + 608 T - 136 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$73$ \( ( 2048 + 1024 T - 64 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$79$ \( ( -1751 + 408 T + 122 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$83$ \( 12544 + 13056 T^{2} + 3424 T^{4} + 176 T^{6} + T^{8} \)
$89$ \( ( -8696 + 1168 T + 156 T^{2} - 28 T^{3} + T^{4} )^{2} \)
$97$ \( ( -10287 + 3564 T - 270 T^{2} - 12 T^{3} + T^{4} )^{2} \)
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