Properties

Label 230.2.e.b
Level $230$
Weight $2$
Character orbit 230.e
Analytic conductor $1.837$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
Defining polynomial: \(x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} + 10 \nu^{4} + 9 \nu^{3} + 18 \nu^{2} + 9 \nu + 5 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - \nu^{5} + 10 \nu^{4} - 9 \nu^{3} + 18 \nu^{2} - 9 \nu + 5 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} + 11 \nu^{2} + 4 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + \nu^{6} + 10 \nu^{5} + 9 \nu^{4} + 18 \nu^{3} + 9 \nu^{2} + 5 \nu - 4 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 18 \nu^{3} - 9 \nu^{2} + 5 \nu \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{7} + 19 \nu^{5} + 29 \nu^{3} + 9 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{4} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 20\)
\(\nu^{5}\)\(=\)\(-9 \beta_{7} + 18 \beta_{6} + 18 \beta_{5} + 7 \beta_{3} - 7 \beta_{2} + 27 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(52 \beta_{6} - 52 \beta_{5} + 72 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} - 151\)
\(\nu^{7}\)\(=\)\(72 \beta_{7} - 142 \beta_{6} - 142 \beta_{5} - 52 \beta_{3} + 52 \beta_{2} - 203 \beta_{1} - 142\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
0.360409i
2.77462i
0.814115i
1.22833i
0.360409i
2.77462i
0.814115i
1.22833i
−0.707107 0.707107i −1.96195 + 1.96195i 1.00000i −1.77462 + 1.36041i 2.77462 −0.105561 + 0.105561i 0.707107 0.707107i 4.69853i 2.21680 + 0.292893i
137.2 −0.707107 0.707107i 0.254848 0.254848i 1.00000i 1.36041 1.77462i −0.360409 0.812668 0.812668i 0.707107 0.707107i 2.87011i −2.21680 + 0.292893i
137.3 0.707107 + 0.707107i −0.868559 + 0.868559i 1.00000i 2.22833 + 0.185885i −1.22833 1.38978 1.38978i −0.707107 + 0.707107i 1.49121i 1.44423 + 1.70711i
137.4 0.707107 + 0.707107i 0.575666 0.575666i 1.00000i 0.185885 + 2.22833i 0.814115 −2.09689 + 2.09689i −0.707107 + 0.707107i 2.33722i −1.44423 + 1.70711i
183.1 −0.707107 + 0.707107i −1.96195 1.96195i 1.00000i −1.77462 1.36041i 2.77462 −0.105561 0.105561i 0.707107 + 0.707107i 4.69853i 2.21680 0.292893i
183.2 −0.707107 + 0.707107i 0.254848 + 0.254848i 1.00000i 1.36041 + 1.77462i −0.360409 0.812668 + 0.812668i 0.707107 + 0.707107i 2.87011i −2.21680 0.292893i
183.3 0.707107 0.707107i −0.868559 0.868559i 1.00000i 2.22833 0.185885i −1.22833 1.38978 + 1.38978i −0.707107 0.707107i 1.49121i 1.44423 1.70711i
183.4 0.707107 0.707107i 0.575666 + 0.575666i 1.00000i 0.185885 2.22833i 0.814115 −2.09689 2.09689i −0.707107 0.707107i 2.33722i −1.44423 1.70711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 183.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.e.b yes 8
5.b even 2 1 1150.2.e.b 8
5.c odd 4 1 230.2.e.a 8
5.c odd 4 1 1150.2.e.c 8
23.b odd 2 1 230.2.e.a 8
115.c odd 2 1 1150.2.e.c 8
115.e even 4 1 inner 230.2.e.b yes 8
115.e even 4 1 1150.2.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.e.a 8 5.c odd 4 1
230.2.e.a 8 23.b odd 2 1
230.2.e.b yes 8 1.a even 1 1 trivial
230.2.e.b yes 8 115.e even 4 1 inner
1150.2.e.b 8 5.b even 2 1
1150.2.e.b 8 115.e even 4 1
1150.2.e.c 8 5.c odd 4 1
1150.2.e.c 8 115.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\):

\( T_{3}^{8} + 4 T_{3}^{7} + 8 T_{3}^{6} - T_{3}^{4} + 8 T_{3}^{2} - 4 T_{3} + 1 \)
\( T_{7}^{8} - 8 T_{7}^{5} + 47 T_{7}^{4} - 56 T_{7}^{3} + 32 T_{7}^{2} + 8 T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( 1 - 4 T + 8 T^{2} - T^{4} + 8 T^{6} + 4 T^{7} + T^{8} \)
$5$ \( 625 - 500 T + 200 T^{2} - 60 T^{3} + 14 T^{4} - 12 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( 1 + 8 T + 32 T^{2} - 56 T^{3} + 47 T^{4} - 8 T^{5} + T^{8} \)
$11$ \( 289 + 886 T^{2} + 471 T^{4} + 50 T^{6} + T^{8} \)
$13$ \( 5041 - 4260 T + 1800 T^{2} + 1860 T^{3} + 819 T^{4} + 60 T^{5} + T^{8} \)
$17$ \( 30625 - 59500 T + 57800 T^{2} - 30140 T^{3} + 9851 T^{4} - 2084 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$19$ \( ( -49 - 34 T + 3 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$23$ \( 279841 + 194672 T + 67712 T^{2} + 18768 T^{3} + 4418 T^{4} + 816 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$29$ \( 16 + 384 T^{2} + 760 T^{4} + 64 T^{6} + T^{8} \)
$31$ \( ( -49 - 86 T - 35 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$37$ \( 795664 - 21408 T + 288 T^{2} + 5536 T^{3} + 4940 T^{4} + 352 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$41$ \( ( -119 + 82 T - 5 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$43$ \( 3083536 - 1475040 T + 352800 T^{2} - 6560 T^{3} - 2356 T^{4} + 160 T^{5} + 200 T^{6} + 20 T^{7} + T^{8} \)
$47$ \( 4624 + 1088 T + 128 T^{2} + 576 T^{3} + 1160 T^{4} + 528 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 614656 + 627200 T + 320000 T^{2} + 86400 T^{3} + 13232 T^{4} + 800 T^{5} + T^{8} \)
$59$ \( 226576 + 53008 T^{2} + 3868 T^{4} + 108 T^{6} + T^{8} \)
$61$ \( 13315201 + 1217618 T^{2} + 35179 T^{4} + 338 T^{6} + T^{8} \)
$67$ \( 9265936 + 6404576 T + 2213408 T^{2} + 403520 T^{3} + 40684 T^{4} + 1360 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$71$ \( ( -1193 - 594 T + 67 T^{2} + 22 T^{3} + T^{4} )^{2} \)
$73$ \( 1882384 - 2403744 T + 1534752 T^{2} - 48928 T^{3} + 2780 T^{4} - 1584 T^{5} + 392 T^{6} - 28 T^{7} + T^{8} \)
$79$ \( ( -5348 + 2080 T - 210 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$83$ \( 10000 + 236000 T + 2784800 T^{2} + 111360 T^{3} + 2316 T^{4} + 1072 T^{5} + 392 T^{6} + 28 T^{7} + T^{8} \)
$89$ \( ( -2500 - 2000 T - 50 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$97$ \( 246584209 - 28328212 T + 1627208 T^{2} + 420988 T^{3} + 60403 T^{4} - 620 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
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