Properties

Label 136.2.c.a
Level $136$
Weight $2$
Character orbit 136.c
Analytic conductor $1.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
Defining polynomial: \(x^{8} - x^{7} - 2 x^{5} + 4 x^{4} - 4 x^{3} - 8 x + 16\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{5} + 4 x^{4} - 4 x^{3} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} + 4 \nu^{2} - 8 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + 4 \nu^{2} + 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 4 \nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 2 \nu^{5} + 4 \nu^{3} - 4 \nu^{2} - 8 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 2 \nu^{5} + 4 \nu^{4} + 4 \nu^{2} - 8 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3\)
\(\nu^{7}\)\(=\)\(-\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
1.40014 + 0.199044i
1.40014 0.199044i
0.814732 + 1.15595i
0.814732 1.15595i
−0.578647 + 1.29041i
−0.578647 1.29041i
−1.13622 + 0.842022i
−1.13622 0.842022i
−1.40014 0.199044i 1.72010i 1.92076 + 0.557378i 3.30391i −0.342376 + 2.40838i 3.45790 −2.57839 1.16272i 0.0412530 0.657624 4.62592i
69.2 −1.40014 + 0.199044i 1.72010i 1.92076 0.557378i 3.30391i −0.342376 2.40838i 3.45790 −2.57839 + 1.16272i 0.0412530 0.657624 + 4.62592i
69.3 −0.814732 1.15595i 2.64089i −0.672424 + 1.88357i 1.77580i −3.05273 + 2.15162i −0.423267 2.72515 0.757320i −3.97431 −2.05273 + 1.44680i
69.4 −0.814732 + 1.15595i 2.64089i −0.672424 1.88357i 1.77580i −3.05273 2.15162i −0.423267 2.72515 + 0.757320i −3.97431 −2.05273 1.44680i
69.5 0.578647 1.29041i 2.34593i −1.33034 1.49339i 3.12087i 3.02722 + 1.35746i 2.86993 −2.69688 + 0.852541i −2.50338 4.02722 + 1.80588i
69.6 0.578647 + 1.29041i 2.34593i −1.33034 + 1.49339i 3.12087i 3.02722 1.35746i 2.86993 −2.69688 0.852541i −2.50338 4.02722 1.80588i
69.7 1.13622 0.842022i 0.750707i 0.581998 1.91345i 0.436910i −0.632112 0.852970i −1.90455 −0.949886 2.66415i 2.43644 0.367888 + 0.496427i
69.8 1.13622 + 0.842022i 0.750707i 0.581998 + 1.91345i 0.436910i −0.632112 + 0.852970i −1.90455 −0.949886 + 2.66415i 2.43644 0.367888 0.496427i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.8
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 136.2.c.a 8
3.b odd 2 1 1224.2.f.d 8
4.b odd 2 1 544.2.c.a 8
8.b even 2 1 inner 136.2.c.a 8
8.d odd 2 1 544.2.c.a 8
12.b even 2 1 4896.2.f.c 8
16.e even 4 2 4352.2.a.bc 8
16.f odd 4 2 4352.2.a.be 8
24.f even 2 1 4896.2.f.c 8
24.h odd 2 1 1224.2.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.c.a 8 1.a even 1 1 trivial
136.2.c.a 8 8.b even 2 1 inner
544.2.c.a 8 4.b odd 2 1
544.2.c.a 8 8.d odd 2 1
1224.2.f.d 8 3.b odd 2 1
1224.2.f.d 8 24.h odd 2 1
4352.2.a.bc 8 16.e even 4 2
4352.2.a.be 8 16.f odd 4 2
4896.2.f.c 8 12.b even 2 1
4896.2.f.c 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16 T_{3}^{6} + 84 T_{3}^{4} + 156 T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 8 T + 4 T^{3} + 4 T^{4} + 2 T^{5} + T^{7} + T^{8} \)
$3$ \( 64 + 156 T^{2} + 84 T^{4} + 16 T^{6} + T^{8} \)
$5$ \( 64 + 368 T^{2} + 176 T^{4} + 24 T^{6} + T^{8} \)
$7$ \( ( 8 + 18 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$11$ \( 4096 + 2748 T^{2} + 572 T^{4} + 44 T^{6} + T^{8} \)
$13$ \( 9216 + 4976 T^{2} + 896 T^{4} + 60 T^{6} + T^{8} \)
$17$ \( ( 1 + T )^{8} \)
$19$ \( 20736 + 42752 T^{2} + 3936 T^{4} + 112 T^{6} + T^{8} \)
$23$ \( ( 48 + 70 T - 12 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$29$ \( 1032256 + 163056 T^{2} + 7952 T^{4} + 152 T^{6} + T^{8} \)
$31$ \( ( 64 - 230 T - 40 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$37$ \( 5184 + 4016 T^{2} + 928 T^{4} + 64 T^{6} + T^{8} \)
$41$ \( ( -48 - 112 T - 64 T^{2} + T^{4} )^{2} \)
$43$ \( 589824 + 155648 T^{2} + 8192 T^{4} + 156 T^{6} + T^{8} \)
$47$ \( ( -192 - 56 T + 36 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$53$ \( 1048576 + 175872 T^{2} + 9152 T^{4} + 176 T^{6} + T^{8} \)
$59$ \( 4096 + 57344 T^{2} + 5568 T^{4} + 140 T^{6} + T^{8} \)
$61$ \( 25644096 + 4364528 T^{2} + 81968 T^{4} + 504 T^{6} + T^{8} \)
$67$ \( 2304 + 169472 T^{2} + 12320 T^{4} + 240 T^{6} + T^{8} \)
$71$ \( ( 4608 - 182 T - 208 T^{2} + T^{4} )^{2} \)
$73$ \( ( 30704 + 944 T - 368 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$79$ \( ( 216 + 378 T + 156 T^{2} + 22 T^{3} + T^{4} )^{2} \)
$83$ \( 34668544 + 3227648 T^{2} + 64256 T^{4} + 444 T^{6} + T^{8} \)
$89$ \( ( -72 + 476 T - 88 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$97$ \( ( -1168 + 704 T - 104 T^{2} - 4 T^{3} + T^{4} )^{2} \)
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