Properties

Label 57.2.f.a
Level $57$
Weight $2$
Character orbit 57.f
Analytic conductor $0.455$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
−0.965926 + 1.67303i 1.57313 0.724745i −0.866025 1.50000i 1.22474 + 0.707107i −0.307007 + 3.33195i −3.73205 −0.517638 1.94949 2.28024i −2.36603 + 1.36603i
8.2 −0.258819 + 0.448288i −1.57313 + 0.724745i 0.866025 + 1.50000i 1.22474 + 0.707107i 0.0822623 0.892794i −0.267949 −1.93185 1.94949 2.28024i −0.633975 + 0.366025i
8.3 0.258819 0.448288i −0.158919 1.72474i 0.866025 + 1.50000i −1.22474 0.707107i −0.814313 0.375156i −0.267949 1.93185 −2.94949 + 0.548188i −0.633975 + 0.366025i
8.4 0.965926 1.67303i 0.158919 + 1.72474i −0.866025 1.50000i −1.22474 0.707107i 3.03906 + 1.40010i −3.73205 0.517638 −2.94949 + 0.548188i −2.36603 + 1.36603i
50.1 −0.965926 1.67303i 1.57313 + 0.724745i −0.866025 + 1.50000i 1.22474 0.707107i −0.307007 3.33195i −3.73205 −0.517638 1.94949 + 2.28024i −2.36603 1.36603i
50.2 −0.258819 0.448288i −1.57313 0.724745i 0.866025 1.50000i 1.22474 0.707107i 0.0822623 + 0.892794i −0.267949 −1.93185 1.94949 + 2.28024i −0.633975 0.366025i
50.3 0.258819 + 0.448288i −0.158919 + 1.72474i 0.866025 1.50000i −1.22474 + 0.707107i −0.814313 + 0.375156i −0.267949 1.93185 −2.94949 0.548188i −0.633975 0.366025i
50.4 0.965926 + 1.67303i 0.158919 1.72474i −0.866025 + 1.50000i −1.22474 + 0.707107i 3.03906 1.40010i −3.73205 0.517638 −2.94949 0.548188i −2.36603 1.36603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.f.a 8
3.b odd 2 1 inner 57.2.f.a 8
4.b odd 2 1 912.2.bn.m 8
12.b even 2 1 912.2.bn.m 8
19.c even 3 1 1083.2.d.b 8
19.d odd 6 1 inner 57.2.f.a 8
19.d odd 6 1 1083.2.d.b 8
57.f even 6 1 inner 57.2.f.a 8
57.f even 6 1 1083.2.d.b 8
57.h odd 6 1 1083.2.d.b 8
76.f even 6 1 912.2.bn.m 8
228.n odd 6 1 912.2.bn.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 1.a even 1 1 trivial
57.2.f.a 8 3.b odd 2 1 inner
57.2.f.a 8 19.d odd 6 1 inner
57.2.f.a 8 57.f even 6 1 inner
912.2.bn.m 8 4.b odd 2 1
912.2.bn.m 8 12.b even 2 1
912.2.bn.m 8 76.f even 6 1
912.2.bn.m 8 228.n odd 6 1
1083.2.d.b 8 19.c even 3 1
1083.2.d.b 8 19.d odd 6 1
1083.2.d.b 8 57.f even 6 1
1083.2.d.b 8 57.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 26 T^{2} + 361)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 28 T^{6} + 780 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 64 T^{6} + 3840 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( (T^{4} + 26 T^{2} + 121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 78 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + 51 T^{2} + 104 T + 169)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 112 T^{6} + 12480 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 156 T^{6} + \cdots + 18974736 \) Copy content Toggle raw display
$59$ \( T^{8} + 196 T^{6} + \cdots + 78074896 \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + 159 T^{2} - 518 T + 1369)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 192 T^{6} + 34560 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + 11 T^{2} + 444 T + 1369)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16)^{2} \) Copy content Toggle raw display
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