Properties

Label 510.2.l.f
Level $510$
Weight $2$
Character orbit 510.l
Analytic conductor $4.072$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [510,2,Mod(137,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.l (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(307\) \(341\)
\(\chi(n)\) \(1\) \(\zeta_{24}^{6}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
−0.707107 + 0.707107i 0.599900 1.62484i 1.00000i −1.73205 + 1.41421i 0.724745 + 1.57313i −1.00000 1.00000i 0.707107 + 0.707107i −2.28024 1.94949i 0.224745 2.22474i
137.2 −0.707107 + 0.707107i 1.10721 + 1.33195i 1.00000i 1.73205 + 1.41421i −1.72474 0.158919i −1.00000 1.00000i 0.707107 + 0.707107i −0.548188 + 2.94949i −2.22474 + 0.224745i
137.3 0.707107 0.707107i −1.33195 1.10721i 1.00000i −1.73205 1.41421i −1.72474 + 0.158919i −1.00000 1.00000i −0.707107 0.707107i 0.548188 + 2.94949i −2.22474 + 0.224745i
137.4 0.707107 0.707107i 1.62484 0.599900i 1.00000i 1.73205 1.41421i 0.724745 1.57313i −1.00000 1.00000i −0.707107 0.707107i 2.28024 1.94949i 0.224745 2.22474i
443.1 −0.707107 0.707107i 0.599900 + 1.62484i 1.00000i −1.73205 1.41421i 0.724745 1.57313i −1.00000 + 1.00000i 0.707107 0.707107i −2.28024 + 1.94949i 0.224745 + 2.22474i
443.2 −0.707107 0.707107i 1.10721 1.33195i 1.00000i 1.73205 1.41421i −1.72474 + 0.158919i −1.00000 + 1.00000i 0.707107 0.707107i −0.548188 2.94949i −2.22474 0.224745i
443.3 0.707107 + 0.707107i −1.33195 + 1.10721i 1.00000i −1.73205 + 1.41421i −1.72474 0.158919i −1.00000 + 1.00000i −0.707107 + 0.707107i 0.548188 2.94949i −2.22474 0.224745i
443.4 0.707107 + 0.707107i 1.62484 + 0.599900i 1.00000i 1.73205 + 1.41421i 0.724745 + 1.57313i −1.00000 + 1.00000i −0.707107 + 0.707107i 2.28024 + 1.94949i 0.224745 + 2.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.l.f 8
3.b odd 2 1 inner 510.2.l.f 8
5.c odd 4 1 inner 510.2.l.f 8
15.e even 4 1 inner 510.2.l.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.l.f 8 1.a even 1 1 trivial
510.2.l.f 8 3.b odd 2 1 inner
510.2.l.f 8 5.c odd 4 1 inner
510.2.l.f 8 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\):

\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{23}^{4} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{3} + 72 T^{2} + 180 T + 225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 62 T^{2} + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 58 T^{2} + 625)^{2} \) Copy content Toggle raw display
$31$ \( (T - 3)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 320 T + 1600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 100 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + 200 T^{2} + 760 T + 1444)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 46802 T^{4} + \cdots + 492884401 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{2} + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T - 53)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 32 T^{3} + 512 T^{2} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 22 T^{2} + 25)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 80 T + 25)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 56 T^{2} + 400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 406 T^{2} + 38809)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 536 T + 4489)^{2} \) Copy content Toggle raw display
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