Properties

Label 440.2.bi.b
Level $440$
Weight $2$
Character orbit 440.bi
Analytic conductor $3.513$
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] = [440,2,Mod(141,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.141");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.bi (of order \(10\), degree \(4\), 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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\zeta_{20}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
−0.642040 1.26007i −0.363271 0.500000i −1.17557 + 1.61803i −0.951057 + 0.309017i −0.396802 + 0.778768i 0.618034 + 0.449028i 2.79360 + 0.442463i 0.809017 2.48990i 1.00000 + 1.00000i
141.2 1.26007 0.642040i 0.363271 + 0.500000i 1.17557 1.61803i 0.951057 0.309017i 0.778768 + 0.396802i 0.618034 + 0.449028i 0.442463 2.79360i 0.809017 2.48990i 1.00000 1.00000i
181.1 −0.642040 + 1.26007i −0.363271 + 0.500000i −1.17557 1.61803i −0.951057 0.309017i −0.396802 0.778768i 0.618034 0.449028i 2.79360 0.442463i 0.809017 + 2.48990i 1.00000 1.00000i
181.2 1.26007 + 0.642040i 0.363271 0.500000i 1.17557 + 1.61803i 0.951057 + 0.309017i 0.778768 0.396802i 0.618034 0.449028i 0.442463 + 2.79360i 0.809017 + 2.48990i 1.00000 + 1.00000i
301.1 −1.39680 + 0.221232i −1.53884 0.500000i 1.90211 0.618034i −0.587785 0.809017i 2.26007 + 0.357960i −1.61803 4.97980i −2.52015 + 1.28408i −0.309017 0.224514i 1.00000 + 1.00000i
301.2 −0.221232 1.39680i 1.53884 + 0.500000i −1.90211 + 0.618034i 0.587785 + 0.809017i 0.357960 2.26007i −1.61803 4.97980i 1.28408 + 2.52015i −0.309017 0.224514i 1.00000 1.00000i
421.1 −1.39680 0.221232i −1.53884 + 0.500000i 1.90211 + 0.618034i −0.587785 + 0.809017i 2.26007 0.357960i −1.61803 + 4.97980i −2.52015 1.28408i −0.309017 + 0.224514i 1.00000 1.00000i
421.2 −0.221232 + 1.39680i 1.53884 0.500000i −1.90211 0.618034i 0.587785 0.809017i 0.357960 + 2.26007i −1.61803 + 4.97980i 1.28408 2.52015i −0.309017 + 0.224514i 1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.c even 5 1 inner
88.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.bi.b 8
8.b even 2 1 inner 440.2.bi.b 8
11.c even 5 1 inner 440.2.bi.b 8
88.o even 10 1 inner 440.2.bi.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.bi.b 8 1.a even 1 1 trivial
440.2.bi.b 8 8.b even 2 1 inner
440.2.bi.b 8 11.c even 5 1 inner
440.2.bi.b 8 88.o even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{6} + 6T_{3}^{4} + T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 41 T^{6} + 661 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 16 T^{6} + 736 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + 6 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 25 T^{6} + 3750 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 4)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{6} + 256 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 64 T^{6} + 4096 T^{4} + \cdots + 16777216 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 76 T^{2} - 781 T + 5041)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 27 T^{2} + 81)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 22 T^{3} + 244 T^{2} + 1368 T + 5776)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 20 T^{6} + 2400 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$59$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{6} + 400 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
$67$ \( (T^{4} + 183 T^{2} + 961)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + 244 T^{2} - 1672 T + 5776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + 216 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 31 T^{6} + 366 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T - 81)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} \) Copy content Toggle raw display
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