Properties

Label 168.2.p.a
Level $168$
Weight $2$
Character orbit 168.p
Analytic conductor $1.341$
Analytic rank $0$
Dimension $16$
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] = [168,2,Mod(139,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.p (of order \(2\), degree \(1\), 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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 4x^{10} + 16x^{8} - 16x^{6} - 64x^{4} + 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{12} - \beta_{11} + \beta_{6}) q^{5} - \beta_{6} q^{6} + (\beta_{14} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{14} - \beta_{5} + \beta_{4} - 1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{12} - \beta_{11} + \beta_{6}) q^{5} - \beta_{6} q^{6} + (\beta_{14} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{14} - \beta_{5} + \beta_{4} - 1) q^{8} - q^{9} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{6} - \beta_{3}) q^{10} + (\beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{11} - \beta_{12} q^{12} + ( - 2 \beta_{15} + \beta_{13} - \beta_{11} + \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 1) q^{14} + ( - \beta_{7} - \beta_{5} - \beta_1) q^{15} + (\beta_{14} - \beta_{13} + \beta_{8} + \beta_{5} + \beta_{2} + 1) q^{16} + ( - \beta_{12} + \beta_{11} + \beta_{6}) q^{17} + \beta_{5} q^{18} + (\beta_{13} - \beta_{11} - \beta_{9} + \beta_{8} + 2 \beta_{6} - 2 \beta_{3}) q^{19} + ( - 2 \beta_{15} + \beta_{13} - \beta_{8} + \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{20} - \beta_{15} q^{21} + ( - \beta_{14} + \beta_{13} + \beta_{10} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{2} + \beta_1 + 1) q^{22} + (\beta_{13} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_1) q^{23} + (\beta_{15} - \beta_{13} - \beta_{6} + \beta_{3}) q^{24} + (\beta_{10} - \beta_{7} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{25} + (2 \beta_{15} + 2 \beta_{8} - 2 \beta_{3}) q^{26} + \beta_{3} q^{27} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{28}+ \cdots + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 2 q^{4} - 10 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 2 q^{4} - 10 q^{8} - 16 q^{9} - 8 q^{11} - 14 q^{14} + 18 q^{16} - 2 q^{18} + 8 q^{22} + 16 q^{25} - 10 q^{28} - 16 q^{30} - 18 q^{32} + 24 q^{35} - 2 q^{36} - 4 q^{42} - 8 q^{43} + 52 q^{46} - 8 q^{49} - 34 q^{50} + 50 q^{56} - 16 q^{57} + 24 q^{58} + 32 q^{60} + 2 q^{64} - 40 q^{67} - 24 q^{70} + 10 q^{72} - 32 q^{74} - 24 q^{78} + 16 q^{81} + 8 q^{84} - 32 q^{86} - 88 q^{88} - 56 q^{91} + 44 q^{92} + 66 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 4x^{10} + 16x^{8} - 16x^{6} - 64x^{4} + 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + \nu^{12} - 4\nu^{10} - 4\nu^{8} + 16\nu^{6} - 16\nu^{4} - 64\nu^{2} + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 4\nu^{11} - 4\nu^{9} - 8\nu^{7} + 32\nu^{6} + 64\nu^{5} + 32\nu^{4} + 32\nu^{3} - 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - \nu^{13} - 2\nu^{11} + 8\nu^{9} + 8\nu^{7} - 32\nu^{5} + 64\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + \nu^{13} - 4\nu^{11} + 4\nu^{9} + 8\nu^{7} + 32\nu^{6} - 64\nu^{5} + 32\nu^{4} - 32\nu^{3} + 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - \nu^{12} - 2\nu^{10} + 8\nu^{8} + 8\nu^{6} - 32\nu^{4} + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{15} - \nu^{13} + 4\nu^{11} + 4\nu^{9} - 16\nu^{7} + 16\nu^{5} + 64\nu^{3} - 64\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} + \nu^{12} + 6\nu^{10} - 4\nu^{8} - 8\nu^{6} + 32\nu^{4} + 64\nu^{2} - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + \nu^{13} + 4 \nu^{11} - 8 \nu^{10} - 4 \nu^{9} + 8 \nu^{8} - 8 \nu^{7} + 16 \nu^{6} - 64 \nu^{4} + 32 \nu^{3} - 64 \nu^{2} - 128 \nu + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{15} + \nu^{13} + 6\nu^{11} - 4\nu^{9} - 8\nu^{7} + 32\nu^{5} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{14} - \nu^{12} - 6\nu^{10} + 4\nu^{8} + 8\nu^{6} - 32\nu^{4} + 64\nu^{2} + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} + \nu^{13} + 6\nu^{11} - 4\nu^{9} - 8\nu^{7} + 32\nu^{5} - 192\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 3\nu^{13} + 4\nu^{9} - 16\nu^{7} - 48\nu^{5} + 64\nu^{3} + 64\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + \nu^{13} + 4 \nu^{11} + 8 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} - 8 \nu^{7} - 16 \nu^{6} + 64 \nu^{4} + 32 \nu^{3} + 64 \nu^{2} - 128 \nu - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} + \nu^{13} - 4 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 8 \nu^{7} - 16 \nu^{6} - 64 \nu^{5} - 32 \nu^{3} + 128 \nu^{2} + 128 \nu - 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - \nu^{13} + \nu^{11} + 4 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{6} - 16 \nu^{5} + 32 \nu^{4} + 48 \nu^{3} + 32 \nu^{2} + 32 \nu - 128 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} - 2\beta_{12} - \beta_{11} - \beta_{9} + 2\beta_{8} + 2\beta_{6} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{14} + 2\beta_{13} + \beta_{10} - 2\beta_{8} - 3\beta_{7} - 2\beta_{5} + 2\beta_{4} - 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{9} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{14} - 2 \beta_{13} - \beta_{10} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{2} + 2 \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{15} - 6 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} - \beta_{9} + 4 \beta_{8} - 10 \beta_{6} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10 \beta_{14} + 2 \beta_{13} + \beta_{10} - 2 \beta_{8} - 3 \beta_{7} + 6 \beta_{5} + 6 \beta_{4} - 4 \beta_{2} - 10 \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{15} + 6 \beta_{13} - 10 \beta_{12} - 3 \beta_{11} + \beta_{9} - 4 \beta_{8} + 26 \beta_{6} + 2 \beta_{4} + 26 \beta_{3} - 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10 \beta_{14} - 2 \beta_{13} - 17 \beta_{10} + 2 \beta_{8} + 19 \beta_{7} + 26 \beta_{5} - 6 \beta_{4} + 4 \beta_{2} + 10 \beta _1 + 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10 \beta_{15} - 6 \beta_{13} + 10 \beta_{12} + 3 \beta_{11} + 31 \beta_{9} + 4 \beta_{8} - 26 \beta_{6} - 2 \beta_{4} + 38 \beta_{3} + 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 42 \beta_{14} + 2 \beta_{13} + 17 \beta_{10} - 2 \beta_{8} + 45 \beta_{7} + 6 \beta_{5} + 6 \beta_{4} - 36 \beta_{2} + 22 \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10 \beta_{15} + 38 \beta_{13} - 74 \beta_{12} - 35 \beta_{11} + \beta_{9} + 28 \beta_{8} + 90 \beta_{6} + 34 \beta_{4} - 38 \beta_{3} - 34 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22 \beta_{14} + 62 \beta_{13} + 15 \beta_{10} - 62 \beta_{8} - 13 \beta_{7} + 58 \beta_{5} - 6 \beta_{4} - 28 \beta_{2} + 42 \beta _1 + 70 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 54 \beta_{15} + 90 \beta_{13} + 10 \beta_{12} - 61 \beta_{11} + 31 \beta_{9} + 36 \beta_{8} - 26 \beta_{6} - 34 \beta_{4} + 102 \beta_{3} + 34 \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
0.310478 + 1.37971i
−0.310478 1.37971i
−0.310478 + 1.37971i
0.310478 1.37971i
1.40199 + 0.185533i
−1.40199 0.185533i
−1.40199 + 0.185533i
1.40199 0.185533i
1.20933 0.733159i
−1.20933 + 0.733159i
−1.20933 0.733159i
1.20933 + 0.733159i
0.474920 1.33209i
−0.474920 + 1.33209i
−0.474920 1.33209i
0.474920 + 1.33209i
−1.37971 0.310478i 1.00000i 1.80721 + 0.856739i 2.33443 −0.310478 + 1.37971i 0.490487 + 2.59989i −2.22743 1.74315i −1.00000 −3.22084 0.724789i
139.2 −1.37971 0.310478i 1.00000i 1.80721 + 0.856739i −2.33443 0.310478 1.37971i −0.490487 + 2.59989i −2.22743 1.74315i −1.00000 3.22084 + 0.724789i
139.3 −1.37971 + 0.310478i 1.00000i 1.80721 0.856739i −2.33443 0.310478 + 1.37971i −0.490487 2.59989i −2.22743 + 1.74315i −1.00000 3.22084 0.724789i
139.4 −1.37971 + 0.310478i 1.00000i 1.80721 0.856739i 2.33443 −0.310478 1.37971i 0.490487 2.59989i −2.22743 + 1.74315i −1.00000 −3.22084 + 0.724789i
139.5 −0.185533 1.40199i 1.00000i −1.93115 + 0.520231i 3.84444 −1.40199 + 0.185533i 1.62140 2.09071i 1.08765 + 2.61094i −1.00000 −0.713272 5.38987i
139.6 −0.185533 1.40199i 1.00000i −1.93115 + 0.520231i −3.84444 1.40199 0.185533i −1.62140 2.09071i 1.08765 + 2.61094i −1.00000 0.713272 + 5.38987i
139.7 −0.185533 + 1.40199i 1.00000i −1.93115 0.520231i −3.84444 1.40199 + 0.185533i −1.62140 + 2.09071i 1.08765 2.61094i −1.00000 0.713272 5.38987i
139.8 −0.185533 + 1.40199i 1.00000i −1.93115 0.520231i 3.84444 −1.40199 0.185533i 1.62140 + 2.09071i 1.08765 2.61094i −1.00000 −0.713272 + 5.38987i
139.9 0.733159 1.20933i 1.00000i −0.924955 1.77326i −1.12786 −1.20933 0.733159i −2.11337 1.59175i −2.82260 0.181508i −1.00000 −0.826905 + 1.36396i
139.10 0.733159 1.20933i 1.00000i −0.924955 1.77326i 1.12786 1.20933 + 0.733159i 2.11337 1.59175i −2.82260 0.181508i −1.00000 0.826905 1.36396i
139.11 0.733159 + 1.20933i 1.00000i −0.924955 + 1.77326i 1.12786 1.20933 0.733159i 2.11337 + 1.59175i −2.82260 + 0.181508i −1.00000 0.826905 + 1.36396i
139.12 0.733159 + 1.20933i 1.00000i −0.924955 + 1.77326i −1.12786 −1.20933 + 0.733159i −2.11337 + 1.59175i −2.82260 + 0.181508i −1.00000 −0.826905 1.36396i
139.13 1.33209 0.474920i 1.00000i 1.54890 1.26527i −1.58069 −0.474920 1.33209i 2.37995 + 1.15578i 1.46237 2.42105i −1.00000 −2.10562 + 0.750703i
139.14 1.33209 0.474920i 1.00000i 1.54890 1.26527i 1.58069 0.474920 + 1.33209i −2.37995 + 1.15578i 1.46237 2.42105i −1.00000 2.10562 0.750703i
139.15 1.33209 + 0.474920i 1.00000i 1.54890 + 1.26527i 1.58069 0.474920 1.33209i −2.37995 1.15578i 1.46237 + 2.42105i −1.00000 2.10562 + 0.750703i
139.16 1.33209 + 0.474920i 1.00000i 1.54890 + 1.26527i −1.58069 −0.474920 + 1.33209i 2.37995 1.15578i 1.46237 + 2.42105i −1.00000 −2.10562 0.750703i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.p.a 16
3.b odd 2 1 504.2.p.g 16
4.b odd 2 1 672.2.p.a 16
7.b odd 2 1 inner 168.2.p.a 16
8.b even 2 1 672.2.p.a 16
8.d odd 2 1 inner 168.2.p.a 16
12.b even 2 1 2016.2.p.g 16
21.c even 2 1 504.2.p.g 16
24.f even 2 1 504.2.p.g 16
24.h odd 2 1 2016.2.p.g 16
28.d even 2 1 672.2.p.a 16
56.e even 2 1 inner 168.2.p.a 16
56.h odd 2 1 672.2.p.a 16
84.h odd 2 1 2016.2.p.g 16
168.e odd 2 1 504.2.p.g 16
168.i even 2 1 2016.2.p.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.p.a 16 1.a even 1 1 trivial
168.2.p.a 16 7.b odd 2 1 inner
168.2.p.a 16 8.d odd 2 1 inner
168.2.p.a 16 56.e even 2 1 inner
504.2.p.g 16 3.b odd 2 1
504.2.p.g 16 21.c even 2 1
504.2.p.g 16 24.f even 2 1
504.2.p.g 16 168.e odd 2 1
672.2.p.a 16 4.b odd 2 1
672.2.p.a 16 8.b even 2 1
672.2.p.a 16 28.d even 2 1
672.2.p.a 16 56.h odd 2 1
2016.2.p.g 16 12.b even 2 1
2016.2.p.g 16 24.h odd 2 1
2016.2.p.g 16 84.h odd 2 1
2016.2.p.g 16 168.i even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} + 2 T^{5} - 4 T^{4} + 4 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 160 T^{4} - 368 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{14} + 68 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 24 T^{2} - 60 T - 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 68 T^{6} + 1264 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 64 T^{6} + 1088 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 96 T^{6} + 2368 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 96 T^{6} + 1984 T^{4} + 5168 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 68 T^{6} + 1264 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 180 T^{6} + 11824 T^{4} + \cdots + 3444736)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 64 T^{6} + 1024 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 208 T^{6} + 14304 T^{4} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} - 52 T^{2} - 8 T + 160)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 256 T^{6} + 20736 T^{4} + \cdots + 6553600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 164 T^{6} + 7408 T^{4} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 208 T^{6} + 10240 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 164 T^{6} + 2544 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} - 60 T^{2} - 888 T - 2272)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 112 T^{6} + 864 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 256 T^{6} + 17408 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 188 T^{6} + 9856 T^{4} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 544 T^{6} + 98304 T^{4} + \cdots + 147865600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 400 T^{6} + 43296 T^{4} + \cdots + 17572864)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 544 T^{6} + 70400 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
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