Properties

Label 140.2.g.c
Level $140$
Weight $2$
Character orbit 140.g
Analytic conductor $1.118$
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] = [140,2,Mod(111,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("140.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.g (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: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{4} + \beta_1 q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 2) q^{8} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (\beta_{6} - \beta_{4}) q^{3} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{4} + \beta_1 q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 2) q^{8} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{9} - \beta_{2} q^{10} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{11} + ( - 2 \beta_{2} - 4 \beta_1) q^{12} + 2 \beta_1 q^{13} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{14} + ( - \beta_{5} + \beta_{3}) q^{15} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} - 2) q^{16} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 4) q^{18} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{19} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{20} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 3 \beta_1 - 5) q^{21} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 4) q^{22} + (4 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{24} - q^{25} - 2 \beta_{2} q^{26} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2}) q^{27} + (\beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{28} - 2 q^{29} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{30} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{31} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2) q^{32} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{33} + ( - 2 \beta_{6} + 2 \beta_{4} + 4 \beta_{2} - 4 \beta_1) q^{34} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3}) q^{35} + (3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 4) q^{36} + 2 q^{37} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{38} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{39} + (\beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{40} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{41} + (4 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{42} + ( - 4 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{43} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 4) q^{44} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{45} + (4 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 6) q^{46} + ( - 3 \beta_{5} - 3 \beta_{3} + 6 \beta_{2}) q^{47} + ( - 4 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{48} + ( - \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} - 5 \beta_1 + 4) q^{49} + \beta_{7} q^{50} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{51} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{52} + 2 q^{53} + ( - 4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2} + 8 \beta_1) q^{54} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{55} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{56} + ( - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 8) q^{57} + 2 \beta_{7} q^{58} + (\beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - 6 \beta_{2}) q^{59} + (2 \beta_{7} + 4) q^{60} + 2 \beta_1 q^{61} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{62} + ( - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} - 5 \beta_{2}) q^{63} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2) q^{64} - 2 q^{65} + (4 \beta_{6} - 4 \beta_{4} + 4 \beta_{2} + 8 \beta_1) q^{66} + ( - 8 \beta_{7} - 4 \beta_{6} + \beta_{5} - 4 \beta_{4} - \beta_{3}) q^{67} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 8 \beta_{2} - 4 \beta_1) q^{68} + (3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 10 \beta_1) q^{69} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_1 + 1) q^{70} + (2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{71} + (7 \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_{4} + 4 \beta_{3} + 10) q^{72} + ( - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{73} - 2 \beta_{7} q^{74} + ( - \beta_{6} + \beta_{4}) q^{75} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{76} + (2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 4) q^{77} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4) q^{78} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3}) q^{79} + (\beta_{5} + \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{80} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 7) q^{81} + (2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} + 4 \beta_1) q^{82} + (\beta_{6} - \beta_{4}) q^{83} + (2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - 2 \beta_1 + 2) q^{84} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2) q^{85} + ( - 4 \beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} - 2) q^{86} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{87} + (2 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 4) q^{88} + 12 \beta_1 q^{89} + ( - 2 \beta_{6} + 2 \beta_{4} - \beta_{2} - 4 \beta_1) q^{90} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{91} + ( - 2 \beta_{7} + 4 \beta_{6} + 4 \beta_{4} - 4) q^{92} + ( - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4) q^{93} + ( - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{94} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{95} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 12 \beta_1) q^{96} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{97} + ( - 5 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{98} + (2 \beta_{7} + \beta_{6} + 9 \beta_{5} + \beta_{4} - 9 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{4} + 14 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{4} + 14 q^{8} + 16 q^{9} - 6 q^{14} - 14 q^{16} - 30 q^{18} - 36 q^{21} + 28 q^{22} - 8 q^{25} + 14 q^{28} - 16 q^{29} - 12 q^{30} - 18 q^{32} - 30 q^{36} + 16 q^{37} + 8 q^{42} - 20 q^{44} + 44 q^{46} + 36 q^{49} - 2 q^{50} + 16 q^{53} + 2 q^{56} - 48 q^{57} - 4 q^{58} + 28 q^{60} + 2 q^{64} - 16 q^{65} + 4 q^{70} + 62 q^{72} + 4 q^{74} + 24 q^{77} - 24 q^{78} + 48 q^{81} + 8 q^{84} + 24 q^{85} - 20 q^{86} + 36 q^{88} - 12 q^{92} - 16 q^{93} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 16\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{6} + \beta_{5} - 5\beta_{4} + \beta_{3} - 4\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 20\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
1.17915 + 0.780776i
−1.17915 0.780776i
1.17915 0.780776i
−1.17915 + 0.780776i
−0.599676 + 1.28078i
0.599676 1.28078i
−0.599676 1.28078i
0.599676 + 1.28078i
−0.780776 1.17915i −3.02045 −0.780776 + 1.84130i 1.00000i 2.35829 + 3.56155i 2.17238 1.51022i 2.78078 0.516994i 6.12311 1.17915 0.780776i
111.2 −0.780776 1.17915i 3.02045 −0.780776 + 1.84130i 1.00000i −2.35829 3.56155i −2.17238 1.51022i 2.78078 0.516994i 6.12311 −1.17915 + 0.780776i
111.3 −0.780776 + 1.17915i −3.02045 −0.780776 1.84130i 1.00000i 2.35829 3.56155i 2.17238 + 1.51022i 2.78078 + 0.516994i 6.12311 1.17915 + 0.780776i
111.4 −0.780776 + 1.17915i 3.02045 −0.780776 1.84130i 1.00000i −2.35829 + 3.56155i −2.17238 + 1.51022i 2.78078 + 0.516994i 6.12311 −1.17915 0.780776i
111.5 1.28078 0.599676i −0.936426 1.28078 1.53610i 1.00000i −1.19935 + 0.561553i 2.60399 + 0.468213i 0.719224 2.73546i −2.12311 −0.599676 1.28078i
111.6 1.28078 0.599676i 0.936426 1.28078 1.53610i 1.00000i 1.19935 0.561553i −2.60399 + 0.468213i 0.719224 2.73546i −2.12311 0.599676 + 1.28078i
111.7 1.28078 + 0.599676i −0.936426 1.28078 + 1.53610i 1.00000i −1.19935 0.561553i 2.60399 0.468213i 0.719224 + 2.73546i −2.12311 −0.599676 + 1.28078i
111.8 1.28078 + 0.599676i 0.936426 1.28078 + 1.53610i 1.00000i 1.19935 + 0.561553i −2.60399 0.468213i 0.719224 + 2.73546i −2.12311 0.599676 1.28078i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.g.c 8
3.b odd 2 1 1260.2.c.c 8
4.b odd 2 1 inner 140.2.g.c 8
5.b even 2 1 700.2.g.j 8
5.c odd 4 1 700.2.c.i 8
5.c odd 4 1 700.2.c.j 8
7.b odd 2 1 inner 140.2.g.c 8
7.c even 3 2 980.2.o.e 16
7.d odd 6 2 980.2.o.e 16
8.b even 2 1 2240.2.k.e 8
8.d odd 2 1 2240.2.k.e 8
12.b even 2 1 1260.2.c.c 8
20.d odd 2 1 700.2.g.j 8
20.e even 4 1 700.2.c.i 8
20.e even 4 1 700.2.c.j 8
21.c even 2 1 1260.2.c.c 8
28.d even 2 1 inner 140.2.g.c 8
28.f even 6 2 980.2.o.e 16
28.g odd 6 2 980.2.o.e 16
35.c odd 2 1 700.2.g.j 8
35.f even 4 1 700.2.c.i 8
35.f even 4 1 700.2.c.j 8
56.e even 2 1 2240.2.k.e 8
56.h odd 2 1 2240.2.k.e 8
84.h odd 2 1 1260.2.c.c 8
140.c even 2 1 700.2.g.j 8
140.j odd 4 1 700.2.c.i 8
140.j odd 4 1 700.2.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c 8 1.a even 1 1 trivial
140.2.g.c 8 4.b odd 2 1 inner
140.2.g.c 8 7.b odd 2 1 inner
140.2.g.c 8 28.d even 2 1 inner
700.2.c.i 8 5.c odd 4 1
700.2.c.i 8 20.e even 4 1
700.2.c.i 8 35.f even 4 1
700.2.c.i 8 140.j odd 4 1
700.2.c.j 8 5.c odd 4 1
700.2.c.j 8 20.e even 4 1
700.2.c.j 8 35.f even 4 1
700.2.c.j 8 140.j odd 4 1
700.2.g.j 8 5.b even 2 1
700.2.g.j 8 20.d odd 2 1
700.2.g.j 8 35.c odd 2 1
700.2.g.j 8 140.c even 2 1
980.2.o.e 16 7.c even 3 2
980.2.o.e 16 7.d odd 6 2
980.2.o.e 16 28.f even 6 2
980.2.o.e 16 28.g odd 6 2
1260.2.c.c 8 3.b odd 2 1
1260.2.c.c 8 12.b even 2 1
1260.2.c.c 8 21.c even 2 1
1260.2.c.c 8 84.h odd 2 1
2240.2.k.e 8 8.b even 2 1
2240.2.k.e 8 8.d odd 2 1
2240.2.k.e 8 56.e even 2 1
2240.2.k.e 8 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{19}^{4} - 28T_{19}^{2} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + 162 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 74 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 126 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 124 T^{2} + 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 218 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 92 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 324 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
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