Properties

Label 210.2.t.e
Level $210$
Weight $2$
Character orbit 210.t
Analytic conductor $1.677$
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] = [210,2,Mod(59,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.t (of order \(6\), 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: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 7\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{6} - 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{5} - 21\beta_{3} + 29\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(\beta_{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−1.01575 + 1.40294i
−1.72286 + 0.178197i
1.72286 0.178197i
1.01575 1.40294i
−1.01575 1.40294i
−1.72286 0.178197i
1.72286 + 0.178197i
1.01575 + 1.40294i
−0.500000 0.866025i −1.72286 0.178197i −0.500000 + 0.866025i 2.20711 + 0.358719i 0.707107 + 1.58114i −1.41421 + 2.23607i 1.00000 2.93649 + 0.614017i −0.792893 2.09077i
59.2 −0.500000 0.866025i −1.01575 1.40294i −0.500000 + 0.866025i 0.792893 2.09077i −0.707107 + 1.58114i 1.41421 2.23607i 1.00000 −0.936492 + 2.85008i −2.20711 + 0.358719i
59.3 −0.500000 0.866025i 1.01575 + 1.40294i −0.500000 + 0.866025i 2.20711 + 0.358719i 0.707107 1.58114i −1.41421 2.23607i 1.00000 −0.936492 + 2.85008i −0.792893 2.09077i
59.4 −0.500000 0.866025i 1.72286 + 0.178197i −0.500000 + 0.866025i 0.792893 2.09077i −0.707107 1.58114i 1.41421 + 2.23607i 1.00000 2.93649 + 0.614017i −2.20711 + 0.358719i
89.1 −0.500000 + 0.866025i −1.72286 + 0.178197i −0.500000 0.866025i 2.20711 0.358719i 0.707107 1.58114i −1.41421 2.23607i 1.00000 2.93649 0.614017i −0.792893 + 2.09077i
89.2 −0.500000 + 0.866025i −1.01575 + 1.40294i −0.500000 0.866025i 0.792893 + 2.09077i −0.707107 1.58114i 1.41421 + 2.23607i 1.00000 −0.936492 2.85008i −2.20711 0.358719i
89.3 −0.500000 + 0.866025i 1.01575 1.40294i −0.500000 0.866025i 2.20711 0.358719i 0.707107 + 1.58114i −1.41421 + 2.23607i 1.00000 −0.936492 2.85008i −0.792893 + 2.09077i
89.4 −0.500000 + 0.866025i 1.72286 0.178197i −0.500000 0.866025i 0.792893 + 2.09077i −0.707107 + 1.58114i 1.41421 2.23607i 1.00000 2.93649 0.614017i −2.20711 0.358719i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.t.e 8
3.b odd 2 1 210.2.t.f yes 8
5.b even 2 1 210.2.t.f yes 8
5.c odd 4 2 1050.2.s.i 16
7.c even 3 1 1470.2.d.f 8
7.d odd 6 1 inner 210.2.t.e 8
7.d odd 6 1 1470.2.d.f 8
15.d odd 2 1 inner 210.2.t.e 8
15.e even 4 2 1050.2.s.i 16
21.g even 6 1 210.2.t.f yes 8
21.g even 6 1 1470.2.d.e 8
21.h odd 6 1 1470.2.d.e 8
35.i odd 6 1 210.2.t.f yes 8
35.i odd 6 1 1470.2.d.e 8
35.j even 6 1 1470.2.d.e 8
35.k even 12 2 1050.2.s.i 16
105.o odd 6 1 1470.2.d.f 8
105.p even 6 1 inner 210.2.t.e 8
105.p even 6 1 1470.2.d.f 8
105.w odd 12 2 1050.2.s.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.e 8 1.a even 1 1 trivial
210.2.t.e 8 7.d odd 6 1 inner
210.2.t.e 8 15.d odd 2 1 inner
210.2.t.e 8 105.p even 6 1 inner
210.2.t.f yes 8 3.b odd 2 1
210.2.t.f yes 8 5.b even 2 1
210.2.t.f yes 8 21.g even 6 1
210.2.t.f yes 8 35.i odd 6 1
1050.2.s.i 16 5.c odd 4 2
1050.2.s.i 16 15.e even 4 2
1050.2.s.i 16 35.k even 12 2
1050.2.s.i 16 105.w odd 12 2
1470.2.d.e 8 21.g even 6 1
1470.2.d.e 8 21.h odd 6 1
1470.2.d.e 8 35.i odd 6 1
1470.2.d.e 8 35.j even 6 1
1470.2.d.f 8 7.c even 3 1
1470.2.d.f 8 7.d odd 6 1
1470.2.d.f 8 105.o odd 6 1
1470.2.d.f 8 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\):

\( T_{11}^{8} - 22T_{11}^{6} + 483T_{11}^{4} - 22T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} - 46T_{13}^{2} + 49 \) Copy content Toggle raw display
\( T_{23}^{4} - 2T_{23}^{3} + 33T_{23}^{2} + 58T_{23} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + 7 T^{4} - 36 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} + 17 T^{2} - 30 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{6} + 483 T^{4} - 22 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} - 46 T^{2} + 49)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 5 T^{2} + 42 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + 33 T^{2} + 58 T + 841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 52 T^{2} + 196)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + 50 T^{2} + 24 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 58 T^{6} + 3003 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$41$ \( (T^{4} - 46 T^{2} + 49)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 52 T^{2} + 196)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} - 75 T^{2} - 522 T + 7569)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + 136 T^{6} + 15792 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 232 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$71$ \( (T^{4} + 52 T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 156 T^{6} + 22572 T^{4} + \cdots + 3111696 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + 138 T^{2} + 72 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 296 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 136 T^{6} + 15792 T^{4} + \cdots + 7311616 \) Copy content Toggle raw display
$97$ \( (T^{4} - 156 T^{2} + 1764)^{2} \) Copy content Toggle raw display
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