Properties

Label 600.2.d.h
Level $600$
Weight $2$
Character orbit 600.d
Analytic conductor $4.791$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(349,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("600.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.d (of order \(2\), degree \(1\), not 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{4} - 5\nu^{3} - 6\nu^{2} + 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{4} + 5\nu^{3} - 2\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{4} - \nu^{3} - 2\nu^{2} - 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} + 4\nu^{5} + 6\nu^{4} - 11\nu^{3} - 8\nu^{2} + 4\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 4\nu^{6} + 8\nu^{5} + 22\nu^{4} - 35\nu^{3} - 22\nu^{2} - 20\nu + 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{7} - 10\nu^{6} - 12\nu^{5} - 46\nu^{4} + 71\nu^{3} + 32\nu^{2} + 44\nu - 168 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{6} + 5\beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 2\beta_{2} - 5\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{7} - 3\beta_{6} - 5\beta_{5} - \beta_{4} + \beta_{3} + 10\beta_{2} - 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{6} + 3\beta_{5} - 5\beta_{4} - 9\beta_{3} + 6\beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
−0.565036 1.29643i
−0.565036 + 1.29643i
1.23291 + 0.692769i
1.23291 0.692769i
1.41216 + 0.0762223i
1.41216 0.0762223i
−1.08003 0.912978i
−1.08003 + 0.912978i
−0.796431 1.16863i 1.00000 −0.731395 + 1.86147i 0 −0.796431 1.16863i 4.72294i 2.75787 0.627801i 1.00000 0
349.2 −0.796431 + 1.16863i 1.00000 −0.731395 1.86147i 0 −0.796431 + 1.16863i 4.72294i 2.75787 + 0.627801i 1.00000 0
349.3 −0.192769 1.40101i 1.00000 −1.92568 + 0.540143i 0 −0.192769 1.40101i 0.0802864i 1.12796 + 2.59378i 1.00000 0
349.4 −0.192769 + 1.40101i 1.00000 −1.92568 0.540143i 0 −0.192769 + 1.40101i 0.0802864i 1.12796 2.59378i 1.00000 0
349.5 0.576222 1.29150i 1.00000 −1.33594 1.48838i 0 0.576222 1.29150i 1.97676i −2.69204 + 0.867721i 1.00000 0
349.6 0.576222 + 1.29150i 1.00000 −1.33594 + 1.48838i 0 0.576222 + 1.29150i 1.97676i −2.69204 0.867721i 1.00000 0
349.7 1.41298 0.0591148i 1.00000 1.99301 0.167056i 0 1.41298 0.0591148i 1.33411i 2.80620 0.353863i 1.00000 0
349.8 1.41298 + 0.0591148i 1.00000 1.99301 + 0.167056i 0 1.41298 + 0.0591148i 1.33411i 2.80620 + 0.353863i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.d.h 8
3.b odd 2 1 1800.2.d.s 8
4.b odd 2 1 2400.2.d.g 8
5.b even 2 1 600.2.d.g 8
5.c odd 4 1 600.2.k.d 8
5.c odd 4 1 600.2.k.e yes 8
8.b even 2 1 600.2.d.g 8
8.d odd 2 1 2400.2.d.h 8
12.b even 2 1 7200.2.d.s 8
15.d odd 2 1 1800.2.d.t 8
15.e even 4 1 1800.2.k.q 8
15.e even 4 1 1800.2.k.t 8
20.d odd 2 1 2400.2.d.h 8
20.e even 4 1 2400.2.k.d 8
20.e even 4 1 2400.2.k.e 8
24.f even 2 1 7200.2.d.t 8
24.h odd 2 1 1800.2.d.t 8
40.e odd 2 1 2400.2.d.g 8
40.f even 2 1 inner 600.2.d.h 8
40.i odd 4 1 600.2.k.d 8
40.i odd 4 1 600.2.k.e yes 8
40.k even 4 1 2400.2.k.d 8
40.k even 4 1 2400.2.k.e 8
60.h even 2 1 7200.2.d.t 8
60.l odd 4 1 7200.2.k.r 8
60.l odd 4 1 7200.2.k.s 8
120.i odd 2 1 1800.2.d.s 8
120.m even 2 1 7200.2.d.s 8
120.q odd 4 1 7200.2.k.r 8
120.q odd 4 1 7200.2.k.s 8
120.w even 4 1 1800.2.k.q 8
120.w even 4 1 1800.2.k.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 5.b even 2 1
600.2.d.g 8 8.b even 2 1
600.2.d.h 8 1.a even 1 1 trivial
600.2.d.h 8 40.f even 2 1 inner
600.2.k.d 8 5.c odd 4 1
600.2.k.d 8 40.i odd 4 1
600.2.k.e yes 8 5.c odd 4 1
600.2.k.e yes 8 40.i odd 4 1
1800.2.d.s 8 3.b odd 2 1
1800.2.d.s 8 120.i odd 2 1
1800.2.d.t 8 15.d odd 2 1
1800.2.d.t 8 24.h odd 2 1
1800.2.k.q 8 15.e even 4 1
1800.2.k.q 8 120.w even 4 1
1800.2.k.t 8 15.e even 4 1
1800.2.k.t 8 120.w even 4 1
2400.2.d.g 8 4.b odd 2 1
2400.2.d.g 8 40.e odd 2 1
2400.2.d.h 8 8.d odd 2 1
2400.2.d.h 8 20.d odd 2 1
2400.2.k.d 8 20.e even 4 1
2400.2.k.d 8 40.k even 4 1
2400.2.k.e 8 20.e even 4 1
2400.2.k.e 8 40.k even 4 1
7200.2.d.s 8 12.b even 2 1
7200.2.d.s 8 120.m even 2 1
7200.2.d.t 8 24.f even 2 1
7200.2.d.t 8 60.h even 2 1
7200.2.k.r 8 60.l odd 4 1
7200.2.k.r 8 120.q odd 4 1
7200.2.k.s 8 60.l odd 4 1
7200.2.k.s 8 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{8} + 28T_{7}^{6} + 134T_{7}^{4} + 156T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + 32T_{11}^{6} + 336T_{11}^{4} + 1344T_{11}^{2} + 1600 \) Copy content Toggle raw display
\( T_{13}^{4} - 22T_{13}^{2} - 32T_{13} + 9 \) Copy content Toggle raw display
\( T_{37}^{4} - 64T_{37}^{2} - 128T_{37} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 4 T^{6} - 8 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + 134 T^{4} + 156 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{6} + 336 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{2} - 32 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + 1552 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{8} + 116 T^{6} + 4662 T^{4} + \cdots + 380689 \) Copy content Toggle raw display
$23$ \( T^{8} + 128 T^{6} + 4176 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$29$ \( T^{8} + 144 T^{6} + 6288 T^{4} + \cdots + 627264 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 70 T^{2} + 204 T + 673)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{2} - 128 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{2} + 56 T + 328)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 114 T^{2} + 180 T + 2089)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 144 T^{6} + 4832 T^{4} + \cdots + 30976 \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} - 120 T^{2} - 680 T - 152)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 432 T^{6} + \cdots + 31181056 \) Copy content Toggle raw display
$61$ \( T^{8} + 236 T^{6} + 14838 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$67$ \( (T^{4} + 12 T^{3} - 114 T^{2} - 892 T + 5097)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + 96 T^{2} - 72 T - 536)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 400 T^{6} + 41184 T^{4} + \cdots + 186624 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} - 184 T^{2} + 864 T + 8080)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} - 56 T^{2} + 1408 T - 1968)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 224 T^{2} - 64 T + 10880)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 420 T^{6} + \cdots + 78872161 \) Copy content Toggle raw display
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