Properties

Label 525.2.i.h
Level $525$
Weight $2$
Character orbit 525.i
Analytic conductor $4.192$
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] = [525,2,Mod(151,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.i (of order \(3\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\nu^{7} - 215\nu^{6} + 357\nu^{5} + 646\nu^{4} - 1444\nu^{3} + 1156\nu^{2} + 561\nu + 5468 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 84\nu^{7} - 16\nu^{6} + 441\nu^{5} + 798\nu^{4} + 3208\nu^{3} + 1428\nu^{2} + 693\nu + 2262 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -754\nu^{7} + 1760\nu^{6} - 6080\nu^{5} + 1323\nu^{4} - 13440\nu^{3} + 12640\nu^{2} - 16828\nu + 5760 ) / 12729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 17165\nu - 9252 ) / 12729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 5\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 2\beta_{5} - 9\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 20\beta_{6} - 8\beta_{5} - 18\beta_{4} - 33\beta_{3} - 20\beta_{2} - 33\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{7} - 83\beta_{3} - 61\beta_{2} + 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164\beta_{6} + 61\beta_{5} + 146\beta_{4} + 243\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−0.758290 + 1.31340i
−0.276205 + 0.478401i
0.643668 1.11487i
1.39083 2.40898i
−0.758290 1.31340i
−0.276205 0.478401i
0.643668 + 1.11487i
1.39083 + 2.40898i
−1.25829 2.17942i 0.500000 0.866025i −2.16659 + 3.75264i 0 −2.51658 2.29673 + 1.31340i 5.87162 −0.500000 0.866025i 0
151.2 −0.776205 1.34443i 0.500000 0.866025i −0.204988 + 0.355049i 0 −1.55241 −2.60214 + 0.478401i −2.46837 −0.500000 0.866025i 0
151.3 0.143668 + 0.248840i 0.500000 0.866025i 0.958719 1.66055i 0 0.287336 2.39939 1.11487i 1.12562 −0.500000 0.866025i 0
151.4 0.890827 + 1.54296i 0.500000 0.866025i −0.587145 + 1.01696i 0 1.78165 −1.09398 2.40898i 1.47113 −0.500000 0.866025i 0
226.1 −1.25829 + 2.17942i 0.500000 + 0.866025i −2.16659 3.75264i 0 −2.51658 2.29673 1.31340i 5.87162 −0.500000 + 0.866025i 0
226.2 −0.776205 + 1.34443i 0.500000 + 0.866025i −0.204988 0.355049i 0 −1.55241 −2.60214 0.478401i −2.46837 −0.500000 + 0.866025i 0
226.3 0.143668 0.248840i 0.500000 + 0.866025i 0.958719 + 1.66055i 0 0.287336 2.39939 + 1.11487i 1.12562 −0.500000 + 0.866025i 0
226.4 0.890827 1.54296i 0.500000 + 0.866025i −0.587145 1.01696i 0 1.78165 −1.09398 + 2.40898i 1.47113 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.h 8
5.b even 2 1 525.2.i.k 8
5.c odd 4 2 105.2.q.a 16
7.c even 3 1 inner 525.2.i.h 8
7.c even 3 1 3675.2.a.bz 4
7.d odd 6 1 3675.2.a.cb 4
15.e even 4 2 315.2.bf.b 16
20.e even 4 2 1680.2.di.d 16
35.f even 4 2 735.2.q.g 16
35.i odd 6 1 3675.2.a.bn 4
35.j even 6 1 525.2.i.k 8
35.j even 6 1 3675.2.a.bp 4
35.k even 12 2 735.2.d.e 8
35.k even 12 2 735.2.q.g 16
35.l odd 12 2 105.2.q.a 16
35.l odd 12 2 735.2.d.d 8
105.w odd 12 2 2205.2.d.o 8
105.x even 12 2 315.2.bf.b 16
105.x even 12 2 2205.2.d.s 8
140.w even 12 2 1680.2.di.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 5.c odd 4 2
105.2.q.a 16 35.l odd 12 2
315.2.bf.b 16 15.e even 4 2
315.2.bf.b 16 105.x even 12 2
525.2.i.h 8 1.a even 1 1 trivial
525.2.i.h 8 7.c even 3 1 inner
525.2.i.k 8 5.b even 2 1
525.2.i.k 8 35.j even 6 1
735.2.d.d 8 35.l odd 12 2
735.2.d.e 8 35.k even 12 2
735.2.q.g 16 35.f even 4 2
735.2.q.g 16 35.k even 12 2
1680.2.di.d 16 20.e even 4 2
1680.2.di.d 16 140.w even 12 2
2205.2.d.o 8 105.w odd 12 2
2205.2.d.s 8 105.x even 12 2
3675.2.a.bn 4 35.i odd 6 1
3675.2.a.bp 4 35.j even 6 1
3675.2.a.bz 4 7.c even 3 1
3675.2.a.cb 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 4T_{2}^{5} + 26T_{2}^{4} + 16T_{2}^{3} + 44T_{2}^{2} - 12T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 8 T^{6} + 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} - 8 T^{6} + 14 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{6} - 28 T^{5} + \cdots + 900 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 28 T^{2} + 36 T + 127)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + 32 T^{6} + \cdots + 17956 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + 106 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{8} + 10 T^{7} + 132 T^{6} + \cdots + 2268036 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 38 T^{2} - 190 T - 22)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + 70 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + 372 T^{6} + \cdots + 822649 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 50 T^{2} + 146 T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} - 32 T^{2} + 154 T - 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + 84 T^{6} + \cdots + 14400 \) Copy content Toggle raw display
$53$ \( T^{8} + 20 T^{7} + 280 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + 10 T^{6} + 38 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + 164 T^{6} + \cdots + 250000 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + 108 T^{6} + \cdots + 819025 \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} - 90 T^{2} - 1334 T - 3202)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} + 208 T^{6} + \cdots + 1929321 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} + 218 T^{6} + \cdots + 50140561 \) Copy content Toggle raw display
$83$ \( (T^{4} + 6 T^{3} - 56 T^{2} - 238 T + 362)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 8 T^{7} + 258 T^{6} + \cdots + 285156 \) Copy content Toggle raw display
$97$ \( (T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
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