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Properties

Label 144.2.k.b
Level $144$
Weight $2$
Character orbit 144.k
Analytic conductor $1.150$
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] = [144,2,Mod(37,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.k (of order \(4\), 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: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 + 1) q^{8} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{10} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + \beta_1) q^{11} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{13} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_1 - 1) q^{14} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{16} + ( - \beta_{7} - 2 \beta_{4} + 2 \beta_{2} + \beta_1 + 2) q^{17} + (2 \beta_{6} - 2 \beta_{2} - 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{20} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{22} - 2 \beta_{7} q^{23} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{25} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{26} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{28} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{31} + (2 \beta_{7} - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{32} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{34} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 - 4) q^{35} + ( - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{37} + ( - 2 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_1 + 2) q^{38} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{40} + (3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{41} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{43} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{44} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{46} - 2 \beta_1 q^{47} + ( - 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{49} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{4} + 4 \beta_{3} + 5) q^{50} + (2 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3) q^{52} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{53} + ( - 4 \beta_{7} - 4 \beta_{5}) q^{55} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{2} + 2) q^{56} + ( - 3 \beta_{6} + 5 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{58} + ( - 4 \beta_{5} - 4) q^{59} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 1) q^{61} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + 1) q^{62} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{64} + (\beta_{7} + 2 \beta_{4} - 2 \beta_{2} - 5 \beta_1) q^{65} + (4 \beta_{7} + 4 \beta_{4} - 4 \beta_{3} - 4) q^{67} + (4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 6) q^{68} + ( - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 6) q^{70} + ( - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{71} + ( - 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{73} + ( - 2 \beta_{7} + 3 \beta_{6} + \beta_{4} + 4 \beta_{2} - 2 \beta_1 - 5) q^{74} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 4) q^{76} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{77} + ( - 4 \beta_{7} + 5 \beta_{6} - 3 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{79} + ( - 2 \beta_{7} - 2 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{80} + ( - 2 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - \beta_{7} + 4 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{83}+ \cdots + (4 \beta_{7} + \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} + 12 q^{8} - 8 q^{10} + 8 q^{11} - 12 q^{14} - 8 q^{19} - 16 q^{20} - 20 q^{26} + 8 q^{28} + 16 q^{29} + 24 q^{31} - 24 q^{35} - 16 q^{37} + 8 q^{38} + 16 q^{40} - 8 q^{43} + 40 q^{44} - 8 q^{46} - 8 q^{49} + 36 q^{50} - 16 q^{52} - 16 q^{53} - 16 q^{58} - 32 q^{59} + 16 q^{61} + 12 q^{62} + 8 q^{64} + 16 q^{65} - 16 q^{67} - 32 q^{68} + 32 q^{70} - 52 q^{74} + 8 q^{76} - 16 q^{77} - 24 q^{79} - 8 q^{80} + 40 q^{82} + 40 q^{83} - 16 q^{85} + 16 q^{86} + 32 q^{88} - 8 q^{91} + 16 q^{92} + 8 q^{94} + 48 q^{95} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} - 3\nu^{5} + 10\nu^{4} - 15\nu^{3} + 19\nu^{2} - 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 42\nu^{4} - 59\nu^{3} + 48\nu^{2} - 24\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -7\nu^{7} + 25\nu^{6} - 87\nu^{5} + 158\nu^{4} - 231\nu^{3} + 206\nu^{2} - 118\nu + 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 257\nu^{3} - 224\nu^{2} + 130\nu - 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 166\nu - 42 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + 5\beta_{5} - 3\beta_{4} + 5\beta_{3} - 3\beta_{2} + 4\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} - 11\beta_{6} + 11\beta_{5} - 5\beta_{4} - \beta_{3} - 9\beta_{2} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 17\beta_{6} - 13\beta_{5} + 13\beta_{4} - 23\beta_{3} + 3\beta_{2} - 18\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -46\beta_{7} + 29\beta_{6} - 67\beta_{5} + 37\beta_{4} - 15\beta_{3} + 47\beta_{2} - 24\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -76\beta_{7} + 105\beta_{6} - 7\beta_{5} - 31\beta_{4} + 91\beta_{3} + 39\beta_{2} + 68\beta _1 - 83 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 − 0.0297061i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 − 2.10607i
0.500000 + 0.0297061i
0.500000 − 0.691860i
0.500000 − 1.44392i
0.500000 + 2.10607i
−0.874559 + 1.11137i 0 −0.470294 − 1.94392i 0.334904 + 0.334904i 0 4.55765i 2.57172 + 1.17740i 0 −0.665096 + 0.0793096i
37.2 −0.635665 − 1.26330i 0 −1.19186 + 1.60607i 2.68554 + 2.68554i 0 2.15894i 2.78658 + 0.484753i 0 1.68554 − 5.09976i
37.3 0.167452 − 1.40426i 0 −1.94392 − 0.470294i −1.74912 − 1.74912i 0 − 2.55765i −0.985930 + 2.65103i 0 −2.74912 + 2.16333i
37.4 1.34277 − 0.443806i 0 1.60607 − 1.19186i −1.27133 − 1.27133i 0 − 0.158942i 1.62764 − 2.31318i 0 −2.27133 − 1.14288i
109.1 −0.874559 − 1.11137i 0 −0.470294 + 1.94392i 0.334904 − 0.334904i 0 − 4.55765i 2.57172 − 1.17740i 0 −0.665096 − 0.0793096i
109.2 −0.635665 + 1.26330i 0 −1.19186 − 1.60607i 2.68554 − 2.68554i 0 − 2.15894i 2.78658 − 0.484753i 0 1.68554 + 5.09976i
109.3 0.167452 + 1.40426i 0 −1.94392 + 0.470294i −1.74912 + 1.74912i 0 2.55765i −0.985930 − 2.65103i 0 −2.74912 − 2.16333i
109.4 1.34277 + 0.443806i 0 1.60607 + 1.19186i −1.27133 + 1.27133i 0 0.158942i 1.62764 + 2.31318i 0 −2.27133 + 1.14288i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.b 8
3.b odd 2 1 48.2.j.a ✓ 8
4.b odd 2 1 576.2.k.b 8
8.b even 2 1 1152.2.k.c 8
8.d odd 2 1 1152.2.k.f 8
12.b even 2 1 192.2.j.a 8
16.e even 4 1 inner 144.2.k.b 8
16.e even 4 1 1152.2.k.c 8
16.f odd 4 1 576.2.k.b 8
16.f odd 4 1 1152.2.k.f 8
24.f even 2 1 384.2.j.a 8
24.h odd 2 1 384.2.j.b 8
32.g even 8 1 9216.2.a.y 4
32.g even 8 1 9216.2.a.bo 4
32.h odd 8 1 9216.2.a.x 4
32.h odd 8 1 9216.2.a.bn 4
48.i odd 4 1 48.2.j.a ✓ 8
48.i odd 4 1 384.2.j.b 8
48.k even 4 1 192.2.j.a 8
48.k even 4 1 384.2.j.a 8
96.o even 8 1 3072.2.a.n 4
96.o even 8 1 3072.2.a.o 4
96.o even 8 2 3072.2.d.i 8
96.p odd 8 1 3072.2.a.i 4
96.p odd 8 1 3072.2.a.t 4
96.p odd 8 2 3072.2.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a ✓ 8 3.b odd 2 1
48.2.j.a ✓ 8 48.i odd 4 1
144.2.k.b 8 1.a even 1 1 trivial
144.2.k.b 8 16.e even 4 1 inner
192.2.j.a 8 12.b even 2 1
192.2.j.a 8 48.k even 4 1
384.2.j.a 8 24.f even 2 1
384.2.j.a 8 48.k even 4 1
384.2.j.b 8 24.h odd 2 1
384.2.j.b 8 48.i odd 4 1
576.2.k.b 8 4.b odd 2 1
576.2.k.b 8 16.f odd 4 1
1152.2.k.c 8 8.b even 2 1
1152.2.k.c 8 16.e even 4 1
1152.2.k.f 8 8.d odd 2 1
1152.2.k.f 8 16.f odd 4 1
3072.2.a.i 4 96.p odd 8 1
3072.2.a.n 4 96.o even 8 1
3072.2.a.o 4 96.o even 8 1
3072.2.a.t 4 96.p odd 8 1
3072.2.d.f 8 96.p odd 8 2
3072.2.d.i 8 96.o even 8 2
9216.2.a.x 4 32.h odd 8 1
9216.2.a.y 4 32.g even 8 1
9216.2.a.bn 4 32.h odd 8 1
9216.2.a.bo 4 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16T_{5}^{5} + 128T_{5}^{4} + 192T_{5}^{3} + 128T_{5}^{2} - 128T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{6} - 4 T^{5} + 2 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{5} + 128 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} - 64 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 32 T^{6} - 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + 40 T^{2} - 24 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$41$ \( T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216 \) Copy content Toggle raw display
$97$ \( (T^{4} - 224 T^{2} + 768 T + 512)^{2} \) Copy content Toggle raw display
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