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Properties

Label 552.2.f.d
Level $552$
Weight $2$
Character orbit 552.f
Analytic conductor $4.408$
Analytic rank $0$
Dimension $20$
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] = [552,2,Mod(277,552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(552, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("552.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 552.f (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: \(4.40774219157\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + x^{16} - 4 x^{15} + 16 x^{14} - 24 x^{13} + 32 x^{12} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + \beta_{2} q^{4} + \beta_{11} q^{5} - \beta_{4} q^{6} - \beta_{12} q^{7} - \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{5} q^{3} + \beta_{2} q^{4} + \beta_{11} q^{5} - \beta_{4} q^{6} - \beta_{12} q^{7} - \beta_{3} q^{8} - q^{9} + (\beta_{17} - \beta_{12} + \cdots + \beta_{4}) q^{10}+ \cdots + (\beta_{13} - \beta_{10} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 2 q^{6} - 8 q^{7} - 2 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 2 q^{6} - 8 q^{7} - 2 q^{8} - 20 q^{9} - 12 q^{10} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 32 q^{17} + 2 q^{18} + 20 q^{20} + 14 q^{22} + 20 q^{23} + 10 q^{24} - 28 q^{25} + 18 q^{28} - 22 q^{32} + 28 q^{33} - 26 q^{34} + 16 q^{38} + 4 q^{40} - 40 q^{41} + 14 q^{42} + 10 q^{44} - 2 q^{46} + 40 q^{47} + 12 q^{49} - 14 q^{50} + 20 q^{52} - 2 q^{54} - 8 q^{55} + 6 q^{56} - 44 q^{57} - 32 q^{58} - 24 q^{60} + 20 q^{62} + 8 q^{63} - 48 q^{64} + 8 q^{65} + 10 q^{66} + 22 q^{68} + 80 q^{70} - 40 q^{71} + 2 q^{72} - 16 q^{73} - 30 q^{74} + 44 q^{76} - 36 q^{78} - 16 q^{79} + 4 q^{80} + 20 q^{81} - 32 q^{82} + 6 q^{84} - 4 q^{86} + 8 q^{87} - 70 q^{88} - 40 q^{89} + 12 q^{90} + 64 q^{94} - 40 q^{95} + 2 q^{96} + 32 q^{97} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + x^{16} - 4 x^{15} + 16 x^{14} - 24 x^{13} + 32 x^{12} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{17} + 4 \nu^{15} - 2 \nu^{14} + 3 \nu^{13} + 6 \nu^{12} - 14 \nu^{11} - 4 \nu^{10} + \cdots + 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{19} + 2 \nu^{18} - 2 \nu^{17} - 2 \nu^{16} - \nu^{15} + 20 \nu^{14} - 24 \nu^{13} + \cdots + 1536 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{19} - 2 \nu^{16} + 13 \nu^{15} - 14 \nu^{14} + 10 \nu^{13} - 8 \nu^{12} - 4 \nu^{11} + \cdots - 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{18} + 2 \nu^{15} - 5 \nu^{14} + 6 \nu^{13} - 10 \nu^{12} + 8 \nu^{11} - 4 \nu^{10} + \cdots - 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{19} + 2 \nu^{18} - 2 \nu^{17} + 2 \nu^{16} + 5 \nu^{15} + 16 \nu^{14} - 44 \nu^{13} + \cdots + 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{19} - 2 \nu^{17} + 6 \nu^{16} - \nu^{15} + 2 \nu^{14} + 12 \nu^{13} - 16 \nu^{12} + 80 \nu^{10} + \cdots - 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{18} + 4 \nu^{16} - 2 \nu^{15} + 3 \nu^{14} + 6 \nu^{13} - 14 \nu^{12} - 4 \nu^{11} + \cdots + 256 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{19} - 8 \nu^{18} + 6 \nu^{17} + 2 \nu^{16} - 11 \nu^{15} + 6 \nu^{14} + 32 \nu^{13} + \cdots + 1536 ) / 1024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{19} - 2 \nu^{18} + 2 \nu^{17} - 6 \nu^{16} + 7 \nu^{15} + 8 \nu^{14} - 4 \nu^{13} - 16 \nu^{12} + \cdots + 512 ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} + 6 \nu^{16} - 9 \nu^{15} - 10 \nu^{14} + 24 \nu^{13} + \cdots - 4608 ) / 1024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{19} - 2 \nu^{18} - 6 \nu^{17} - 2 \nu^{16} + \nu^{15} - 4 \nu^{14} + 24 \nu^{13} - 8 \nu^{12} + \cdots + 512 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{19} - 3 \nu^{18} + 2 \nu^{16} - 9 \nu^{15} + 11 \nu^{14} + 28 \nu^{13} - 58 \nu^{12} + \cdots - 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( \nu^{19} - 5 \nu^{18} + 6 \nu^{17} - 10 \nu^{16} + 3 \nu^{15} - 3 \nu^{14} + 14 \nu^{13} - 46 \nu^{12} + \cdots - 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3 \nu^{19} - 4 \nu^{18} + 2 \nu^{17} - 2 \nu^{16} + 7 \nu^{15} - 10 \nu^{14} + 8 \nu^{13} - 24 \nu^{12} + \cdots - 512 ) / 512 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 7 \nu^{19} - 2 \nu^{18} + 6 \nu^{17} - 10 \nu^{16} - \nu^{15} - 32 \nu^{14} + 72 \nu^{13} + \cdots - 2560 ) / 1024 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 2 \nu^{19} + 4 \nu^{18} - 3 \nu^{17} + 2 \nu^{16} - 2 \nu^{15} + 10 \nu^{14} - 23 \nu^{13} + \cdots + 1024 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{9} - \beta_{7} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} + \beta_{16} - 2 \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{19} + \beta_{17} - \beta_{16} + \beta_{15} - \beta_{13} - 2 \beta_{11} + \beta_{10} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{18} + 4 \beta_{16} - \beta_{13} - 2 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{18} - \beta_{17} + \beta_{16} - 2 \beta_{15} + \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7 \beta_{19} + 2 \beta_{18} - 6 \beta_{17} - 6 \beta_{16} + \beta_{15} - \beta_{14} - 4 \beta_{13} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - \beta_{19} + 8 \beta_{18} + 5 \beta_{17} - 5 \beta_{16} + 9 \beta_{15} - 8 \beta_{14} - \beta_{13} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16 \beta_{19} - 10 \beta_{18} - 16 \beta_{17} + 4 \beta_{16} - 8 \beta_{15} + 4 \beta_{14} - 5 \beta_{13} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4 \beta_{19} + 4 \beta_{18} - 13 \beta_{17} + 13 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11 \beta_{19} + 10 \beta_{18} - 6 \beta_{17} - 22 \beta_{16} + 21 \beta_{15} - 9 \beta_{14} + \cdots - 43 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 9 \beta_{19} - 24 \beta_{18} - 15 \beta_{17} + 15 \beta_{16} + \beta_{15} - 16 \beta_{14} + 39 \beta_{13} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 40 \beta_{19} + 14 \beta_{18} + 16 \beta_{17} + 20 \beta_{16} - 56 \beta_{15} + 7 \beta_{13} + 14 \beta_{12} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 4 \beta_{18} + 15 \beta_{17} - 79 \beta_{16} + 14 \beta_{15} - 55 \beta_{14} + 12 \beta_{13} + \cdots - 159 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 97 \beta_{19} + 2 \beta_{18} + 58 \beta_{17} + 26 \beta_{16} + 57 \beta_{15} - 33 \beta_{14} - 4 \beta_{13} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 31 \beta_{19} - 56 \beta_{18} + 173 \beta_{17} - 13 \beta_{16} + 105 \beta_{15} - 24 \beta_{14} + \cdots - 192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/552\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(185\) \(277\) \(415\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.38506 + 0.285679i
1.38506 − 0.285679i
1.32157 + 0.503453i
1.32157 − 0.503453i
1.19161 + 0.761625i
1.19161 − 0.761625i
0.585586 + 1.28728i
0.585586 − 1.28728i
0.459619 + 1.33744i
0.459619 − 1.33744i
−0.133806 + 1.40787i
−0.133806 − 1.40787i
−0.238838 + 1.39390i
−0.238838 − 1.39390i
−1.08648 + 0.905299i
−1.08648 − 0.905299i
−1.14290 + 0.832928i
−1.14290 − 0.832928i
−1.34141 + 0.447910i
−1.34141 − 0.447910i
−1.38506 − 0.285679i 1.00000i 1.83677 + 0.791365i − 2.43947i 0.285679 − 1.38506i −3.50435 −2.31796 − 1.62082i −1.00000 −0.696905 + 3.37881i
277.2 −1.38506 + 0.285679i − 1.00000i 1.83677 − 0.791365i 2.43947i 0.285679 + 1.38506i −3.50435 −2.31796 + 1.62082i −1.00000 −0.696905 − 3.37881i
277.3 −1.32157 − 0.503453i − 1.00000i 1.49307 + 1.33069i − 2.69719i −0.503453 + 1.32157i 1.19970 −1.30325 − 2.51029i −1.00000 −1.35791 + 3.56451i
277.4 −1.32157 + 0.503453i 1.00000i 1.49307 − 1.33069i 2.69719i −0.503453 − 1.32157i 1.19970 −1.30325 + 2.51029i −1.00000 −1.35791 − 3.56451i
277.5 −1.19161 − 0.761625i 1.00000i 0.839855 + 1.81512i 1.20415i 0.761625 − 1.19161i 3.87524 0.381661 − 2.80256i −1.00000 0.917114 − 1.43488i
277.6 −1.19161 + 0.761625i − 1.00000i 0.839855 − 1.81512i − 1.20415i 0.761625 + 1.19161i 3.87524 0.381661 + 2.80256i −1.00000 0.917114 + 1.43488i
277.7 −0.585586 − 1.28728i 1.00000i −1.31418 + 1.50763i − 1.66345i 1.28728 − 0.585586i −0.540858 2.71030 + 0.808871i −1.00000 −2.14133 + 0.974093i
277.8 −0.585586 + 1.28728i − 1.00000i −1.31418 − 1.50763i 1.66345i 1.28728 + 0.585586i −0.540858 2.71030 − 0.808871i −1.00000 −2.14133 − 0.974093i
277.9 −0.459619 − 1.33744i − 1.00000i −1.57750 + 1.22943i − 2.04946i −1.33744 + 0.459619i −4.47395 2.36933 + 1.54475i −1.00000 −2.74104 + 0.941972i
277.10 −0.459619 + 1.33744i 1.00000i −1.57750 − 1.22943i 2.04946i −1.33744 − 0.459619i −4.47395 2.36933 − 1.54475i −1.00000 −2.74104 − 0.941972i
277.11 0.133806 − 1.40787i − 1.00000i −1.96419 − 0.376764i 2.81880i −1.40787 − 0.133806i 1.52994 −0.793255 + 2.71491i −1.00000 3.96850 + 0.377173i
277.12 0.133806 + 1.40787i 1.00000i −1.96419 + 0.376764i − 2.81880i −1.40787 + 0.133806i 1.52994 −0.793255 − 2.71491i −1.00000 3.96850 − 0.377173i
277.13 0.238838 − 1.39390i 1.00000i −1.88591 − 0.665833i − 4.08024i 1.39390 + 0.238838i −1.92513 −1.37853 + 2.46975i −1.00000 −5.68744 − 0.974516i
277.14 0.238838 + 1.39390i − 1.00000i −1.88591 + 0.665833i 4.08024i 1.39390 − 0.238838i −1.92513 −1.37853 − 2.46975i −1.00000 −5.68744 + 0.974516i
277.15 1.08648 − 0.905299i 1.00000i 0.360869 − 1.96717i 3.71494i 0.905299 + 1.08648i 1.13753 −1.38880 − 2.46399i −1.00000 3.36313 + 4.03621i
277.16 1.08648 + 0.905299i − 1.00000i 0.360869 + 1.96717i − 3.71494i 0.905299 − 1.08648i 1.13753 −1.38880 + 2.46399i −1.00000 3.36313 − 4.03621i
277.17 1.14290 − 0.832928i − 1.00000i 0.612462 − 1.90391i − 1.03589i −0.832928 − 1.14290i −3.71260 −0.885838 − 2.68613i −1.00000 −0.862822 − 1.18392i
277.18 1.14290 + 0.832928i 1.00000i 0.612462 + 1.90391i 1.03589i −0.832928 + 1.14290i −3.71260 −0.885838 + 2.68613i −1.00000 −0.862822 + 1.18392i
277.19 1.34141 − 0.447910i 1.00000i 1.59875 − 1.20166i − 1.69969i 0.447910 + 1.34141i 2.41449 1.60635 − 2.32801i −1.00000 −0.761306 − 2.27997i
277.20 1.34141 + 0.447910i − 1.00000i 1.59875 + 1.20166i 1.69969i 0.447910 − 1.34141i 2.41449 1.60635 + 2.32801i −1.00000 −0.761306 + 2.27997i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.f.d ✓ 20
4.b odd 2 1 2208.2.f.d 20
8.b even 2 1 inner 552.2.f.d ✓ 20
8.d odd 2 1 2208.2.f.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.f.d ✓ 20 1.a even 1 1 trivial
552.2.f.d ✓ 20 8.b even 2 1 inner
2208.2.f.d 20 4.b odd 2 1
2208.2.f.d 20 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 64 T_{5}^{18} + 1720 T_{5}^{16} + 25520 T_{5}^{14} + 231268 T_{5}^{12} + 1335504 T_{5}^{10} + \cdots + 4129024 \) acting on \(S_{2}^{\mathrm{new}}(552, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 2 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 64 T^{18} + \cdots + 4129024 \) Copy content Toggle raw display
$7$ \( (T^{10} + 4 T^{9} + \cdots - 1184)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 124 T^{18} + \cdots + 12390400 \) Copy content Toggle raw display
$13$ \( T^{20} + 144 T^{18} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{10} - 16 T^{9} + \cdots + 320)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 45741232384 \) Copy content Toggle raw display
$23$ \( (T - 1)^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + 184 T^{18} + \cdots + 4194304 \) Copy content Toggle raw display
$31$ \( (T^{10} - 108 T^{8} + \cdots + 149504)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1507276754944 \) Copy content Toggle raw display
$41$ \( (T^{10} + 20 T^{9} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 2766181017856 \) Copy content Toggle raw display
$47$ \( (T^{10} - 20 T^{9} + \cdots - 5888)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 17615950602496 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1504920469504 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 425864306790400 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 259375405578496 \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} + \cdots + 512000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 8 T^{9} + \cdots + 507904)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 8 T^{9} + \cdots - 37745440)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{10} + 20 T^{9} + \cdots - 272960)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 16 T^{9} + \cdots - 1078295552)^{2} \) Copy content Toggle raw display
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