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Properties

Label 484.2.c.d
Level $484$
Weight $2$
Character orbit 484.c
Analytic conductor $3.865$
Analytic rank $0$
Dimension $16$
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] = [484,2,Mod(483,484)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(484, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("484.483");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.c (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: \(3.86475945783\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 44)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} - \beta_{14} q^{3} + ( - \beta_{15} - \beta_{14} - \beta_1) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_1) q^{5} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{6}+ \cdots + ( - \beta_{15} - \beta_{11} - \beta_{8} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} - \beta_{14} q^{3} + ( - \beta_{15} - \beta_{14} - \beta_1) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_1) q^{5} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{6}+ \cdots + ( - 3 \beta_{13} - \beta_{12} + \cdots - 2 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 4 q^{5} - 22 q^{12} - 12 q^{14} - 12 q^{16} + 16 q^{20} - 4 q^{25} - 24 q^{26} - 6 q^{34} + 50 q^{36} - 12 q^{37} + 42 q^{38} + 4 q^{42} + 40 q^{45} + 74 q^{48} - 44 q^{49} - 52 q^{53} - 12 q^{56} - 60 q^{58} - 8 q^{60} + 28 q^{64} + 24 q^{69} - 68 q^{70} + 4 q^{78} - 8 q^{80} - 24 q^{81} + 26 q^{82} - 14 q^{86} - 36 q^{89} + 36 q^{92} + 72 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19 \nu^{15} - 43 \nu^{14} + 95 \nu^{13} - 139 \nu^{12} + 121 \nu^{11} - 18 \nu^{10} - 242 \nu^{9} + \cdots - 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79 \nu^{15} + 271 \nu^{14} - 595 \nu^{13} + 1023 \nu^{12} - 1125 \nu^{11} + 562 \nu^{10} + \cdots + 12544 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 175 \nu^{15} - 567 \nu^{14} + 1251 \nu^{13} - 2111 \nu^{12} + 2277 \nu^{11} - 1058 \nu^{10} + \cdots - 23680 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 101 \nu^{15} + 317 \nu^{14} - 701 \nu^{13} + 1173 \nu^{12} - 1247 \nu^{11} + 558 \nu^{10} + \cdots + 12352 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 215 \nu^{15} - 755 \nu^{14} + 1671 \nu^{13} - 2891 \nu^{12} + 3217 \nu^{11} - 1646 \nu^{10} + \cdots - 36480 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 219 \nu^{15} + 775 \nu^{14} - 1715 \nu^{13} + 2975 \nu^{12} - 3325 \nu^{11} + 1718 \nu^{10} + \cdots + 38144 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 113 \nu^{15} + 413 \nu^{14} - 913 \nu^{13} + 1597 \nu^{12} - 1807 \nu^{11} + 962 \nu^{10} + \cdots + 21440 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 341 \nu^{15} + 1069 \nu^{14} - 2361 \nu^{13} + 3949 \nu^{12} - 4191 \nu^{11} + 1870 \nu^{10} + \cdots + 41600 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 349 \nu^{15} - 1221 \nu^{14} + 2697 \nu^{13} - 4669 \nu^{12} + 5199 \nu^{11} - 2662 \nu^{10} + \cdots - 59264 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 199 \nu^{15} - 703 \nu^{14} + 1551 \nu^{13} - 2687 \nu^{12} + 2997 \nu^{11} - 1538 \nu^{10} + \cdots - 34368 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 367 \nu^{15} - 1383 \nu^{14} + 3051 \nu^{13} - 5383 \nu^{12} + 6141 \nu^{11} - 3338 \nu^{10} + \cdots - 74624 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 239 \nu^{15} - 901 \nu^{14} + 1989 \nu^{13} - 3509 \nu^{12} + 4003 \nu^{11} - 2180 \nu^{10} + \cdots - 48640 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 195 \nu^{15} - 703 \nu^{14} + 1551 \nu^{13} - 2703 \nu^{12} + 3037 \nu^{11} - 1590 \nu^{10} + \cdots - 35520 ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 819 \nu^{15} + 2871 \nu^{14} - 6339 \nu^{13} + 10967 \nu^{12} - 12197 \nu^{11} + 6230 \nu^{10} + \cdots + 138880 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 326 \nu^{15} - 1175 \nu^{14} + 2593 \nu^{13} - 4519 \nu^{12} + 5075 \nu^{11} - 2655 \nu^{10} + \cdots - 59296 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{12} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{13} + 4\beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4 \beta_{15} + 6 \beta_{14} - \beta_{13} - \beta_{12} + 4 \beta_{11} - 6 \beta_{10} - 4 \beta_{9} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4 \beta_{15} - 2 \beta_{14} - 7 \beta_{13} + \beta_{12} + 4 \beta_{11} - 2 \beta_{10} - 8 \beta_{9} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7 \beta_{15} + 3 \beta_{14} - 5 \beta_{13} - 3 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} + 11 \beta_{9} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3\beta_{15} + 9\beta_{14} + 10\beta_{11} - 8\beta_{8} + 8\beta_{7} + 16\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13 \beta_{15} + 41 \beta_{14} + 3 \beta_{13} - 19 \beta_{12} + 18 \beta_{11} + 10 \beta_{10} + 15 \beta_{9} + \cdots + 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 20 \beta_{15} + 22 \beta_{14} + 15 \beta_{13} + 23 \beta_{12} + 4 \beta_{11} + 34 \beta_{10} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16 \beta_{15} + 14 \beta_{14} + 35 \beta_{13} + 11 \beta_{12} - 92 \beta_{11} - 6 \beta_{10} - 32 \beta_{9} + \cdots - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 29 \beta_{15} + 45 \beta_{14} - 53 \beta_{13} + 9 \beta_{12} + 42 \beta_{11} - 6 \beta_{10} - 11 \beta_{9} + \cdots - 86 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 22\beta_{13} - 20\beta_{12} - 28\beta_{10} + 29\beta_{9} - 95\beta_{6} - 115\beta_{5} - 23\beta_{4} - 41\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
483.1
−0.204982 − 1.39928i
−0.204982 + 1.39928i
1.40958 − 0.114404i
1.40958 + 0.114404i
0.0737040 + 1.41229i
0.0737040 − 1.41229i
1.06665 − 0.928579i
1.06665 + 0.928579i
1.40874 + 0.124276i
1.40874 − 0.124276i
−1.36594 + 0.366325i
−1.36594 − 0.366325i
−0.544389 + 1.30524i
−0.544389 − 1.30524i
0.656642 + 1.25253i
0.656642 − 1.25253i
−1.39414 − 0.237451i − 0.918459i 1.88723 + 0.662079i 1.71472 −0.218089 + 1.28046i 2.39055 −2.47385 − 1.37116i 2.15643 −2.39055 − 0.407162i
483.2 −1.39414 + 0.237451i 0.918459i 1.88723 − 0.662079i 1.71472 −0.218089 − 1.28046i 2.39055 −2.47385 + 1.37116i 2.15643 −2.39055 + 0.407162i
483.3 −1.20762 − 0.735975i − 0.740676i 0.916683 + 1.77755i −1.34841 −0.545118 + 0.894453i −1.62837 0.201231 − 2.82126i 2.45140 1.62837 + 0.992398i
483.4 −1.20762 + 0.735975i 0.740676i 0.916683 − 1.77755i −1.34841 −0.545118 − 0.894453i −1.62837 0.201231 + 2.82126i 2.45140 1.62837 − 0.992398i
483.5 −0.889752 − 1.09925i 1.79313i −0.416683 + 1.95611i 2.96645 1.97110 − 1.59545i 2.63940 2.52099 − 1.28242i −0.215332 −2.63940 − 3.26086i
483.6 −0.889752 + 1.09925i − 1.79313i −0.416683 − 1.95611i 2.96645 1.97110 + 1.59545i 2.63940 2.52099 + 1.28242i −0.215332 −2.63940 + 3.26086i
483.7 −0.553519 − 1.30139i 2.71892i −1.38723 + 1.44069i −2.33275 3.53837 − 1.50497i −1.29122 2.64276 + 1.00788i −4.39250 1.29122 + 3.03582i
483.8 −0.553519 + 1.30139i − 2.71892i −1.38723 − 1.44069i −2.33275 3.53837 + 1.50497i −1.29122 2.64276 − 1.00788i −4.39250 1.29122 − 3.03582i
483.9 0.553519 − 1.30139i − 2.71892i −1.38723 − 1.44069i −2.33275 −3.53837 − 1.50497i 1.29122 −2.64276 + 1.00788i −4.39250 −1.29122 + 3.03582i
483.10 0.553519 + 1.30139i 2.71892i −1.38723 + 1.44069i −2.33275 −3.53837 + 1.50497i 1.29122 −2.64276 − 1.00788i −4.39250 −1.29122 − 3.03582i
483.11 0.889752 − 1.09925i − 1.79313i −0.416683 − 1.95611i 2.96645 −1.97110 − 1.59545i −2.63940 −2.52099 − 1.28242i −0.215332 2.63940 − 3.26086i
483.12 0.889752 + 1.09925i 1.79313i −0.416683 + 1.95611i 2.96645 −1.97110 + 1.59545i −2.63940 −2.52099 + 1.28242i −0.215332 2.63940 + 3.26086i
483.13 1.20762 − 0.735975i 0.740676i 0.916683 − 1.77755i −1.34841 0.545118 + 0.894453i 1.62837 −0.201231 − 2.82126i 2.45140 −1.62837 + 0.992398i
483.14 1.20762 + 0.735975i − 0.740676i 0.916683 + 1.77755i −1.34841 0.545118 − 0.894453i 1.62837 −0.201231 + 2.82126i 2.45140 −1.62837 − 0.992398i
483.15 1.39414 − 0.237451i 0.918459i 1.88723 − 0.662079i 1.71472 0.218089 + 1.28046i −2.39055 2.47385 − 1.37116i 2.15643 2.39055 − 0.407162i
483.16 1.39414 + 0.237451i − 0.918459i 1.88723 + 0.662079i 1.71472 0.218089 − 1.28046i −2.39055 2.47385 + 1.37116i 2.15643 2.39055 + 0.407162i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 483.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 484.2.c.d 16
4.b odd 2 1 inner 484.2.c.d 16
11.b odd 2 1 inner 484.2.c.d 16
11.c even 5 1 44.2.g.a ✓ 16
11.c even 5 1 484.2.g.f 16
11.c even 5 1 484.2.g.i 16
11.c even 5 1 484.2.g.j 16
11.d odd 10 1 44.2.g.a ✓ 16
11.d odd 10 1 484.2.g.f 16
11.d odd 10 1 484.2.g.i 16
11.d odd 10 1 484.2.g.j 16
33.f even 10 1 396.2.r.a 16
33.h odd 10 1 396.2.r.a 16
44.c even 2 1 inner 484.2.c.d 16
44.g even 10 1 44.2.g.a ✓ 16
44.g even 10 1 484.2.g.f 16
44.g even 10 1 484.2.g.i 16
44.g even 10 1 484.2.g.j 16
44.h odd 10 1 44.2.g.a ✓ 16
44.h odd 10 1 484.2.g.f 16
44.h odd 10 1 484.2.g.i 16
44.h odd 10 1 484.2.g.j 16
88.k even 10 1 704.2.u.c 16
88.l odd 10 1 704.2.u.c 16
88.o even 10 1 704.2.u.c 16
88.p odd 10 1 704.2.u.c 16
132.n odd 10 1 396.2.r.a 16
132.o even 10 1 396.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.g.a ✓ 16 11.c even 5 1
44.2.g.a ✓ 16 11.d odd 10 1
44.2.g.a ✓ 16 44.g even 10 1
44.2.g.a ✓ 16 44.h odd 10 1
396.2.r.a 16 33.f even 10 1
396.2.r.a 16 33.h odd 10 1
396.2.r.a 16 132.n odd 10 1
396.2.r.a 16 132.o even 10 1
484.2.c.d 16 1.a even 1 1 trivial
484.2.c.d 16 4.b odd 2 1 inner
484.2.c.d 16 11.b odd 2 1 inner
484.2.c.d 16 44.c even 2 1 inner
484.2.g.f 16 11.c even 5 1
484.2.g.f 16 11.d odd 10 1
484.2.g.f 16 44.g even 10 1
484.2.g.f 16 44.h odd 10 1
484.2.g.i 16 11.c even 5 1
484.2.g.i 16 11.d odd 10 1
484.2.g.i 16 44.g even 10 1
484.2.g.i 16 44.h odd 10 1
484.2.g.j 16 11.c even 5 1
484.2.g.j 16 11.d odd 10 1
484.2.g.j 16 44.g even 10 1
484.2.g.j 16 44.h odd 10 1
704.2.u.c 16 88.k even 10 1
704.2.u.c 16 88.l odd 10 1
704.2.u.c 16 88.o even 10 1
704.2.u.c 16 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(484, [\chi])\):

\( T_{3}^{8} + 12T_{3}^{6} + 39T_{3}^{4} + 38T_{3}^{2} + 11 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{3} - 9T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} + 12 T^{6} + \cdots + 11)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} - 9 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 17 T^{6} + \cdots + 176)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 21 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 46 T^{6} + 351 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 72 T^{6} + \cdots + 1331)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 88 T^{6} + \cdots + 2816)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 85 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 137 T^{6} + \cdots + 654896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{3} + \cdots + 116)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 186 T^{6} + \cdots + 2588881)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 143 T^{6} + \cdots + 148016)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 183 T^{6} + \cdots + 1098416)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 13 T^{3} + \cdots - 124)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 272 T^{6} + \cdots + 130691)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 229 T^{6} + \cdots + 633616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 185 T^{6} + \cdots + 1760000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 93 T^{6} + \cdots + 148016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 126 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 147 T^{6} + \cdots + 169136)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 270 T^{6} + \cdots + 660275)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} + \cdots + 116)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T - 61)^{8} \) Copy content Toggle raw display
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