↑

Properties

Label 392.3.k.o
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(67,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.k (of order \(6\), degree \(2\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 3 x^{14} + 6 x^{13} - 22 x^{12} + 44 x^{11} - 20 x^{10} - 112 x^{9} + 368 x^{8} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + (7 \beta_{15} + 7 \beta_{12} + \cdots + 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 8 q^{3} - 5 q^{4} - 44 q^{6} + 26 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} + 8 q^{3} - 5 q^{4} - 44 q^{6} + 26 q^{8} - 48 q^{9} - 16 q^{10} + 32 q^{11} - 30 q^{12} + 71 q^{16} + 80 q^{17} + 29 q^{18} - 56 q^{19} - 216 q^{20} + 132 q^{22} - 22 q^{24} + 16 q^{25} - 24 q^{26} - 64 q^{27} - 96 q^{30} + 19 q^{32} - 32 q^{33} + 148 q^{34} - 66 q^{36} + 14 q^{38} - 84 q^{40} + 256 q^{41} - 50 q^{44} + 152 q^{46} + 268 q^{48} + 66 q^{50} + 368 q^{51} - 132 q^{52} + 228 q^{54} + 112 q^{57} - 24 q^{58} - 104 q^{59} - 192 q^{60} + 240 q^{62} - 110 q^{64} + 72 q^{65} + 276 q^{66} - 304 q^{67} + 190 q^{68} + 209 q^{72} + 112 q^{73} - 8 q^{74} - 72 q^{75} + 140 q^{76} - 608 q^{78} - 124 q^{80} - 48 q^{81} - 450 q^{82} + 144 q^{83} - 210 q^{86} + 486 q^{88} + 512 q^{89} - 368 q^{90} - 944 q^{92} - 472 q^{94} - 558 q^{96} + 128 q^{97} + 512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 3 x^{14} + 6 x^{13} - 22 x^{12} + 44 x^{11} - 20 x^{10} - 112 x^{9} + 368 x^{8} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9 \nu^{15} - 71 \nu^{14} + 165 \nu^{13} - 246 \nu^{12} - 234 \nu^{11} + 852 \nu^{10} + \cdots + 49152 ) / 802816 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{15} - 71 \nu^{14} + 165 \nu^{13} - 246 \nu^{12} - 234 \nu^{11} + 852 \nu^{10} + \cdots + 49152 ) / 802816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{15} - 19 \nu^{14} - 79 \nu^{13} - 74 \nu^{12} - 50 \nu^{11} - 356 \nu^{10} - 572 \nu^{9} + \cdots + 98304 ) / 401408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43 \nu^{15} + 173 \nu^{14} + 361 \nu^{13} - 1174 \nu^{12} + 3326 \nu^{11} - 4300 \nu^{10} + \cdots + 7880704 ) / 802816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4 \nu^{15} + 13 \nu^{14} - 43 \nu^{13} + 105 \nu^{12} - 108 \nu^{11} - 178 \nu^{10} + 1024 \nu^{9} + \cdots - 92160 ) / 100352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{15} - 81 \nu^{14} + 35 \nu^{13} + 226 \nu^{12} - 446 \nu^{11} + 1204 \nu^{10} + \cdots - 851968 ) / 401408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{15} - 12 \nu^{14} + 12 \nu^{13} + 27 \nu^{12} - 78 \nu^{11} + 162 \nu^{10} - 60 \nu^{9} + \cdots - 36864 ) / 50176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 103 \nu^{15} - 23 \nu^{14} + 213 \nu^{13} - 758 \nu^{12} + 694 \nu^{11} + 212 \nu^{10} + \cdots + 4489216 ) / 802816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 117 \nu^{15} + 13 \nu^{14} - 215 \nu^{13} + 74 \nu^{12} - 738 \nu^{11} + 1588 \nu^{10} + \cdots - 4767744 ) / 802816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19 \nu^{15} + 78 \nu^{14} - 42 \nu^{13} - 91 \nu^{12} + 738 \nu^{11} - 1050 \nu^{10} + \cdots + 964608 ) / 100352 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23 \nu^{15} + 24 \nu^{14} + 48 \nu^{13} - 245 \nu^{12} + 618 \nu^{11} - 318 \nu^{10} + \cdots + 612352 ) / 100352 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 233 \nu^{15} + 17 \nu^{14} - 611 \nu^{13} - 670 \nu^{12} + 2326 \nu^{11} - 4444 \nu^{10} + \cdots - 2146304 ) / 802816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 107 \nu^{15} + 111 \nu^{14} + 91 \nu^{13} - 854 \nu^{12} + 1642 \nu^{11} - 1596 \nu^{10} + \cdots + 425984 ) / 401408 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 131 \nu^{15} - 11 \nu^{14} - 335 \nu^{13} + 1466 \nu^{12} - 2994 \nu^{11} + 724 \nu^{10} + \cdots - 1425408 ) / 401408 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{8} + \beta_{7} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + 2\beta_{5} - \beta_{4} - 10\beta_{2} - 2\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - \beta_{10} + 6\beta_{8} - 7\beta_{7} - 2\beta_{5} - \beta_{3} - 4\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} - 11 \beta_{14} - 2 \beta_{12} + 2 \beta_{11} - 5 \beta_{9} - 8 \beta_{8} - 11 \beta_{7} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{14} - 6 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} - \beta_{10} - 9 \beta_{9} + \beta_{6} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{15} + 6 \beta_{13} + 55 \beta_{10} + 20 \beta_{8} + 23 \beta_{7} + 46 \beta_{5} + \cdots + 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 31 \beta_{15} - 77 \beta_{14} + 30 \beta_{12} - 26 \beta_{11} - 11 \beta_{9} - 44 \beta_{8} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 137 \beta_{14} - 46 \beta_{13} + 46 \beta_{12} + 62 \beta_{11} + 225 \beta_{10} + 121 \beta_{9} + \cdots + 230 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 201 \beta_{15} - 22 \beta_{13} - 99 \beta_{10} + 244 \beta_{8} + 197 \beta_{7} - 70 \beta_{5} + \cdots + 244 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 75 \beta_{15} + 609 \beta_{14} - 46 \beta_{12} + 402 \beta_{11} - 273 \beta_{9} + 1780 \beta_{8} + \cdots + 842 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 237 \beta_{14} - 554 \beta_{13} + 554 \beta_{12} - 150 \beta_{11} - 1141 \beta_{10} - 1405 \beta_{9} + \cdots - 3166 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 291 \beta_{15} - 1986 \beta_{13} + 1167 \beta_{10} - 2676 \beta_{8} - 1065 \beta_{7} + \cdots - 2676 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 537 \beta_{15} - 885 \beta_{14} - 282 \beta_{12} + 582 \beta_{11} + 4197 \beta_{9} + 1436 \beta_{8} + \cdots - 26514 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.94657 + 0.459219i
1.56937 + 1.23978i
0.944308 − 1.76303i
0.288997 + 1.97901i
0.109554 − 1.99700i
−0.575587 + 1.91538i
−1.78423 − 0.903622i
−1.99898 − 0.0637211i
1.94657 − 0.459219i
1.56937 − 1.23978i
0.944308 + 1.76303i
0.288997 − 1.97901i
0.109554 + 1.99700i
−0.575587 − 1.91538i
−1.78423 + 0.903622i
−1.99898 + 0.0637211i
−1.94657 − 0.459219i 2.61182 − 4.52380i 3.57824 + 1.78780i −5.42814 + 3.13394i −7.16148 + 7.60647i 0 −6.14428 − 5.12327i −9.14316 − 15.8364i 12.0054 − 3.60771i
67.2 −1.56937 − 1.23978i −0.0487183 + 0.0843825i 0.925871 + 3.89137i 3.00119 − 1.73274i 0.181073 − 0.0720276i 0 3.37142 − 7.25490i 4.49525 + 7.78601i −6.85820 − 1.00151i
67.3 −0.944308 + 1.76303i 1.72064 − 2.98023i −2.21656 − 3.32969i 4.22869 − 2.44143i 3.62943 + 5.84780i 0 7.96347 − 0.763618i −1.42120 − 2.46158i 0.311142 + 9.76078i
67.4 −0.288997 − 1.97901i −0.0487183 + 0.0843825i −3.83296 + 1.14386i −3.00119 + 1.73274i 0.181073 + 0.0720276i 0 3.37142 + 7.25490i 4.49525 + 7.78601i 4.29644 + 5.43862i
67.5 −0.109554 + 1.99700i −2.28374 + 3.95555i −3.97600 − 0.437557i 4.96451 − 2.86626i −7.64902 − 4.99396i 0 1.30939 − 7.89212i −5.93090 − 10.2726i 5.18003 + 10.2281i
67.6 0.575587 − 1.91538i 2.61182 − 4.52380i −3.33740 − 2.20494i 5.42814 − 3.13394i −7.16148 − 7.60647i 0 −6.14428 + 5.12327i −9.14316 − 15.8364i −2.87833 − 12.2008i
67.7 1.78423 + 0.903622i −2.28374 + 3.95555i 2.36693 + 3.22453i −4.96451 + 2.86626i −7.64902 + 4.99396i 0 1.30939 + 7.89212i −5.93090 − 10.2726i −11.4478 + 0.628019i
67.8 1.99898 + 0.0637211i 1.72064 − 2.98023i 3.99188 + 0.254755i −4.22869 + 2.44143i 3.62943 − 5.84780i 0 7.96347 + 0.763618i −1.42120 − 2.46158i −8.60865 + 4.61093i
275.1 −1.94657 + 0.459219i 2.61182 + 4.52380i 3.57824 − 1.78780i −5.42814 − 3.13394i −7.16148 − 7.60647i 0 −6.14428 + 5.12327i −9.14316 + 15.8364i 12.0054 + 3.60771i
275.2 −1.56937 + 1.23978i −0.0487183 − 0.0843825i 0.925871 − 3.89137i 3.00119 + 1.73274i 0.181073 + 0.0720276i 0 3.37142 + 7.25490i 4.49525 − 7.78601i −6.85820 + 1.00151i
275.3 −0.944308 − 1.76303i 1.72064 + 2.98023i −2.21656 + 3.32969i 4.22869 + 2.44143i 3.62943 − 5.84780i 0 7.96347 + 0.763618i −1.42120 + 2.46158i 0.311142 − 9.76078i
275.4 −0.288997 + 1.97901i −0.0487183 − 0.0843825i −3.83296 − 1.14386i −3.00119 − 1.73274i 0.181073 − 0.0720276i 0 3.37142 − 7.25490i 4.49525 − 7.78601i 4.29644 − 5.43862i
275.5 −0.109554 − 1.99700i −2.28374 − 3.95555i −3.97600 + 0.437557i 4.96451 + 2.86626i −7.64902 + 4.99396i 0 1.30939 + 7.89212i −5.93090 + 10.2726i 5.18003 − 10.2281i
275.6 0.575587 + 1.91538i 2.61182 + 4.52380i −3.33740 + 2.20494i 5.42814 + 3.13394i −7.16148 + 7.60647i 0 −6.14428 − 5.12327i −9.14316 + 15.8364i −2.87833 + 12.2008i
275.7 1.78423 − 0.903622i −2.28374 − 3.95555i 2.36693 − 3.22453i −4.96451 − 2.86626i −7.64902 − 4.99396i 0 1.30939 − 7.89212i −5.93090 + 10.2726i −11.4478 − 0.628019i
275.8 1.99898 − 0.0637211i 1.72064 + 2.98023i 3.99188 − 0.254755i −4.22869 − 2.44143i 3.62943 + 5.84780i 0 7.96347 − 0.763618i −1.42120 + 2.46158i −8.60865 − 4.61093i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.o 16
7.b odd 2 1 392.3.k.n 16
7.c even 3 1 56.3.g.b ✓ 8
7.c even 3 1 inner 392.3.k.o 16
7.d odd 6 1 392.3.g.m 8
7.d odd 6 1 392.3.k.n 16
8.d odd 2 1 inner 392.3.k.o 16
21.h odd 6 1 504.3.g.b 8
28.f even 6 1 1568.3.g.m 8
28.g odd 6 1 224.3.g.b 8
56.e even 2 1 392.3.k.n 16
56.j odd 6 1 1568.3.g.m 8
56.k odd 6 1 56.3.g.b ✓ 8
56.k odd 6 1 inner 392.3.k.o 16
56.m even 6 1 392.3.g.m 8
56.m even 6 1 392.3.k.n 16
56.p even 6 1 224.3.g.b 8
84.n even 6 1 2016.3.g.b 8
112.u odd 12 2 1792.3.d.j 16
112.w even 12 2 1792.3.d.j 16
168.s odd 6 1 2016.3.g.b 8
168.v even 6 1 504.3.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b ✓ 8 7.c even 3 1
56.3.g.b ✓ 8 56.k odd 6 1
224.3.g.b 8 28.g odd 6 1
224.3.g.b 8 56.p even 6 1
392.3.g.m 8 7.d odd 6 1
392.3.g.m 8 56.m even 6 1
392.3.k.n 16 7.b odd 2 1
392.3.k.n 16 7.d odd 6 1
392.3.k.n 16 56.e even 2 1
392.3.k.n 16 56.m even 6 1
392.3.k.o 16 1.a even 1 1 trivial
392.3.k.o 16 7.c even 3 1 inner
392.3.k.o 16 8.d odd 2 1 inner
392.3.k.o 16 56.k odd 6 1 inner
504.3.g.b 8 21.h odd 6 1
504.3.g.b 8 168.v even 6 1
1568.3.g.m 8 28.f even 6 1
1568.3.g.m 8 56.j odd 6 1
1792.3.d.j 16 112.u odd 12 2
1792.3.d.j 16 112.w even 12 2
2016.3.g.b 8 84.n even 6 1
2016.3.g.b 8 168.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{8} - 4T_{3}^{7} + 38T_{3}^{6} - 72T_{3}^{5} + 796T_{3}^{4} - 1696T_{3}^{3} + 6576T_{3}^{2} + 640T_{3} + 64 \) Copy content Toggle raw display
\( T_{5}^{16} - 108 T_{5}^{14} + 7500 T_{5}^{12} - 315824 T_{5}^{10} + 9739280 T_{5}^{8} + \cdots + 136651472896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{8} - 4 T^{7} + 38 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 136651472896 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 16 T^{7} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 908 T^{6} + \cdots + 133448704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 40 T^{7} + \cdots + 565504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 28 T^{7} + \cdots + 1658944)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + 3344 T^{6} + \cdots + 13389266944)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{3} + \cdots + 837776)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2716 T^{2} + \cdots - 217952)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 281474976710656 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 126626768156736)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} + 152 T^{7} + \cdots + 626959908864)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 221437256269824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 2981506703616)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{4} - 36 T^{3} + \cdots + 3180872)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 256 T^{7} + \cdots + 382834237696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 32 T^{3} + \cdots + 9539216)^{4} \) Copy content Toggle raw display
show more
show less