Properties

Label 441.4.e.z
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(226,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.226");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), 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: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{7} + 9 \beta_{2} - 9) q^{4} - \beta_{6} q^{5} + (2 \beta_{15} - 11 \beta_{12} + 11 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{7} + 9 \beta_{2} - 9) q^{4} - \beta_{6} q^{5} + (2 \beta_{15} - 11 \beta_{12} + 11 \beta_{4}) q^{8} + ( - 3 \beta_{14} - 2 \beta_{13}) q^{10} + (\beta_{15} + 7 \beta_{12} + \beta_{9}) q^{11} + ( - 2 \beta_{14} - 3 \beta_{13} - 2 \beta_{10} + 3 \beta_{8}) q^{13} + ( - 9 \beta_{7} - 105 \beta_{2} + 9 \beta_1) q^{16} + ( - 2 \beta_{11} + \beta_{6} + 2 \beta_{5} + \beta_{3}) q^{17} + (6 \beta_{10} + 2 \beta_{8}) q^{19} + (5 \beta_{5} - 14 \beta_{3}) q^{20} + ( - 4 \beta_1 + 124) q^{22} + ( - 3 \beta_{9} + 7 \beta_{4}) q^{23} + (4 \beta_{7} + 69 \beta_{2} - 69) q^{25} + (5 \beta_{11} + 18 \beta_{6}) q^{26} + ( - 9 \beta_{15} - 13 \beta_{12} + 13 \beta_{4}) q^{29} + ( - 2 \beta_{14} + 10 \beta_{13}) q^{31} + ( - 2 \beta_{15} + 107 \beta_{12} - 2 \beta_{9}) q^{32} + (\beta_{14} - 18 \beta_{13} + \beta_{10} + 18 \beta_{8}) q^{34} + (24 \beta_{7} + 4 \beta_{2} - 24 \beta_1) q^{37} + ( - 4 \beta_{11} - 32 \beta_{6} + 4 \beta_{5} - 32 \beta_{3}) q^{38} + ( - 23 \beta_{10} + 62 \beta_{8}) q^{40} + (10 \beta_{5} - \beta_{3}) q^{41} + (20 \beta_1 + 260) q^{43} + (16 \beta_{9} - 108 \beta_{4}) q^{44} + ( - 16 \beta_{7} - 104 \beta_{2} + 104) q^{46} + ( - 2 \beta_{11} - 30 \beta_{6}) q^{47} + (8 \beta_{15} - 109 \beta_{12} + 109 \beta_{4}) q^{50} + (43 \beta_{14} + 62 \beta_{13}) q^{52} + ( - 14 \beta_{15} + 34 \beta_{12} - 14 \beta_{9}) q^{53} + (26 \beta_{14} - 6 \beta_{13} + 26 \beta_{10} + 6 \beta_{8}) q^{55} + (14 \beta_{7} - 266 \beta_{2} - 14 \beta_1) q^{58} + ( - 10 \beta_{11} - 2 \beta_{6} + 10 \beta_{5} - 2 \beta_{3}) q^{59} + (22 \beta_{10} + 55 \beta_{8}) q^{61} + ( - 8 \beta_{5} + 8 \beta_{3}) q^{62} + ( - 41 \beta_1 + 969) q^{64} + ( - 17 \beta_{9} + 191 \beta_{4}) q^{65} + ( - 40 \beta_{7} - 20 \beta_{2} + 20) q^{67} + (\beta_{11} + 38 \beta_{6}) q^{68} + (9 \beta_{15} + 163 \beta_{12} - 163 \beta_{4}) q^{71} + ( - 28 \beta_{14} + 23 \beta_{13}) q^{73} + (48 \beta_{15} - 244 \beta_{12} + 48 \beta_{9}) q^{74} + ( - 52 \beta_{14} - 120 \beta_{13} - 52 \beta_{10} + 120 \beta_{8}) q^{76} + ( - 28 \beta_{7} - 688 \beta_{2} + 28 \beta_1) q^{79} + (45 \beta_{11} + 150 \beta_{6} - 45 \beta_{5} + 150 \beta_{3}) q^{80} + ( - 13 \beta_{10} + 102 \beta_{8}) q^{82} + ( - 16 \beta_{5} - 52 \beta_{3}) q^{83} + (64 \beta_1 + 326) q^{85} + ( - 40 \beta_{9} - 60 \beta_{4}) q^{86} + (124 \beta_{7} + 764 \beta_{2} - 764) q^{88} + ( - 10 \beta_{11} - 7 \beta_{6}) q^{89} + ( - 8 \beta_{15} + 208 \beta_{12} - 208 \beta_{4}) q^{92} + ( - 92 \beta_{14} - 80 \beta_{13}) q^{94} + ( - 26 \beta_{15} - 342 \beta_{12} - 26 \beta_{9}) q^{95} + (28 \beta_{14} + 57 \beta_{13} + 28 \beta_{10} - 57 \beta_{8}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 68 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 68 q^{4} - 804 q^{16} + 1952 q^{22} - 536 q^{25} - 64 q^{37} + 4320 q^{43} + 768 q^{46} - 2184 q^{58} + 15176 q^{64} - 5392 q^{79} + 5728 q^{85} - 5616 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5955326 \nu^{14} + 420756261 \nu^{12} + 21224761594 \nu^{10} + 399577053316 \nu^{8} + 4077769899992 \nu^{6} + \cdots + 20\!\cdots\!85 ) / 156438029994849 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2249798084357 \nu^{14} + 165523687535043 \nu^{12} + \cdots + 48\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5596355476439 \nu^{14} + 378608216872794 \nu^{12} + \cdots - 12\!\cdots\!68 ) / 10\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11310865459517 \nu^{15} + 773166243923574 \nu^{13} + \cdots - 47\!\cdots\!76 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 81945585249437 \nu^{14} + \cdots - 11\!\cdots\!80 ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!39 \nu^{14} + \cdots + 81\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!57 \nu^{14} + \cdots - 10\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 79176058216619 \nu^{15} + \cdots - 23\!\cdots\!32 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 38872184755339 \nu^{15} + \cdots - 21\!\cdots\!40 \nu ) / 52\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4417491293917 \nu^{15} + 300026796677634 \nu^{13} + \cdots - 15\!\cdots\!96 \nu ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 63\!\cdots\!73 \nu^{14} + \cdots - 14\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!73 \nu^{15} + \cdots - 30\!\cdots\!00 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 69\!\cdots\!11 \nu^{15} + \cdots - 15\!\cdots\!00 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!33 \nu^{15} + \cdots + 10\!\cdots\!00 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 560405796289847 \nu^{15} + \cdots - 58\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - 7\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + 7\beta_{7} + 4\beta_{6} - 2\beta_{5} + 4\beta_{3} + 133\beta_{2} - 133 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{15} + 21\beta_{14} + 44\beta_{13} - 231\beta_{12} + 21\beta_{10} - 44\beta_{8} + 231\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -92\beta_{11} - 315\beta_{7} - 296\beta_{6} - 4599\beta_{2} + 315\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -742\beta_{15} - 1295\beta_{14} - 2034\beta_{13} + 9373\beta_{12} - 742\beta_{9} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4350\beta_{5} - 15364\beta_{3} - 14189\beta _1 + 191877 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -64631\beta_{10} + 35042\beta_{9} + 95558\beta_{8} - 414659\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 204408 \beta_{11} + 649047 \beta_{7} + 737424 \beta_{6} - 204408 \beta_{5} + 737424 \beta_{3} + 8599591 \beta_{2} - 8599591 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1626702 \beta_{15} + 3065727 \beta_{14} + 4469278 \beta_{13} - 18937597 \beta_{12} + 3065727 \beta_{10} - 4469278 \beta_{8} + 18937597 \beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -9543218\beta_{11} - 29949157\beta_{7} - 34602748\beta_{6} - 394769893\beta_{2} + 29949157\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -75414626\beta_{15} - 143300619\beta_{14} - 208198022\beta_{13} + 874415451\beta_{12} - 75414626\beta_{9} ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 444094580\beta_{5} - 1612220888\beta_{3} - 1387260567\beta _1 + 18262565559 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -6668017811\beta_{10} + 3498552862\beta_{9} + 9679785270\beta_{8} - 40524590533\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2948235786 \beta_{11} + 9193754963 \beta_{7} + 10706353996 \beta_{6} - 2948235786 \beta_{5} + 10706353996 \beta_{3} + 120992910771 \beta_{2} + \cdots - 120992910771 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 162381746450 \beta_{15} + 309827369159 \beta_{14} + 449677106846 \beta_{13} - 1880447860739 \beta_{12} + 309827369159 \beta_{10} - 449677106846 \beta_{8} + \cdots + 1880447860739 \beta_{4} ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−3.40706 + 5.90120i
−1.99285 + 3.45171i
−0.272818 + 0.472535i
−1.68703 + 2.92202i
0.272818 0.472535i
1.68703 2.92202i
3.40706 5.90120i
1.99285 3.45171i
−3.40706 5.90120i
−1.99285 3.45171i
−0.272818 0.472535i
−1.68703 2.92202i
0.272818 + 0.472535i
1.68703 + 2.92202i
3.40706 + 5.90120i
1.99285 + 3.45171i
−2.69995 + 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 71.0573 0 −42.0566 72.8441i
226.2 −2.69995 + 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 71.0573 0 42.0566 + 72.8441i
226.3 −0.979925 + 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 −23.8298 0 −11.6510 20.1801i
226.4 −0.979925 + 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 −23.8298 0 11.6510 + 20.1801i
226.5 0.979925 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 23.8298 0 11.6510 + 20.1801i
226.6 0.979925 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 23.8298 0 −11.6510 20.1801i
226.7 2.69995 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 −71.0573 0 42.0566 + 72.8441i
226.8 2.69995 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 −71.0573 0 −42.0566 72.8441i
361.1 −2.69995 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 71.0573 0 −42.0566 + 72.8441i
361.2 −2.69995 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 71.0573 0 42.0566 72.8441i
361.3 −0.979925 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 −23.8298 0 −11.6510 + 20.1801i
361.4 −0.979925 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 −23.8298 0 11.6510 20.1801i
361.5 0.979925 + 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 23.8298 0 11.6510 20.1801i
361.6 0.979925 + 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 23.8298 0 −11.6510 + 20.1801i
361.7 2.69995 + 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 −71.0573 0 42.0566 72.8441i
361.8 2.69995 + 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 −71.0573 0 −42.0566 + 72.8441i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.z 16
3.b odd 2 1 inner 441.4.e.z 16
7.b odd 2 1 inner 441.4.e.z 16
7.c even 3 1 441.4.a.x 8
7.c even 3 1 inner 441.4.e.z 16
7.d odd 6 1 441.4.a.x 8
7.d odd 6 1 inner 441.4.e.z 16
21.c even 2 1 inner 441.4.e.z 16
21.g even 6 1 441.4.a.x 8
21.g even 6 1 inner 441.4.e.z 16
21.h odd 6 1 441.4.a.x 8
21.h odd 6 1 inner 441.4.e.z 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.x 8 7.c even 3 1
441.4.a.x 8 7.d odd 6 1
441.4.a.x 8 21.g even 6 1
441.4.a.x 8 21.h odd 6 1
441.4.e.z 16 1.a even 1 1 trivial
441.4.e.z 16 3.b odd 2 1 inner
441.4.e.z 16 7.b odd 2 1 inner
441.4.e.z 16 7.c even 3 1 inner
441.4.e.z 16 7.d odd 6 1 inner
441.4.e.z 16 21.c even 2 1 inner
441.4.e.z 16 21.g even 6 1 inner
441.4.e.z 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} + 33T_{2}^{6} + 977T_{2}^{4} + 3696T_{2}^{2} + 12544 \) Copy content Toggle raw display
\( T_{5}^{8} + 384T_{5}^{6} + 113156T_{5}^{4} + 13171200T_{5}^{2} + 1176490000 \) Copy content Toggle raw display
\( T_{13}^{4} - 5268T_{13}^{2} + 4900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 33 T^{6} + 977 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 384 T^{6} + 113156 T^{4} + \cdots + 1176490000)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 2348 T^{6} + \cdots + 1836567040000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 5268 T^{2} + 4900)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 10496 T^{6} + \cdots + 66171521929744)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 23080 T^{6} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 6012 T^{6} + \cdots + 81603616374784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 53088 T^{2} + 16601200)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 19464 T^{6} + \cdots + 26\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + 92496 T^{2} + \cdots + 8508217600)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 233264 T^{2} + \cdots + 10038958300)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 540 T + 8800)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 335128 T^{6} + \cdots + 36\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 133376 T^{6} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 231096 T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 745844 T^{6} + \cdots + 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 256400 T^{2} + \cdots + 65740960000)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 959388 T^{2} + \cdots + 187904819200)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 535668 T^{6} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1348 T^{3} + \cdots + 108004249600)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1919232 T^{2} + \cdots + 351934815232)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 231776 T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1382388 T^{2} + \cdots + 35588822500)^{4} \) Copy content Toggle raw display
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