Properties

Label 61.2.e.a
Level $61$
Weight $2$
Character orbit 61.e
Analytic conductor $0.487$
Analytic rank $0$
Dimension $12$
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] = [61,2,Mod(9,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 61.e (of order \(5\), degree \(4\), 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: \(0.487087452330\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} - 17 x^{3} + 14 x^{2} - 5 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{10} + \beta_{8}) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{11} + \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{10} - \beta_{9} - 1) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{10} + \beta_{8}) q^{3} + ( - \beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{4} + (\beta_{11} + \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{10} - \beta_{9} - 1) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{10} + (2 \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_1 - 1) q^{11} + ( - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2}) q^{12} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{3}) q^{13} + (\beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_1 - 1) q^{14} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{4} + \beta_{3} + \beta_{2}) q^{15} + (2 \beta_{9} - 2 \beta_{8} + \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{16} + ( - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + ( - \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2}) q^{18} + ( - 2 \beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - 2 \beta_{2} - \beta_1) q^{19} + (\beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_1) q^{20} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{3} - \beta_1 + 2) q^{21} + (\beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + \cdots - 3) q^{22}+ \cdots + (3 \beta_{11} + 3 \beta_{10} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 10 q^{7} + 4 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 10 q^{7} + 4 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} + 6 q^{12} + 2 q^{13} - 2 q^{14} - 5 q^{15} + 10 q^{16} + 6 q^{17} - 15 q^{18} - q^{19} + 13 q^{20} + 32 q^{21} - 25 q^{22} - 7 q^{23} - 12 q^{25} + 2 q^{26} + 8 q^{27} - 13 q^{28} - 8 q^{29} + 3 q^{30} + 19 q^{31} + 26 q^{32} - q^{33} - 30 q^{34} + 10 q^{35} + 8 q^{36} + 31 q^{37} + 27 q^{38} - 23 q^{39} + 4 q^{41} + 11 q^{42} + q^{43} - 20 q^{44} - 18 q^{45} + 23 q^{46} - 22 q^{47} - 8 q^{48} + 3 q^{49} + 74 q^{50} + 9 q^{52} - 35 q^{53} - 26 q^{54} - 18 q^{56} - 31 q^{57} - 16 q^{58} - 13 q^{59} + 2 q^{60} - 6 q^{61} - 52 q^{62} + 7 q^{63} + 16 q^{64} + 4 q^{65} - 19 q^{66} - 9 q^{67} - 33 q^{68} + 35 q^{69} - 5 q^{70} - 24 q^{71} - 30 q^{72} + 51 q^{73} - 20 q^{74} + 40 q^{75} - 31 q^{76} + q^{77} + 22 q^{78} - 26 q^{79} - 23 q^{80} - 2 q^{81} + 78 q^{82} + 3 q^{83} + 17 q^{84} + 18 q^{85} + 55 q^{86} + 49 q^{87} - q^{88} - 17 q^{89} + 54 q^{90} - 27 q^{91} + 37 q^{92} + 10 q^{93} + 13 q^{94} + 4 q^{95} - 17 q^{96} + 17 q^{97} + 29 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} - 17 x^{3} + 14 x^{2} - 5 x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19891 \nu^{11} - 114974 \nu^{10} + 362529 \nu^{9} - 673717 \nu^{8} + 1202532 \nu^{7} - 1315063 \nu^{6} + 1708502 \nu^{5} + 58800 \nu^{4} + 422521 \nu^{3} + \cdots + 95570 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 147786 \nu^{11} - 379469 \nu^{10} + 771926 \nu^{9} - 4107793 \nu^{8} + 4845187 \nu^{7} - 14767041 \nu^{6} + 5123199 \nu^{5} - 30881178 \nu^{4} + \cdots - 5509490 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 279535 \nu^{11} - 503769 \nu^{10} + 1828536 \nu^{9} - 1989137 \nu^{8} + 5491685 \nu^{7} - 978328 \nu^{6} + 10188488 \nu^{5} + 17315912 \nu^{4} + \cdots - 299426 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 334836 \nu^{11} + 687279 \nu^{10} - 2203888 \nu^{9} + 2614830 \nu^{8} - 6289582 \nu^{7} + 1831517 \nu^{6} - 10607072 \nu^{5} - 17211647 \nu^{4} + \cdots + 279535 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 351124 \nu^{11} - 1148942 \nu^{10} + 3466717 \nu^{9} - 6241964 \nu^{8} + 11978675 \nu^{7} - 12574471 \nu^{6} + 18785684 \nu^{5} + 3002403 \nu^{4} + \cdots - 2574616 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1268916 \nu^{11} + 3238544 \nu^{10} - 9630840 \nu^{9} + 13682648 \nu^{8} - 27496876 \nu^{7} + 15163504 \nu^{6} - 36702332 \nu^{5} - 57918271 \nu^{4} + \cdots + 1339344 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1848387 \nu^{11} - 5413412 \nu^{10} + 15752245 \nu^{9} - 25125343 \nu^{8} + 47506293 \nu^{7} - 37721190 \nu^{6} + 63735272 \nu^{5} + 59672975 \nu^{4} + \cdots + 573314 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2442867 \nu^{11} - 6489486 \nu^{10} + 19422140 \nu^{9} - 29055593 \nu^{8} + 58085028 \nu^{7} - 39215972 \nu^{6} + 82822330 \nu^{5} + 94141734 \nu^{4} + \cdots - 4061746 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3187731 \nu^{11} + 8162528 \nu^{10} - 24567797 \nu^{9} + 35584663 \nu^{8} - 72664621 \nu^{7} + 45047258 \nu^{6} - 106163560 \nu^{5} + \cdots + 544826 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3214409 \nu^{11} - 9807301 \nu^{10} + 29011875 \nu^{9} - 48707038 \nu^{8} + 92737202 \nu^{7} - 84418540 \nu^{6} + 134195500 \nu^{5} + 74090403 \nu^{4} + \cdots - 9454239 ) / 3960529 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} + 8\beta_{9} + 5\beta_{7} - 4\beta_{4} - 4\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{11} + 17\beta_{9} + 16\beta_{7} + \beta_{6} + 16\beta_{5} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{11} - 16\beta_{10} + 23\beta_{6} + 23\beta_{5} + 15\beta_{4} + 26\beta_{3} - 15\beta_{2} - 23\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{10} - 72\beta_{9} - 65\beta_{7} + 65\beta_{6} + 9\beta_{5} + \beta_{4} - 74\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 65 \beta_{11} - 9 \beta_{10} - 146 \beta_{9} - 9 \beta_{8} - 104 \beta_{7} + \beta_{5} - 42 \beta_{4} - 98 \beta_{3} + 65 \beta_{2} - 104 \beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 104 \beta_{11} - \beta_{10} - 50 \beta_{9} - 104 \beta_{8} - 58 \beta_{7} - 267 \beta_{6} - 38 \beta_{4} - 38 \beta_{3} + 103 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 58 \beta_{11} + 197 \beta_{9} - 267 \beta_{8} - 13 \beta_{7} - 467 \beta_{6} - 13 \beta_{5} + 210 \beta_{3} + 467 \beta _1 + 373 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13 \beta_{11} + 13 \beta_{10} - 328 \beta_{6} - 328 \beta_{5} + 187 \beta_{4} + 232 \beta_{3} - 454 \beta_{2} + 1435 \beta _1 + 187 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.622685 + 1.91643i
0.135246 + 0.416243i
−0.566948 1.74489i
1.72363 1.25229i
0.275227 0.199964i
−0.689843 + 0.501200i
0.622685 1.91643i
0.135246 0.416243i
−0.566948 + 1.74489i
1.72363 + 1.25229i
0.275227 + 0.199964i
−0.689843 0.501200i
−1.63021 1.18442i −0.0985128 0.0715738i 0.636708 + 1.95958i −1.25939 3.87601i 0.0758235 + 0.233361i −1.40753 + 1.02263i 0.0376286 0.115809i −0.922469 2.83907i −2.53774 + 7.81036i
9.2 −0.354078 0.257253i 1.34849 + 0.979734i −0.558842 1.71994i 0.423596 + 1.30369i −0.225431 0.693804i 0.0394707 0.0286771i −0.515076 + 1.58524i −0.0685100 0.210852i 0.185393 0.570581i
9.3 1.48429 + 1.07840i −0.940958 0.683646i 0.422134 + 1.29919i 0.144814 + 0.445690i −0.659410 2.02945i −2.24997 + 1.63470i 0.359414 1.10616i −0.509021 1.56661i −0.265687 + 0.817700i
20.1 −0.658369 2.02625i −0.354958 1.09245i −2.05421 + 1.49247i 0.330580 0.240180i −1.97988 + 1.43847i −0.545941 + 1.68023i 0.929291 + 0.675169i 1.35961 0.987811i −0.704310 0.511711i
20.2 −0.105127 0.323548i 0.408342 + 1.25675i 1.52440 1.10754i −1.79963 + 1.30751i 0.363691 0.264237i 0.217359 0.668962i −1.06905 0.776713i 1.01438 0.736990i 0.612232 + 0.444812i
20.3 0.263496 + 0.810959i −0.862402 2.65420i 1.02981 0.748201i −0.339968 + 0.247001i 1.92521 1.39874i −1.05338 + 3.24198i 2.25780 + 1.64039i −3.87399 + 2.81462i −0.289888 0.210616i
34.1 −1.63021 + 1.18442i −0.0985128 + 0.0715738i 0.636708 1.95958i −1.25939 + 3.87601i 0.0758235 0.233361i −1.40753 1.02263i 0.0376286 + 0.115809i −0.922469 + 2.83907i −2.53774 7.81036i
34.2 −0.354078 + 0.257253i 1.34849 0.979734i −0.558842 + 1.71994i 0.423596 1.30369i −0.225431 + 0.693804i 0.0394707 + 0.0286771i −0.515076 1.58524i −0.0685100 + 0.210852i 0.185393 + 0.570581i
34.3 1.48429 1.07840i −0.940958 + 0.683646i 0.422134 1.29919i 0.144814 0.445690i −0.659410 + 2.02945i −2.24997 1.63470i 0.359414 + 1.10616i −0.509021 + 1.56661i −0.265687 0.817700i
58.1 −0.658369 + 2.02625i −0.354958 + 1.09245i −2.05421 1.49247i 0.330580 + 0.240180i −1.97988 1.43847i −0.545941 1.68023i 0.929291 0.675169i 1.35961 + 0.987811i −0.704310 + 0.511711i
58.2 −0.105127 + 0.323548i 0.408342 1.25675i 1.52440 + 1.10754i −1.79963 1.30751i 0.363691 + 0.264237i 0.217359 + 0.668962i −1.06905 + 0.776713i 1.01438 + 0.736990i 0.612232 0.444812i
58.3 0.263496 0.810959i −0.862402 + 2.65420i 1.02981 + 0.748201i −0.339968 0.247001i 1.92521 + 1.39874i −1.05338 3.24198i 2.25780 1.64039i −3.87399 2.81462i −0.289888 + 0.210616i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.2.e.a 12
3.b odd 2 1 549.2.k.a 12
4.b odd 2 1 976.2.v.a 12
61.e even 5 1 inner 61.2.e.a 12
61.e even 5 1 3721.2.a.g 6
61.g even 10 1 3721.2.a.h 6
183.n odd 10 1 549.2.k.a 12
244.n odd 10 1 976.2.v.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.e.a 12 1.a even 1 1 trivial
61.2.e.a 12 61.e even 5 1 inner
549.2.k.a 12 3.b odd 2 1
549.2.k.a 12 183.n odd 10 1
976.2.v.a 12 4.b odd 2 1
976.2.v.a 12 244.n odd 10 1
3721.2.a.g 6 61.e even 5 1
3721.2.a.h 6 61.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + 4 T^{10} + 4 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + 8 T^{10} - 3 T^{9} + 15 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + 5 T^{11} + 26 T^{10} + 51 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 10 T^{11} + 59 T^{10} + 227 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} - 22 T^{4} - 87 T^{3} + \cdots + 505)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - T^{5} - 36 T^{4} + 19 T^{3} + \cdots - 341)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + 19 T^{10} + \cdots + 1394761 \) Copy content Toggle raw display
$19$ \( T^{12} + T^{11} + 34 T^{10} + \cdots + 1234321 \) Copy content Toggle raw display
$23$ \( T^{12} + 7 T^{11} + 27 T^{10} + \cdots + 1104601 \) Copy content Toggle raw display
$29$ \( (T^{6} + 4 T^{5} - 102 T^{4} - 365 T^{3} + \cdots + 9221)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 19 T^{11} + 229 T^{10} + \cdots + 4431025 \) Copy content Toggle raw display
$37$ \( T^{12} - 31 T^{11} + 487 T^{10} + \cdots + 45981961 \) Copy content Toggle raw display
$41$ \( T^{12} - 4 T^{11} + \cdots + 13457160025 \) Copy content Toggle raw display
$43$ \( T^{12} - T^{11} + 169 T^{10} + \cdots + 3094081 \) Copy content Toggle raw display
$47$ \( (T^{6} + 11 T^{5} + 24 T^{4} - 64 T^{3} + \cdots + 59)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 35 T^{11} + \cdots + 18439195681 \) Copy content Toggle raw display
$59$ \( T^{12} + 13 T^{11} + \cdots + 43584495361 \) Copy content Toggle raw display
$61$ \( T^{12} + 6 T^{11} + \cdots + 51520374361 \) Copy content Toggle raw display
$67$ \( T^{12} + 9 T^{11} + 91 T^{10} + \cdots + 10201 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} + \cdots + 892754641 \) Copy content Toggle raw display
$73$ \( T^{12} - 51 T^{11} + \cdots + 722542500625 \) Copy content Toggle raw display
$79$ \( T^{12} + 26 T^{11} + 348 T^{10} + \cdots + 7557001 \) Copy content Toggle raw display
$83$ \( T^{12} - 3 T^{11} + \cdots + 6037134601 \) Copy content Toggle raw display
$89$ \( T^{12} + 17 T^{11} + 314 T^{10} + \cdots + 101761 \) Copy content Toggle raw display
$97$ \( T^{12} - 17 T^{11} + \cdots + 281266441 \) Copy content Toggle raw display
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