Properties

Label 630.2.l.f
Level $630$
Weight $2$
Character orbit 630.l
Analytic conductor $5.031$
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] = [630,2,Mod(331,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.l (of order \(3\), degree \(2\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + 104 x^{3} + 197 x^{2} + 166 x + 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + 104 x^{3} + 197 x^{2} + 166 x + 79 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10 \nu^{11} + 76 \nu^{10} - 632 \nu^{9} + 2093 \nu^{8} - 4654 \nu^{7} + 7286 \nu^{6} - 6541 \nu^{5} + 6581 \nu^{4} + 1877 \nu^{3} + 6580 \nu^{2} + 652 \nu + 8785 ) / 1863 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35 \nu^{11} - 82 \nu^{10} + 1108 \nu^{9} - 4209 \nu^{8} + 10470 \nu^{7} - 18003 \nu^{6} + 18903 \nu^{5} - 22194 \nu^{4} - 210 \nu^{3} - 18844 \nu^{2} - 10010 \nu - 20317 ) / 5589 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21 \nu^{11} - 167 \nu^{10} + 591 \nu^{9} - 1610 \nu^{8} + 3769 \nu^{7} - 6692 \nu^{6} + 9625 \nu^{5} - 10484 \nu^{4} + 4036 \nu^{3} - 5939 \nu^{2} - 5034 \nu - 10796 ) / 1863 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83 \nu^{11} + 119 \nu^{10} + 250 \nu^{9} - 690 \nu^{8} + 1083 \nu^{7} - 453 \nu^{6} - 6846 \nu^{5} - 5814 \nu^{4} - 16437 \nu^{3} - 27589 \nu^{2} - 24980 \nu - 12115 ) / 5589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{11} + 183 \nu^{10} - 511 \nu^{9} + 920 \nu^{8} - 1093 \nu^{7} + 671 \nu^{6} - 1198 \nu^{5} + 1373 \nu^{4} - 4441 \nu^{3} + 3315 \nu^{2} - 2587 \nu + 240 ) / 1863 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 104 \nu^{11} - 608 \nu^{10} + 2066 \nu^{9} - 4899 \nu^{8} + 7746 \nu^{7} - 8286 \nu^{6} + 9249 \nu^{5} - 162 \nu^{4} + 11181 \nu^{3} + 10909 \nu^{2} + 11114 \nu + 8242 ) / 5589 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} - 10 \nu^{10} + 32 \nu^{9} - 80 \nu^{8} + 133 \nu^{7} - 134 \nu^{6} + 121 \nu^{5} + 166 \nu^{4} + 118 \nu^{3} + 353 \nu^{2} + 404 \nu + 191 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 251 \nu^{11} + 1202 \nu^{10} - 3167 \nu^{9} + 5382 \nu^{8} - 3447 \nu^{7} - 6855 \nu^{6} + 13122 \nu^{5} - 37011 \nu^{4} - 14685 \nu^{3} - 36910 \nu^{2} - 57296 \nu - 59101 ) / 5589 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 257 \nu^{11} + 1598 \nu^{10} - 5189 \nu^{9} + 11730 \nu^{8} - 18843 \nu^{7} + 20058 \nu^{6} - 20724 \nu^{5} + 3780 \nu^{4} - 15204 \nu^{3} - 3115 \nu^{2} - 20096 \nu + 2765 ) / 5589 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 424 \nu^{11} + 2293 \nu^{10} - 6544 \nu^{9} + 12972 \nu^{8} - 16575 \nu^{7} + 8883 \nu^{6} - 5235 \nu^{5} - 28362 \nu^{4} - 14688 \nu^{3} - 35744 \nu^{2} - 45916 \nu - 26909 ) / 5589 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + 3\beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} - 4\beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{11} - 3\beta_{10} + 5\beta_{8} + 3\beta_{5} + 3\beta_{4} - 11\beta_{3} - 3\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{11} - 20 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} - 3 \beta_{7} + 9 \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 19 \beta_{3} + 5 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39 \beta_{11} - 64 \beta_{10} - 17 \beta_{9} + \beta_{8} - 16 \beta_{7} + 37 \beta_{6} + 10 \beta_{5} + 6 \beta_{4} - 20 \beta_{3} + 18 \beta_{2} + 13 \beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 94 \beta_{11} - 163 \beta_{10} - 63 \beta_{9} - 23 \beta_{8} - 60 \beta_{7} + 109 \beta_{6} + 12 \beta_{5} + 13 \beta_{4} - 4 \beta_{3} + 58 \beta_{2} - 13 \beta _1 - 191 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 203 \beta_{11} - 396 \beta_{10} - 181 \beta_{9} - 145 \beta_{8} - 178 \beta_{7} + 286 \beta_{6} - 30 \beta_{5} + 3 \beta_{4} + 80 \beta_{3} + 137 \beta_{2} - 168 \beta _1 - 576 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 370 \beta_{11} - 933 \beta_{10} - 434 \beta_{9} - 599 \beta_{8} - 468 \beta_{7} + 693 \beta_{6} - 258 \beta_{5} - 157 \beta_{4} + 519 \beta_{3} + 353 \beta_{2} - 655 \beta _1 - 1580 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 416 \beta_{11} - 1915 \beta_{10} - 903 \beta_{9} - 1947 \beta_{8} - 1161 \beta_{7} + 1483 \beta_{6} - 1065 \beta_{5} - 845 \beta_{4} + 2277 \beta_{3} + 982 \beta_{2} - 1955 \beta _1 - 4045 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 596 \beta_{11} - 2712 \beta_{10} - 1579 \beta_{9} - 5385 \beta_{8} - 2573 \beta_{7} + 2512 \beta_{6} - 3371 \beta_{5} - 2841 \beta_{4} + 7682 \beta_{3} + 2399 \beta_{2} - 5196 \beta _1 - 9214 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3}\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
−0.659665 0.495491i
−0.633121 + 0.576989i
0.142686 + 1.50500i
0.429413 1.63537i
0.737753 1.71208i
2.48293 + 0.894932i
−0.659665 + 0.495491i
−0.633121 0.576989i
0.142686 1.50500i
0.429413 + 1.63537i
0.737753 + 1.71208i
2.48293 0.894932i
−0.500000 0.866025i −1.65967 0.495491i −0.500000 + 0.866025i −1.00000 0.400725 + 1.68506i 2.25729 1.38008i 1.00000 2.50898 + 1.64470i 0.500000 + 0.866025i
331.2 −0.500000 0.866025i −1.63312 + 0.576989i −0.500000 + 0.866025i −1.00000 1.31625 + 1.12583i −1.40545 2.24159i 1.00000 2.33417 1.88459i 0.500000 + 0.866025i
331.3 −0.500000 0.866025i −0.857314 + 1.50500i −0.500000 + 0.866025i −1.00000 1.73202 0.0100417i −1.85185 + 1.88962i 1.00000 −1.53002 2.58051i 0.500000 + 0.866025i
331.4 −0.500000 0.866025i −0.570587 1.63537i −0.500000 + 0.866025i −1.00000 −1.13098 + 1.31183i 2.25729 1.38008i 1.00000 −2.34886 + 1.86624i 0.500000 + 0.866025i
331.5 −0.500000 0.866025i −0.262247 1.71208i −0.500000 + 0.866025i −1.00000 −1.35158 + 1.08315i −1.85185 + 1.88962i 1.00000 −2.86245 + 0.897978i 0.500000 + 0.866025i
331.6 −0.500000 0.866025i 1.48293 + 0.894932i −0.500000 + 0.866025i −1.00000 0.0335666 1.73173i −1.40545 2.24159i 1.00000 1.39819 + 2.65425i 0.500000 + 0.866025i
571.1 −0.500000 + 0.866025i −1.65967 + 0.495491i −0.500000 0.866025i −1.00000 0.400725 1.68506i 2.25729 + 1.38008i 1.00000 2.50898 1.64470i 0.500000 0.866025i
571.2 −0.500000 + 0.866025i −1.63312 0.576989i −0.500000 0.866025i −1.00000 1.31625 1.12583i −1.40545 + 2.24159i 1.00000 2.33417 + 1.88459i 0.500000 0.866025i
571.3 −0.500000 + 0.866025i −0.857314 1.50500i −0.500000 0.866025i −1.00000 1.73202 + 0.0100417i −1.85185 1.88962i 1.00000 −1.53002 + 2.58051i 0.500000 0.866025i
571.4 −0.500000 + 0.866025i −0.570587 + 1.63537i −0.500000 0.866025i −1.00000 −1.13098 1.31183i 2.25729 + 1.38008i 1.00000 −2.34886 1.86624i 0.500000 0.866025i
571.5 −0.500000 + 0.866025i −0.262247 + 1.71208i −0.500000 0.866025i −1.00000 −1.35158 1.08315i −1.85185 1.88962i 1.00000 −2.86245 0.897978i 0.500000 0.866025i
571.6 −0.500000 + 0.866025i 1.48293 0.894932i −0.500000 0.866025i −1.00000 0.0335666 + 1.73173i −1.40545 + 2.24159i 1.00000 1.39819 2.65425i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.l.f yes 12
3.b odd 2 1 1890.2.l.h 12
7.c even 3 1 630.2.i.h 12
9.c even 3 1 630.2.i.h 12
9.d odd 6 1 1890.2.i.f 12
21.h odd 6 1 1890.2.i.f 12
63.g even 3 1 inner 630.2.l.f yes 12
63.n odd 6 1 1890.2.l.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.h 12 7.c even 3 1
630.2.i.h 12 9.c even 3 1
630.2.l.f yes 12 1.a even 1 1 trivial
630.2.l.f yes 12 63.g even 3 1 inner
1890.2.i.f 12 9.d odd 6 1
1890.2.i.f 12 21.h odd 6 1
1890.2.l.h 12 3.b odd 2 1
1890.2.l.h 12 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{6} - 7T_{11}^{5} - 12T_{11}^{4} + 146T_{11}^{3} - 140T_{11}^{2} - 324T_{11} + 363 \) Copy content Toggle raw display
\( T_{13}^{12} - 2 T_{13}^{11} + 78 T_{13}^{10} - 140 T_{13}^{9} + 4390 T_{13}^{8} - 8067 T_{13}^{7} + 111972 T_{13}^{6} - 227754 T_{13}^{5} + 2135322 T_{13}^{4} - 3328074 T_{13}^{3} + 11628711 T_{13}^{2} + \cdots + 5349969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{11} + 25 T^{10} + 57 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 19 T^{3} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 7 T^{5} - 12 T^{4} + 146 T^{3} + \cdots + 363)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + 78 T^{10} + \cdots + 5349969 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{11} + 76 T^{10} + \cdots + 1046529 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} + 180 T^{10} + \cdots + 62869041 \) Copy content Toggle raw display
$23$ \( (T^{6} - 9 T^{5} - 33 T^{4} + 540 T^{3} + \cdots - 891)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 9 T^{11} + 105 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$31$ \( T^{12} - 9 T^{11} + 171 T^{10} + \cdots + 296080849 \) Copy content Toggle raw display
$37$ \( T^{12} + 12 T^{11} + 147 T^{10} + \cdots + 185761 \) Copy content Toggle raw display
$41$ \( T^{12} - T^{11} + 49 T^{10} + \cdots + 349281 \) Copy content Toggle raw display
$43$ \( T^{12} - 7 T^{11} + \cdots + 5930694121 \) Copy content Toggle raw display
$47$ \( T^{12} - 7 T^{11} + 139 T^{10} + \cdots + 1447209 \) Copy content Toggle raw display
$53$ \( T^{12} - 2 T^{11} + 151 T^{10} + \cdots + 41177889 \) Copy content Toggle raw display
$59$ \( T^{12} - 29 T^{11} + 598 T^{10} + \cdots + 4173849 \) Copy content Toggle raw display
$61$ \( T^{12} + 11 T^{11} + 125 T^{10} + \cdots + 537289 \) Copy content Toggle raw display
$67$ \( T^{12} + 22 T^{11} + 321 T^{10} + \cdots + 3470769 \) Copy content Toggle raw display
$71$ \( (T^{6} - 5 T^{5} - 144 T^{4} + 238 T^{3} + \cdots - 43461)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 6 T^{11} + 174 T^{10} + \cdots + 131079601 \) Copy content Toggle raw display
$79$ \( T^{12} - T^{11} + 422 T^{10} + \cdots + 9983806561 \) Copy content Toggle raw display
$83$ \( T^{12} + 26 T^{11} + \cdots + 1938758266449 \) Copy content Toggle raw display
$89$ \( T^{12} - 2 T^{11} + \cdots + 851631819921 \) Copy content Toggle raw display
$97$ \( T^{12} - 6 T^{11} + 414 T^{10} + \cdots + 212955649 \) Copy content Toggle raw display
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