Properties

Label 8022.2.a.x
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.a (trivial)

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: yes
Analytic conductor: \(64.0559925015\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 35 x^{12} + 132 x^{11} + 481 x^{10} - 1626 x^{9} - 3297 x^{8} + 9311 x^{7} + 11681 x^{6} - 25107 x^{5} - 20641 x^{4} + 28863 x^{3} + 19453 x^{2} + \cdots - 7663 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 35 x^{12} + 132 x^{11} + 481 x^{10} - 1626 x^{9} - 3297 x^{8} + 9311 x^{7} + 11681 x^{6} - 25107 x^{5} - 20641 x^{4} + 28863 x^{3} + 19453 x^{2} + \cdots - 7663 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35323092726 \nu^{13} + 168487738835 \nu^{12} + 1019289884269 \nu^{11} - 4972622030106 \nu^{10} - 10996862510644 \nu^{9} + \cdots + 128532032771355 ) / 3673013164859 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 152709989075 \nu^{13} - 833860731113 \nu^{12} - 4248843441488 \nu^{11} + 26710536614072 \nu^{10} + 39368149780047 \nu^{9} + \cdots - 16\!\cdots\!51 ) / 7346026329718 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 363208293737 \nu^{13} + 1577212591837 \nu^{12} + 12358489916358 \nu^{11} - 53793026249422 \nu^{10} + \cdots + 30\!\cdots\!19 ) / 14692052659436 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26691319371 \nu^{13} - 133076865887 \nu^{12} - 802709258830 \nu^{11} + 4298885664214 \nu^{10} + 8685808324629 \nu^{9} + \cdots - 260157246235357 ) / 773265929444 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 309533908915 \nu^{13} + 1356645752253 \nu^{12} + 10217116410550 \nu^{11} - 44765065259242 \nu^{10} + \cdots + 20\!\cdots\!55 ) / 7346026329718 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 164135475709 \nu^{13} + 733886946572 \nu^{12} + 5380889687846 \nu^{11} - 24308799269087 \nu^{10} - 65841509241424 \nu^{9} + \cdots + 13\!\cdots\!08 ) / 3673013164859 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 740624817489 \nu^{13} + 3772162249193 \nu^{12} + 21769741077870 \nu^{11} - 120956578686234 \nu^{10} + \cdots + 66\!\cdots\!07 ) / 14692052659436 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 400370957747 \nu^{13} - 1910367622567 \nu^{12} - 12533389600610 \nu^{11} + 62560988628490 \nu^{10} + 143703425277429 \nu^{9} + \cdots - 35\!\cdots\!99 ) / 7346026329718 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 418925411031 \nu^{13} + 2015773249985 \nu^{12} + 13033243099082 \nu^{11} - 65944030048542 \nu^{10} + \cdots + 38\!\cdots\!23 ) / 7346026329718 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 211853213082 \nu^{13} + 983147080316 \nu^{12} + 6827745827080 \nu^{11} - 32697809422193 \nu^{10} - 81273396820160 \nu^{9} + \cdots + 21\!\cdots\!41 ) / 3673013164859 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1695634119093 \nu^{13} + 8323538030917 \nu^{12} + 51765741700874 \nu^{11} - 270570605066026 \nu^{10} + \cdots + 14\!\cdots\!67 ) / 14692052659436 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} - 2 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 10 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{13} - 3 \beta_{12} - \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 13 \beta_{2} + 18 \beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{13} - 31 \beta_{12} - 4 \beta_{11} - 28 \beta_{10} + 37 \beta_{9} + 32 \beta_{8} - 17 \beta_{7} - 20 \beta_{6} + 5 \beta_{5} + 36 \beta_{4} - 32 \beta_{3} + 26 \beta_{2} + 124 \beta _1 + 71 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 63 \beta_{13} - 70 \beta_{12} - 32 \beta_{11} - 83 \beta_{10} + 94 \beta_{9} + 74 \beta_{8} - 28 \beta_{7} - 34 \beta_{6} + 51 \beta_{5} + 11 \beta_{4} - 48 \beta_{3} + 176 \beta_{2} + 296 \beta _1 + 725 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 239 \beta_{13} - 457 \beta_{12} - 138 \beta_{11} - 396 \beta_{10} + 600 \beta_{9} + 499 \beta_{8} - 272 \beta_{7} - 343 \beta_{6} + 136 \beta_{5} + 554 \beta_{4} - 472 \beta_{3} + 477 \beta_{2} + 1666 \beta _1 + 1330 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1042 \beta_{13} - 1288 \beta_{12} - 727 \beta_{11} - 1384 \beta_{10} + 1735 \beta_{9} + 1457 \beta_{8} - 608 \beta_{7} - 775 \beta_{6} + 964 \beta_{5} + 391 \beta_{4} - 918 \beta_{3} + 2528 \beta_{2} + 4708 \beta _1 + 9681 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3577 \beta_{13} - 6839 \beta_{12} - 3169 \beta_{11} - 5805 \beta_{10} + 9425 \beta_{9} + 7955 \beta_{8} - 4368 \beta_{7} - 5567 \beta_{6} + 2795 \beta_{5} + 8163 \beta_{4} - 6913 \beta_{3} + 7946 \beta_{2} + \cdots + 22865 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16046 \beta_{13} - 21885 \beta_{12} - 14347 \beta_{11} - 21689 \beta_{10} + 29664 \beta_{9} + 26388 \beta_{8} - 11895 \beta_{7} - 14878 \beta_{6} + 16638 \beta_{5} + 9320 \beta_{4} + \cdots + 137060 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 53845 \beta_{13} - 104023 \beta_{12} - 61959 \beta_{11} - 86617 \beta_{10} + 146845 \beta_{9} + 128475 \beta_{8} - 70951 \beta_{7} - 87873 \beta_{6} + 52231 \beta_{5} + 119131 \beta_{4} + \cdots + 378441 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 240711 \beta_{13} - 358979 \beta_{12} - 262841 \beta_{11} - 332392 \beta_{10} + 490926 \beta_{9} + 458018 \beta_{8} - 220217 \beta_{7} - 261199 \beta_{6} + 278316 \beta_{5} + \cdots + 2009029 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 813002 \beta_{13} - 1600949 \beta_{12} - 1119266 \beta_{11} - 1302208 \beta_{10} + 2287088 \beta_{9} + 2086049 \beta_{8} - 1164292 \beta_{7} - 1364934 \beta_{6} + 933748 \beta_{5} + \cdots + 6143881 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.43278
−2.80057
−2.39614
−2.37835
−0.879372
−0.840453
−0.744629
1.04642
1.26594
1.50425
2.54364
3.22476
3.88590
4.00137
−1.00000 −1.00000 1.00000 −4.43278 1.00000 1.00000 −1.00000 1.00000 4.43278
1.2 −1.00000 −1.00000 1.00000 −3.80057 1.00000 1.00000 −1.00000 1.00000 3.80057
1.3 −1.00000 −1.00000 1.00000 −3.39614 1.00000 1.00000 −1.00000 1.00000 3.39614
1.4 −1.00000 −1.00000 1.00000 −3.37835 1.00000 1.00000 −1.00000 1.00000 3.37835
1.5 −1.00000 −1.00000 1.00000 −1.87937 1.00000 1.00000 −1.00000 1.00000 1.87937
1.6 −1.00000 −1.00000 1.00000 −1.84045 1.00000 1.00000 −1.00000 1.00000 1.84045
1.7 −1.00000 −1.00000 1.00000 −1.74463 1.00000 1.00000 −1.00000 1.00000 1.74463
1.8 −1.00000 −1.00000 1.00000 0.0464237 1.00000 1.00000 −1.00000 1.00000 −0.0464237
1.9 −1.00000 −1.00000 1.00000 0.265939 1.00000 1.00000 −1.00000 1.00000 −0.265939
1.10 −1.00000 −1.00000 1.00000 0.504252 1.00000 1.00000 −1.00000 1.00000 −0.504252
1.11 −1.00000 −1.00000 1.00000 1.54364 1.00000 1.00000 −1.00000 1.00000 −1.54364
1.12 −1.00000 −1.00000 1.00000 2.22476 1.00000 1.00000 −1.00000 1.00000 −2.22476
1.13 −1.00000 −1.00000 1.00000 2.88590 1.00000 1.00000 −1.00000 1.00000 −2.88590
1.14 −1.00000 −1.00000 1.00000 3.00137 1.00000 1.00000 −1.00000 1.00000 −3.00137
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8022.2.a.x 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.x 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{14} + 10 T_{5}^{13} + 4 T_{5}^{12} - 236 T_{5}^{11} - 520 T_{5}^{10} + 1886 T_{5}^{9} + 6024 T_{5}^{8} - 5473 T_{5}^{7} - 26249 T_{5}^{6} + 2332 T_{5}^{5} + 44644 T_{5}^{4} + 4538 T_{5}^{3} - 22606 T_{5}^{2} + \cdots - 216 \) Copy content Toggle raw display
\( T_{11}^{14} + 8 T_{11}^{13} - 72 T_{11}^{12} - 727 T_{11}^{11} + 1287 T_{11}^{10} + 24400 T_{11}^{9} + 16563 T_{11}^{8} - 363472 T_{11}^{7} - 752537 T_{11}^{6} + 2104246 T_{11}^{5} + 7693826 T_{11}^{4} + 536336 T_{11}^{3} + \cdots - 12635712 \) Copy content Toggle raw display
\( T_{13}^{14} - 5 T_{13}^{13} - 88 T_{13}^{12} + 428 T_{13}^{11} + 2739 T_{13}^{10} - 13385 T_{13}^{9} - 35815 T_{13}^{8} + 188002 T_{13}^{7} + 152454 T_{13}^{6} - 1118570 T_{13}^{5} + 272054 T_{13}^{4} + 1844292 T_{13}^{3} + \cdots + 68768 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 10 T^{13} + 4 T^{12} - 236 T^{11} + \cdots - 216 \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 8 T^{13} - 72 T^{12} + \cdots - 12635712 \) Copy content Toggle raw display
$13$ \( T^{14} - 5 T^{13} - 88 T^{12} + \cdots + 68768 \) Copy content Toggle raw display
$17$ \( T^{14} + 16 T^{13} - 11 T^{12} + \cdots - 4493696 \) Copy content Toggle raw display
$19$ \( T^{14} - 7 T^{13} - 135 T^{12} + \cdots + 582372 \) Copy content Toggle raw display
$23$ \( T^{14} + 8 T^{13} + \cdots - 1312254464 \) Copy content Toggle raw display
$29$ \( T^{14} + 21 T^{13} + \cdots + 179968648 \) Copy content Toggle raw display
$31$ \( T^{14} - 7 T^{13} - 172 T^{12} + \cdots + 4419072 \) Copy content Toggle raw display
$37$ \( T^{14} - 8 T^{13} - 315 T^{12} + \cdots + 219308032 \) Copy content Toggle raw display
$41$ \( T^{14} + 20 T^{13} + \cdots + 2882134944 \) Copy content Toggle raw display
$43$ \( T^{14} - 22 T^{13} + \cdots - 13806254592 \) Copy content Toggle raw display
$47$ \( T^{14} + 18 T^{13} + \cdots + 6794776576 \) Copy content Toggle raw display
$53$ \( T^{14} + 25 T^{13} + 5 T^{12} + \cdots + 18222048 \) Copy content Toggle raw display
$59$ \( T^{14} + 15 T^{13} + \cdots + 1638770432 \) Copy content Toggle raw display
$61$ \( T^{14} - 5 T^{13} - 370 T^{12} + \cdots - 50415776 \) Copy content Toggle raw display
$67$ \( T^{14} - 12 T^{13} + \cdots + 7316168544 \) Copy content Toggle raw display
$71$ \( T^{14} + 20 T^{13} + \cdots + 362273299488 \) Copy content Toggle raw display
$73$ \( T^{14} + T^{13} + \cdots - 11406227387464 \) Copy content Toggle raw display
$79$ \( T^{14} - 22 T^{13} + \cdots + 8963645056 \) Copy content Toggle raw display
$83$ \( T^{14} + 33 T^{13} + \cdots - 563197854976 \) Copy content Toggle raw display
$89$ \( T^{14} + 21 T^{13} + \cdots + 1598223968 \) Copy content Toggle raw display
$97$ \( T^{14} + 10 T^{13} + \cdots + 10975123472896 \) Copy content Toggle raw display
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