Properties

Label 2736.3.o.r
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
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] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (of order \(2\), degree \(1\), 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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + 597327 x^{12} - 837564 x^{11} + 2838686 x^{10} - 2898276 x^{9} + \cdots + 36100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{37} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + (\beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + (\beta_{3} + 1) q^{7} + ( - \beta_{4} + \beta_1 + 1) q^{11} - \beta_{13} q^{13} + (\beta_{2} - 2) q^{17} + ( - \beta_{15} + \beta_{3} - 2) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{23} + (\beta_{10} - \beta_{4} + \beta_{3} + \beta_1 + 4) q^{25} + ( - \beta_{17} - \beta_{15} - \beta_{13} - \beta_{9}) q^{29} + ( - \beta_{19} + \beta_{18} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{8}) q^{31} + ( - \beta_{17} + \beta_{15} - \beta_{10} - \beta_{7} - 2 \beta_{4} - \beta_{2} - \beta_1 - 10) q^{35} + (\beta_{18} + \beta_{17} + \beta_{15} - \beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{8}) q^{37} + ( - \beta_{18} - 2 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8}) q^{41} + ( - \beta_{17} + \beta_{15} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{43} + ( - \beta_{17} + \beta_{15} - \beta_{10} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{47}+ \cdots + (2 \beta_{18} - 3 \beta_{17} - \beta_{16} - 3 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} - \beta_{12} + \cdots - 3 \beta_{9}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + 597327 x^{12} - 837564 x^{11} + 2838686 x^{10} - 2898276 x^{9} + \cdots + 36100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16\!\cdots\!99 \nu^{19} + \cdots + 42\!\cdots\!16 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38\!\cdots\!16 \nu^{19} + \cdots - 81\!\cdots\!16 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 55\!\cdots\!12 \nu^{19} + \cdots + 19\!\cdots\!08 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58\!\cdots\!16 \nu^{19} + \cdots + 21\!\cdots\!12 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 90\!\cdots\!71 \nu^{19} + \cdots + 59\!\cdots\!52 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!43 \nu^{19} + \cdots - 32\!\cdots\!64 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40\!\cdots\!32 \nu^{19} + \cdots + 25\!\cdots\!40 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!97 \nu^{19} + \cdots + 50\!\cdots\!60 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!41 \nu^{19} + \cdots + 98\!\cdots\!80 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!25 \nu^{19} + \cdots + 18\!\cdots\!12 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 72\!\cdots\!84 \nu^{19} + \cdots + 91\!\cdots\!60 ) / 33\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!36 \nu^{19} + \cdots + 27\!\cdots\!40 ) / 72\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{19} + \cdots + 20\!\cdots\!60 ) / 33\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{19} + \cdots + 14\!\cdots\!80 ) / 24\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 87\!\cdots\!19 \nu^{19} + \cdots - 16\!\cdots\!20 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 18\!\cdots\!99 \nu^{19} + \cdots - 41\!\cdots\!00 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 32\!\cdots\!03 \nu^{19} + \cdots - 27\!\cdots\!00 ) / 45\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 28\!\cdots\!81 \nu^{19} + \cdots + 43\!\cdots\!40 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!85 \nu^{19} + \cdots + 19\!\cdots\!40 ) / 67\!\cdots\!41 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{18} - \beta_{17} - \beta_{15} + \beta_{13} - \beta_{12} - 2\beta_{11} - \beta_{9} - 4\beta_{4} + 4\beta_{3} + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 10 \beta_{19} + 7 \beta_{18} + 3 \beta_{17} - 6 \beta_{16} - 9 \beta_{15} + 14 \beta_{14} + 11 \beta_{13} - 29 \beta_{12} - 10 \beta_{11} + 7 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 4 \beta _1 - 118 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{17} - 5 \beta_{15} - 9 \beta_{10} + 6 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 102 \beta_{4} - 75 \beta_{3} + 2 \beta_{2} - 8 \beta _1 - 112 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 600 \beta_{19} - 333 \beta_{18} + 481 \beta_{17} + 338 \beta_{16} - 191 \beta_{15} - 910 \beta_{14} - 769 \beta_{13} + 1281 \beta_{12} + 732 \beta_{11} - 60 \beta_{10} - 469 \beta_{9} - 70 \beta_{8} + 280 \beta_{7} - 84 \beta_{6} + \cdots - 5780 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1480 \beta_{19} + 867 \beta_{18} + 1379 \beta_{17} + 756 \beta_{16} + 3179 \beta_{15} - 3794 \beta_{14} - 2659 \beta_{13} + 3847 \beta_{12} + 9050 \beta_{11} + 1256 \beta_{10} - 833 \beta_{9} - 464 \beta_{8} + \cdots + 17968 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9863 \beta_{17} + 9863 \beta_{15} + 2644 \beta_{10} - 8802 \beta_{7} + 1613 \beta_{6} + 2813 \beta_{5} - 18710 \beta_{4} + 4499 \beta_{3} - 7962 \beta_{2} - 3966 \beta _1 + 169367 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 123774 \beta_{19} - 28069 \beta_{18} - 209999 \beta_{17} - 63720 \beta_{16} - 65531 \beta_{15} + 281280 \beta_{14} + 204821 \beta_{13} - 292735 \beta_{12} - 572686 \beta_{11} + \cdots + 1380264 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2157980 \beta_{19} + 963553 \beta_{18} + 483971 \beta_{17} - 1182358 \beta_{16} - 1904885 \beta_{15} + 3534834 \beta_{14} + 2940433 \beta_{13} - 4438749 \beta_{12} + \cdots - 20645344 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2708064 \beta_{17} - 2708064 \beta_{15} - 2471014 \beta_{10} + 3002705 \beta_{7} + 1847372 \beta_{6} + 150332 \beta_{5} + 21589639 \beta_{4} - 13551088 \beta_{3} + 2004278 \beta_{2} + \cdots - 50524736 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 132802650 \beta_{19} - 54027105 \beta_{18} + 123398467 \beta_{17} + 72435042 \beta_{16} - 23966185 \beta_{15} - 222678954 \beta_{14} - 183401109 \beta_{13} + \cdots - 1281689150 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 682394814 \beta_{19} - 31995467 \beta_{18} + 151603019 \beta_{17} + 359353008 \beta_{16} + 929726799 \beta_{15} - 1378101024 \beta_{14} - 1050118429 \beta_{13} + \cdots + 7154649976 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1152279790 \beta_{17} + 1152279790 \beta_{15} + 482830935 \beta_{10} - 1093657394 \beta_{7} - 32850285 \beta_{6} + 238841537 \beta_{5} - 3760559113 \beta_{4} + \cdots + 20151252679 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 47924797136 \beta_{19} + 5273905181 \beta_{18} - 61643806723 \beta_{17} - 25392750068 \beta_{16} - 7258461387 \beta_{15} + 93910481090 \beta_{14} + \cdots + 495414629696 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 523121794890 \beta_{19} + 181381395201 \beta_{18} + 59297990013 \beta_{17} - 283635645014 \beta_{16} - 523777450319 \beta_{15} + 907210986478 \beta_{14} + \cdots - 5120627804022 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 933244928353 \beta_{17} - 933244928353 \beta_{15} - 634835552577 \beta_{10} + 965165474152 \beta_{7} + 349047400658 \beta_{6} - 63781030873 \beta_{5} + \cdots - 16895883956052 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 33328220141620 \beta_{19} - 10837840970449 \beta_{18} + 34234246792029 \beta_{17} + 18032892033126 \beta_{16} - 2974689881899 \beta_{15} + \cdots - 327898917272952 ) / 16 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 224219509291284 \beta_{19} - 40523183971793 \beta_{18} + 16225053128667 \beta_{17} + 119626753439604 \beta_{16} + 269314701997483 \beta_{15} + \cdots + 22\!\cdots\!68 ) / 16 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 597776987190383 \beta_{17} + 597776987190383 \beta_{15} + 297148023316908 \beta_{10} - 582588423789262 \beta_{7} - 78740530663111 \beta_{6} + \cdots + 10\!\cdots\!95 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 15\!\cdots\!70 \beta_{19} + \cdots + 15\!\cdots\!40 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.0545091 0.0944125i
0.0545091 + 0.0944125i
0.735188 + 1.27338i
0.735188 1.27338i
−0.861157 + 1.49157i
−0.861157 1.49157i
−3.49479 6.05315i
−3.49479 + 6.05315i
−1.34238 + 2.32507i
−1.34238 2.32507i
1.76236 + 3.05250i
1.76236 3.05250i
0.567379 0.982730i
0.567379 + 0.982730i
0.855886 + 1.48244i
0.855886 1.48244i
−0.330374 0.572225i
−0.330374 + 0.572225i
4.05338 + 7.02066i
4.05338 7.02066i
0 0 0 −7.51170 0 7.72974 0 0 0
721.2 0 0 0 −7.51170 0 7.72974 0 0 0
721.3 0 0 0 −6.80804 0 9.74880 0 0 0
721.4 0 0 0 −6.80804 0 9.74880 0 0 0
721.5 0 0 0 −5.26532 0 1.82069 0 0 0
721.6 0 0 0 −5.26532 0 1.82069 0 0 0
721.7 0 0 0 −2.73712 0 −11.2420 0 0 0
721.8 0 0 0 −2.73712 0 −11.2420 0 0 0
721.9 0 0 0 0.0118390 0 −5.38136 0 0 0
721.10 0 0 0 0.0118390 0 −5.38136 0 0 0
721.11 0 0 0 0.816130 0 6.23331 0 0 0
721.12 0 0 0 0.816130 0 6.23331 0 0 0
721.13 0 0 0 1.70658 0 0.562937 0 0 0
721.14 0 0 0 1.70658 0 0.562937 0 0 0
721.15 0 0 0 3.78754 0 −0.363991 0 0 0
721.16 0 0 0 3.78754 0 −0.363991 0 0 0
721.17 0 0 0 7.79518 0 −9.11667 0 0 0
721.18 0 0 0 7.79518 0 −9.11667 0 0 0
721.19 0 0 0 8.20492 0 8.00860 0 0 0
721.20 0 0 0 8.20492 0 8.00860 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.r 20
3.b odd 2 1 912.3.o.e 20
4.b odd 2 1 1368.3.o.c 20
12.b even 2 1 456.3.o.a 20
19.b odd 2 1 inner 2736.3.o.r 20
57.d even 2 1 912.3.o.e 20
76.d even 2 1 1368.3.o.c 20
228.b odd 2 1 456.3.o.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.3.o.a 20 12.b even 2 1
456.3.o.a 20 228.b odd 2 1
912.3.o.e 20 3.b odd 2 1
912.3.o.e 20 57.d even 2 1
1368.3.o.c 20 4.b odd 2 1
1368.3.o.c 20 76.d even 2 1
2736.3.o.r 20 1.a even 1 1 trivial
2736.3.o.r 20 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{10} - 142 T_{5}^{8} - 60 T_{5}^{7} + 6433 T_{5}^{6} + 3716 T_{5}^{5} - 96156 T_{5}^{4} - 7072 T_{5}^{3} + 370656 T_{5}^{2} - 253056 T_{5} + 2944 \) Copy content Toggle raw display
\( T_{7}^{10} - 8 T_{7}^{9} - 218 T_{7}^{8} + 1976 T_{7}^{7} + 12441 T_{7}^{6} - 140592 T_{7}^{5} - 51568 T_{7}^{4} + 2619680 T_{7}^{3} - 4274352 T_{7}^{2} + 223616 T_{7} + 774016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 142 T^{8} - 60 T^{7} + 6433 T^{6} + \cdots + 2944)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 8 T^{9} - 218 T^{8} + \cdots + 774016)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 8 T^{9} - 800 T^{8} + \cdots - 185601120)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 1872 T^{18} + \cdots + 92\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{10} + 16 T^{9} + \cdots - 166822288832)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 40 T^{19} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{10} - 32 T^{9} + \cdots - 11287201510400)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 7316 T^{18} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{20} + 11944 T^{18} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + 12528 T^{18} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + 11364 T^{18} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + 32 T^{9} + \cdots + 896271935895552)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 24 T^{9} + \cdots - 152806724137280)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 32868 T^{18} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{20} + 30992 T^{18} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} - 92 T^{9} + \cdots - 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 32424 T^{18} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + 64976 T^{18} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{10} - 52 T^{9} + \cdots + 27\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 57984 T^{18} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{10} - 112 T^{9} + \cdots - 55\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 82564 T^{18} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + 83296 T^{18} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
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