Properties

Label 1001.2.a.n
Level $1001$
Weight $2$
Character orbit 1001.a
Self dual yes
Analytic conductor $7.993$
Analytic rank $0$
Dimension $11$
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] = [1001,2,Mod(1,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, 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("1001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.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: \(7.99302524233\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 18x^{9} + 15x^{8} + 117x^{7} - 78x^{6} - 326x^{5} + 167x^{4} + 348x^{3} - 143x^{2} - 74x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 18x^{9} + 15x^{8} + 117x^{7} - 78x^{6} - 326x^{5} + 167x^{4} + 348x^{3} - 143x^{2} - 74x + 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - \nu^{8} - 13\nu^{7} + 10\nu^{6} + 52\nu^{5} - 28\nu^{4} - 58\nu^{3} + 19\nu^{2} - 14\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 2 \nu^{9} - 16 \nu^{8} + 31 \nu^{7} + 86 \nu^{6} - 148 \nu^{5} - 178 \nu^{4} + 201 \nu^{3} + 131 \nu^{2} + 14 \nu - 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 2 \nu^{9} - 8 \nu^{8} + 23 \nu^{7} - 18 \nu^{6} - 84 \nu^{5} + 254 \nu^{4} + 105 \nu^{3} - 461 \nu^{2} - 10 \nu + 88 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{10} + 2 \nu^{9} + 16 \nu^{8} - 31 \nu^{7} - 86 \nu^{6} + 164 \nu^{5} + 162 \nu^{4} - 329 \nu^{3} - 35 \nu^{2} + 178 \nu - 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} - 18\nu^{8} - 3\nu^{7} + 122\nu^{6} + 28\nu^{5} - 370\nu^{4} - 67\nu^{3} + 433\nu^{2} + 26\nu - 88 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{10} + \nu^{9} + 17 \nu^{8} - 14 \nu^{7} - 104 \nu^{6} + 68 \nu^{5} + 274 \nu^{4} - 131 \nu^{3} - 282 \nu^{2} + 76 \nu + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{10} + \nu^{9} + 35 \nu^{8} - 11 \nu^{7} - 218 \nu^{6} + 32 \nu^{5} + 564 \nu^{4} - 8 \nu^{3} - 515 \nu^{2} - 14 \nu + 64 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{5} + 8\beta_{3} + 28\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + 10\beta_{9} + 13\beta_{8} - 10\beta_{7} + \beta_{5} + \beta_{3} + 35\beta_{2} + \beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{10} + 12 \beta_{9} + 16 \beta_{8} - 3 \beta_{7} + 13 \beta_{5} + 2 \beta_{4} + 55 \beta_{3} + \beta_{2} + 164 \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{10} + 80 \beta_{9} + 123 \beta_{8} - 79 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 16 \beta_{3} + 206 \beta_{2} + 16 \beta _1 + 524 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31 \beta_{10} + 112 \beta_{9} + 177 \beta_{8} - 46 \beta_{7} + 2 \beta_{6} + 123 \beta_{5} + 36 \beta_{4} + 363 \beta_{3} + 18 \beta_{2} + 986 \beta _1 + 253 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 154 \beta_{10} + 598 \beta_{9} + 1034 \beta_{8} - 581 \beta_{7} + 36 \beta_{6} + 177 \beta_{5} + 42 \beta_{4} + 174 \beta_{3} + 1228 \beta_{2} + 183 \beta _1 + 3301 \) 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
2.69905
2.42100
2.14071
1.35489
0.610570
0.282248
−0.501958
−1.46230
−1.72345
−2.37413
−2.44663
−2.69905 −3.23405 5.28489 −1.09400 8.72887 1.00000 −8.86610 7.45906 2.95277
1.2 −2.42100 2.69707 3.86122 3.65517 −6.52959 1.00000 −4.50601 4.27418 −8.84916
1.3 −2.14071 0.698070 2.58264 −3.13099 −1.49437 1.00000 −1.24727 −2.51270 6.70256
1.4 −1.35489 −0.152758 −0.164270 3.54377 0.206971 1.00000 2.93235 −2.97666 −4.80142
1.5 −0.610570 −1.93031 −1.62720 1.75578 1.17859 1.00000 2.21466 0.726080 −1.07202
1.6 −0.282248 3.11943 −1.92034 0.477922 −0.880455 1.00000 1.10651 6.73086 −0.134893
1.7 0.501958 0.372196 −1.74804 −1.99673 0.186827 1.00000 −1.88136 −2.86147 −1.00228
1.8 1.46230 −2.15409 0.138330 0.136889 −3.14993 1.00000 −2.72233 1.64009 0.200174
1.9 1.72345 1.94436 0.970287 2.87921 3.35102 1.00000 −1.77466 0.780553 4.96218
1.10 2.37413 2.26106 3.63647 −2.08089 5.36804 1.00000 3.88519 2.11240 −4.94030
1.11 2.44663 −1.62099 3.98600 2.85388 −3.96597 1.00000 4.85901 −0.372378 6.98239
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 1001.2.a.n 11
3.b odd 2 1 9009.2.a.bs 11
7.b odd 2 1 7007.2.a.w 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.2.a.n 11 1.a even 1 1 trivial
7007.2.a.w 11 7.b odd 2 1
9009.2.a.bs 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\):

\( T_{2}^{11} + T_{2}^{10} - 18 T_{2}^{9} - 15 T_{2}^{8} + 117 T_{2}^{7} + 78 T_{2}^{6} - 326 T_{2}^{5} - 167 T_{2}^{4} + 348 T_{2}^{3} + 143 T_{2}^{2} - 74 T_{2} - 24 \) Copy content Toggle raw display
\( T_{3}^{11} - 2 T_{3}^{10} - 22 T_{3}^{9} + 42 T_{3}^{8} + 165 T_{3}^{7} - 289 T_{3}^{6} - 514 T_{3}^{5} + 795 T_{3}^{4} + 547 T_{3}^{3} - 745 T_{3}^{2} + 86 T_{3} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + T^{10} - 18 T^{9} - 15 T^{8} + \cdots - 24 \) Copy content Toggle raw display
$3$ \( T^{11} - 2 T^{10} - 22 T^{9} + 42 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( T^{11} - 7 T^{10} - 8 T^{9} + 138 T^{8} + \cdots - 174 \) Copy content Toggle raw display
$7$ \( (T - 1)^{11} \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( (T - 1)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} - 7 T^{10} - 75 T^{9} + \cdots - 4272 \) Copy content Toggle raw display
$19$ \( T^{11} - 22 T^{10} + 138 T^{9} + \cdots + 93440 \) Copy content Toggle raw display
$23$ \( T^{11} - 3 T^{10} - 155 T^{9} + \cdots + 1092096 \) Copy content Toggle raw display
$29$ \( T^{11} + 6 T^{10} - 183 T^{9} + \cdots - 1261440 \) Copy content Toggle raw display
$31$ \( T^{11} - 28 T^{10} + 205 T^{9} + \cdots - 8046592 \) Copy content Toggle raw display
$37$ \( T^{11} - T^{10} - 174 T^{9} + \cdots - 10120192 \) Copy content Toggle raw display
$41$ \( T^{11} + 4 T^{10} - 214 T^{9} + \cdots + 897024 \) Copy content Toggle raw display
$43$ \( T^{11} + 8 T^{10} - 89 T^{9} + \cdots + 774352 \) Copy content Toggle raw display
$47$ \( T^{11} - 22 T^{10} - 139 T^{9} + \cdots + 12413952 \) Copy content Toggle raw display
$53$ \( T^{11} - 9 T^{10} - 212 T^{9} + \cdots + 9351948 \) Copy content Toggle raw display
$59$ \( T^{11} + 2 T^{10} - 440 T^{9} + \cdots - 634513920 \) Copy content Toggle raw display
$61$ \( T^{11} - 303 T^{9} + \cdots - 104898944 \) Copy content Toggle raw display
$67$ \( T^{11} - 23 T^{10} - 51 T^{9} + \cdots + 20600344 \) Copy content Toggle raw display
$71$ \( T^{11} - 3 T^{10} - 405 T^{9} + \cdots + 3265536 \) Copy content Toggle raw display
$73$ \( T^{11} - 29 T^{10} + \cdots + 272272384 \) Copy content Toggle raw display
$79$ \( T^{11} - 26 T^{10} + \cdots - 789727040 \) Copy content Toggle raw display
$83$ \( T^{11} - 9 T^{10} - 361 T^{9} + \cdots - 17918904 \) Copy content Toggle raw display
$89$ \( T^{11} - 9 T^{10} - 238 T^{9} + \cdots - 122805330 \) Copy content Toggle raw display
$97$ \( T^{11} - 40 T^{10} + \cdots - 6467409952 \) Copy content Toggle raw display
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