Properties

Label 8048.2.a.p
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} -\beta_{2} q^{5} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{11} + ( -2 + \beta_{3} + \beta_{7} ) q^{13} -\beta_{3} q^{15} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{17} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{19} + ( 1 + \beta_{2} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{21} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{23} + ( -2 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{25} + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{27} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{29} + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + ( \beta_{1} - \beta_{4} - 2 \beta_{9} ) q^{33} + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{35} + ( -6 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{37} + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{39} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{8} - \beta_{9} ) q^{41} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{43} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{45} + ( -2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{47} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{51} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{53} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{9} ) q^{55} + ( -2 - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{57} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} + ( 3 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 6 \beta_{7} + \beta_{8} + \beta_{9} ) q^{61} + ( -1 + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{63} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{65} + ( -4 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 5 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{67} + ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{69} + ( 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} ) q^{71} + ( -4 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 4 \beta_{9} ) q^{73} + ( -4 + 4 \beta_{1} + \beta_{3} + \beta_{5} ) q^{75} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} ) q^{77} + ( 4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} ) q^{79} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{81} + ( -5 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 5 \beta_{6} + \beta_{7} + 4 \beta_{8} + 3 \beta_{9} ) q^{83} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{85} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{87} + ( -4 + 4 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 5 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{89} + ( 1 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} ) q^{91} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{8} ) q^{93} + ( -5 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{95} + ( -7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} ) q^{97} + ( -4 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{6} - 4 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + O(q^{10}) \) \( 10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + 3q^{11} - 18q^{13} + 2q^{15} - 11q^{17} + q^{21} + 2q^{23} - 27q^{25} + 2q^{27} - 9q^{29} + 22q^{31} - 10q^{33} + 6q^{35} - 35q^{37} - 8q^{39} - 4q^{41} + 20q^{43} + 2q^{45} - 7q^{47} - 27q^{49} - 9q^{51} - 24q^{53} + 11q^{55} - 23q^{57} - 17q^{59} - 4q^{61} - 10q^{63} - 16q^{65} + 6q^{67} - 2q^{69} + q^{71} - 31q^{73} - 30q^{75} + 3q^{77} + 10q^{79} - 6q^{81} - 22q^{83} - 6q^{85} - 25q^{87} + q^{89} - 10q^{91} - 6q^{93} - 39q^{95} - 57q^{97} - 35q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( 3 \nu^{9} - 8 \nu^{8} - 22 \nu^{7} + 57 \nu^{6} + 46 \nu^{5} - 113 \nu^{4} - 34 \nu^{3} + 65 \nu^{2} + 12 \nu - 5 \)
\(\beta_{3}\)\(=\)\( 5 \nu^{9} - 13 \nu^{8} - 37 \nu^{7} + 92 \nu^{6} + 78 \nu^{5} - 181 \nu^{4} - 57 \nu^{3} + 104 \nu^{2} + 20 \nu - 8 \)
\(\beta_{4}\)\(=\)\( -6 \nu^{9} + 15 \nu^{8} + 47 \nu^{7} - 108 \nu^{6} - 113 \nu^{5} + 221 \nu^{4} + 108 \nu^{3} - 139 \nu^{2} - 44 \nu + 13 \)
\(\beta_{5}\)\(=\)\( 10 \nu^{9} - 25 \nu^{8} - 78 \nu^{7} + 180 \nu^{6} + 184 \nu^{5} - 368 \nu^{4} - 165 \nu^{3} + 230 \nu^{2} + 60 \nu - 21 \)
\(\beta_{6}\)\(=\)\( -11 \nu^{9} + 28 \nu^{8} + 84 \nu^{7} - 200 \nu^{6} - 191 \nu^{5} + 402 \nu^{4} + 165 \nu^{3} - 244 \nu^{2} - 63 \nu + 22 \)
\(\beta_{7}\)\(=\)\( -18 \nu^{9} + 45 \nu^{8} + 139 \nu^{7} - 321 \nu^{6} - 321 \nu^{5} + 645 \nu^{4} + 278 \nu^{3} - 394 \nu^{2} - 102 \nu + 38 \)
\(\beta_{8}\)\(=\)\( -21 \nu^{9} + 53 \nu^{8} + 162 \nu^{7} - 380 \nu^{6} - 375 \nu^{5} + 770 \nu^{4} + 331 \nu^{3} - 475 \nu^{2} - 127 \nu + 45 \)
\(\beta_{9}\)\(=\)\( -22 \nu^{9} + 55 \nu^{8} + 170 \nu^{7} - 392 \nu^{6} - 394 \nu^{5} + 785 \nu^{4} + 346 \nu^{3} - 473 \nu^{2} - 130 \nu + 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{4} - \beta_{3} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 10 \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\(-\beta_{9} + 8 \beta_{8} + 3 \beta_{7} - 22 \beta_{6} + 9 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 5 \beta_{2} + 34 \beta_{1} - 12\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 12 \beta_{8} + 19 \beta_{7} - 73 \beta_{6} + 22 \beta_{5} + 58 \beta_{4} - 37 \beta_{3} - 24 \beta_{2} + 83 \beta_{1} - 28\)
\(\nu^{7}\)\(=\)\(-12 \beta_{9} + 58 \beta_{8} + 37 \beta_{7} - 192 \beta_{6} + 73 \beta_{5} + 113 \beta_{4} - 71 \beta_{3} - 63 \beta_{2} + 252 \beta_{1} - 107\)
\(\nu^{8}\)\(=\)\(-58 \beta_{9} + 113 \beta_{8} + 152 \beta_{7} - 578 \beta_{6} + 192 \beta_{5} + 425 \beta_{4} - 247 \beta_{3} - 220 \beta_{2} + 661 \beta_{1} - 278\)
\(\nu^{9}\)\(=\)\(-113 \beta_{9} + 425 \beta_{8} + 345 \beta_{7} - 1565 \beta_{6} + 578 \beta_{5} + 967 \beta_{4} - 554 \beta_{3} - 591 \beta_{2} + 1920 \beta_{1} - 879\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.78533
1.95007
1.31567
1.07636
0.208270
−0.489003
−0.510671
−0.858231
−1.40552
−2.07227
0 −1.78533 0 −0.701114 0 2.02991 0 0.187388 0
1.2 0 −0.950069 0 −2.28693 0 −2.71022 0 −2.09737 0
1.3 0 −0.315672 0 2.25024 0 3.20647 0 −2.90035 0
1.4 0 −0.0763625 0 1.17276 0 −0.469303 0 −2.99417 0
1.5 0 0.791730 0 0.178789 0 0.0809018 0 −2.37316 0
1.6 0 1.48900 0 −1.79865 0 −0.552233 0 −0.782869 0
1.7 0 1.51067 0 −2.23445 0 3.60329 0 −0.717874 0
1.8 0 1.85823 0 1.44291 0 1.96509 0 0.453023 0
1.9 0 2.40552 0 0.590303 0 −1.95900 0 2.78655 0
1.10 0 3.07227 0 0.386144 0 −0.194914 0 6.43884 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

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 8048.2.a.p 10
4.b odd 2 1 503.2.a.e 10
12.b even 2 1 4527.2.a.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.e 10 4.b odd 2 1
4527.2.a.k 10 12.b even 2 1
8048.2.a.p 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{13}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 1 + 14 T + 5 T^{2} - 86 T^{3} + 54 T^{4} + 75 T^{5} - 85 T^{6} + 10 T^{7} + 18 T^{8} - 8 T^{9} + T^{10} \)
$5$ \( 1 - 9 T + 18 T^{2} + 16 T^{3} - 59 T^{4} + 7 T^{5} + 41 T^{6} - 7 T^{7} - 11 T^{8} + T^{9} + T^{10} \)
$7$ \( -1 + 4 T + 86 T^{2} + 214 T^{3} + 23 T^{4} - 228 T^{5} + 8 T^{6} + 64 T^{7} - 9 T^{8} - 5 T^{9} + T^{10} \)
$11$ \( 311 - 1920 T + 1973 T^{2} + 1781 T^{3} - 1961 T^{4} - 537 T^{5} + 515 T^{6} + 80 T^{7} - 41 T^{8} - 3 T^{9} + T^{10} \)
$13$ \( -19 + 4107 T + 5985 T^{2} - 481 T^{3} - 4882 T^{4} - 2777 T^{5} - 212 T^{6} + 307 T^{7} + 120 T^{8} + 18 T^{9} + T^{10} \)
$17$ \( 30151 - 12996 T - 29792 T^{2} + 3190 T^{3} + 11026 T^{4} + 1710 T^{5} - 1167 T^{6} - 367 T^{7} + 2 T^{8} + 11 T^{9} + T^{10} \)
$19$ \( 8863 + 26147 T - 7633 T^{2} - 23773 T^{3} - 3789 T^{4} + 4568 T^{5} + 1396 T^{6} - 144 T^{7} - 73 T^{8} + T^{10} \)
$23$ \( -2281 - 9002 T + 28798 T^{2} + 21377 T^{3} - 12083 T^{4} - 5432 T^{5} + 2080 T^{6} + 275 T^{7} - 102 T^{8} - 2 T^{9} + T^{10} \)
$29$ \( -1397 - 8766 T - 15421 T^{2} - 187 T^{3} + 20531 T^{4} + 13633 T^{5} + 487 T^{6} - 702 T^{7} - 65 T^{8} + 9 T^{9} + T^{10} \)
$31$ \( 8207 + 5501 T - 57982 T^{2} - 31755 T^{3} + 27453 T^{4} + 2931 T^{5} - 3545 T^{6} + 233 T^{7} + 127 T^{8} - 22 T^{9} + T^{10} \)
$37$ \( -3774629 - 929679 T + 1565189 T^{2} + 560598 T^{3} - 111154 T^{4} - 71008 T^{5} - 5712 T^{6} + 1859 T^{7} + 437 T^{8} + 35 T^{9} + T^{10} \)
$41$ \( -17357 - 111711 T + 143286 T^{2} + 149773 T^{3} - 84240 T^{4} - 7004 T^{5} + 6310 T^{6} - 59 T^{7} - 150 T^{8} + 4 T^{9} + T^{10} \)
$43$ \( -147629 - 16858 T + 511389 T^{2} - 368059 T^{3} - 18426 T^{4} + 78150 T^{5} - 24232 T^{6} + 2646 T^{7} + 5 T^{8} - 20 T^{9} + T^{10} \)
$47$ \( 34183 - 173434 T + 234373 T^{2} + 1951 T^{3} - 115790 T^{4} + 17637 T^{5} + 6796 T^{6} - 757 T^{7} - 148 T^{8} + 7 T^{9} + T^{10} \)
$53$ \( 30585517 - 10539391 T - 9677101 T^{2} + 827393 T^{3} + 972179 T^{4} + 78887 T^{5} - 20053 T^{6} - 2987 T^{7} + 40 T^{8} + 24 T^{9} + T^{10} \)
$59$ \( -3373 - 22898 T - 4680 T^{2} + 157745 T^{3} + 106944 T^{4} - 453 T^{5} - 14454 T^{6} - 3243 T^{7} - 123 T^{8} + 17 T^{9} + T^{10} \)
$61$ \( 160395869 + 192168049 T + 30652592 T^{2} - 9861764 T^{3} - 2056331 T^{4} + 177043 T^{5} + 43009 T^{6} - 1337 T^{7} - 358 T^{8} + 4 T^{9} + T^{10} \)
$67$ \( -52161527 + 188372124 T + 68241481 T^{2} - 3632530 T^{3} - 2954276 T^{4} - 44201 T^{5} + 47943 T^{6} + 1216 T^{7} - 352 T^{8} - 6 T^{9} + T^{10} \)
$71$ \( 14183807 + 195049108 T + 61525134 T^{2} - 13001388 T^{3} - 3793615 T^{4} + 182955 T^{5} + 66864 T^{6} - 594 T^{7} - 448 T^{8} - T^{9} + T^{10} \)
$73$ \( -3955559 - 6719488 T - 1983330 T^{2} + 1173628 T^{3} + 568868 T^{4} - 14845 T^{5} - 33800 T^{6} - 3742 T^{7} + 127 T^{8} + 31 T^{9} + T^{10} \)
$79$ \( 8912581 - 43253952 T + 49214936 T^{2} - 2538597 T^{3} - 5921284 T^{4} - 334500 T^{5} + 91361 T^{6} + 4173 T^{7} - 542 T^{8} - 10 T^{9} + T^{10} \)
$83$ \( 40035623 + 2541538 T - 15595528 T^{2} - 801664 T^{3} + 1491862 T^{4} + 182619 T^{5} - 33621 T^{6} - 6610 T^{7} - 161 T^{8} + 22 T^{9} + T^{10} \)
$89$ \( -789547 - 7972537 T + 3494049 T^{2} + 1459657 T^{3} - 583997 T^{4} - 71001 T^{5} + 27890 T^{6} + 739 T^{7} - 380 T^{8} - T^{9} + T^{10} \)
$97$ \( 3229338523 + 3026058216 T + 985216534 T^{2} + 111741584 T^{3} - 7165096 T^{4} - 2939025 T^{5} - 240656 T^{6} - 276 T^{7} + 1029 T^{8} + 57 T^{9} + T^{10} \)
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