Properties

Label 325.2.e.e
Level $325$
Weight $2$
Character orbit 325.e
Analytic conductor $2.595$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 8 x^{10} + 54 x^{8} + 78 x^{6} + 92 x^{4} + 10 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( \beta_{6} - \beta_{11} ) q^{3} + ( -1 + \beta_{2} + \beta_{7} - \beta_{8} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{7} - \beta_{8} ) q^{6} + ( \beta_{1} - \beta_{5} ) q^{7} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{10} + \beta_{11} ) q^{8} + ( -1 + \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( \beta_{6} - \beta_{11} ) q^{3} + ( -1 + \beta_{2} + \beta_{7} - \beta_{8} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{7} - \beta_{8} ) q^{6} + ( \beta_{1} - \beta_{5} ) q^{7} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{10} + \beta_{11} ) q^{8} + ( -1 + \beta_{7} - \beta_{9} ) q^{9} + ( -\beta_{3} - \beta_{9} ) q^{11} + ( -3 \beta_{1} - \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{10} - \beta_{11} ) q^{12} + ( \beta_{1} + \beta_{5} + \beta_{10} - \beta_{11} ) q^{13} + ( 4 - \beta_{2} ) q^{14} + ( \beta_{3} - 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{16} + ( \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} - \beta_{4} - \beta_{6} + \beta_{10} ) q^{18} + ( -2 + 2 \beta_{7} + \beta_{9} ) q^{19} + ( -2 \beta_{2} + \beta_{3} ) q^{21} -\beta_{4} q^{22} + ( -\beta_{6} + \beta_{11} ) q^{23} + ( \beta_{3} - 6 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{24} + ( 3 - 2 \beta_{2} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{26} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} ) q^{27} + ( 5 \beta_{6} - \beta_{10} - 3 \beta_{11} ) q^{28} -3 \beta_{7} q^{29} + ( -2 + 2 \beta_{2} ) q^{31} + ( 2 \beta_{1} - \beta_{5} ) q^{32} + ( -\beta_{1} - 3 \beta_{5} ) q^{33} + ( -1 - \beta_{2} - \beta_{3} ) q^{34} + ( -\beta_{3} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{36} + ( \beta_{10} + \beta_{11} ) q^{37} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{6} - \beta_{10} ) q^{38} + ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{39} + ( 3 \beta_{7} - 2 \beta_{8} ) q^{41} + ( 6 \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{42} + ( \beta_{1} + 3 \beta_{5} ) q^{43} + ( \beta_{2} - \beta_{3} ) q^{44} + ( 2 - \beta_{2} - 2 \beta_{7} + \beta_{8} ) q^{46} + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( 6 \beta_{1} + \beta_{4} + 4 \beta_{5} ) q^{48} + ( \beta_{3} - \beta_{7} + \beta_{9} ) q^{49} + ( 2 - \beta_{3} ) q^{51} + ( 4 \beta_{1} + 3 \beta_{5} + 7 \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{52} + ( -\beta_{1} + \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{10} + 4 \beta_{11} ) q^{53} + ( 4 \beta_{7} - \beta_{8} ) q^{54} + ( -4 + 4 \beta_{2} + 4 \beta_{7} - 4 \beta_{8} - \beta_{9} ) q^{56} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{11} ) q^{57} + 3 \beta_{1} q^{58} + ( 2 \beta_{2} - 2 \beta_{8} - \beta_{9} ) q^{59} + ( 1 - \beta_{7} + 2 \beta_{9} ) q^{61} + ( -8 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{62} + ( -2 \beta_{10} + 4 \beta_{11} ) q^{63} + ( 1 + 2 \beta_{3} ) q^{64} + \beta_{2} q^{66} + ( -3 \beta_{6} - \beta_{11} ) q^{67} + ( 2 \beta_{6} + \beta_{11} ) q^{68} + ( 4 - 4 \beta_{7} + \beta_{9} ) q^{69} + ( -2 + 2 \beta_{7} - \beta_{9} ) q^{71} + ( 5 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{72} + ( -5 \beta_{1} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - \beta_{10} + 2 \beta_{11} ) q^{73} + ( -1 - \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{74} + ( \beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{76} + ( -\beta_{1} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{10} + 3 \beta_{11} ) q^{77} + ( 4 \beta_{1} + \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{78} + ( 8 + 2 \beta_{2} ) q^{79} + ( -2 \beta_{3} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{81} + ( -9 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 4 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{10} - 4 \beta_{11} ) q^{83} + ( -16 + 3 \beta_{2} + 16 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{84} -\beta_{2} q^{86} + ( 3 \beta_{1} + 3 \beta_{5} ) q^{87} + ( -3 \beta_{6} - 2 \beta_{10} + \beta_{11} ) q^{88} + ( -\beta_{3} - 2 \beta_{7} - 4 \beta_{8} - \beta_{9} ) q^{89} + ( -4 + 2 \beta_{3} + 2 \beta_{8} - \beta_{9} ) q^{91} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{10} + \beta_{11} ) q^{92} + ( -6 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -2 \beta_{3} - 10 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{94} + ( 2 - 3 \beta_{2} + \beta_{3} ) q^{96} + ( 5 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{97} + ( \beta_{1} + \beta_{4} ) q^{98} + ( -8 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{4} - 10q^{6} - 6q^{9} + O(q^{10}) \) \( 12q - 4q^{4} - 10q^{6} - 6q^{9} + 44q^{14} - 16q^{16} - 12q^{19} - 8q^{21} - 32q^{24} + 24q^{26} - 18q^{29} - 16q^{31} - 16q^{34} - 2q^{36} + 32q^{39} + 14q^{41} + 4q^{44} + 10q^{46} - 6q^{49} + 24q^{51} + 22q^{54} - 16q^{56} + 4q^{59} + 6q^{61} + 12q^{64} + 4q^{66} + 24q^{69} - 12q^{71} - 8q^{74} - 10q^{76} + 104q^{79} + 14q^{81} - 90q^{84} - 4q^{86} - 20q^{89} - 44q^{91} - 56q^{94} + 12q^{96} - 104q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 8 x^{10} + 54 x^{8} + 78 x^{6} + 92 x^{4} + 10 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -16 \nu^{10} - 108 \nu^{8} - 729 \nu^{6} - 184 \nu^{4} - 20 \nu^{2} + 3531 \)\()/1222\)
\(\beta_{3}\)\(=\)\((\)\( 92 \nu^{10} + 621 \nu^{8} + 4039 \nu^{6} + 1058 \nu^{4} + 115 \nu^{2} - 3959 \)\()/1222\)
\(\beta_{4}\)\(=\)\((\)\( -92 \nu^{11} - 621 \nu^{9} - 4039 \nu^{7} - 1058 \nu^{5} - 115 \nu^{3} + 3959 \nu \)\()/1222\)
\(\beta_{5}\)\(=\)\((\)\( 108 \nu^{11} + 729 \nu^{9} + 4768 \nu^{7} + 1242 \nu^{5} + 135 \nu^{3} - 11156 \nu \)\()/1222\)
\(\beta_{6}\)\(=\)\((\)\( -135 \nu^{11} - 1064 \nu^{9} - 7182 \nu^{7} - 9801 \nu^{5} - 12236 \nu^{3} - 1330 \nu \)\()/1222\)
\(\beta_{7}\)\(=\)\((\)\( -135 \nu^{10} - 1064 \nu^{8} - 7182 \nu^{6} - 9801 \nu^{4} - 12236 \nu^{2} - 108 \)\()/1222\)
\(\beta_{8}\)\(=\)\((\)\( -405 \nu^{10} - 3192 \nu^{8} - 21546 \nu^{6} - 29403 \nu^{4} - 35486 \nu^{2} - 324 \)\()/1222\)
\(\beta_{9}\)\(=\)\((\)\( 428 \nu^{10} + 3500 \nu^{8} + 23625 \nu^{6} + 36694 \nu^{4} + 40250 \nu^{2} + 4375 \)\()/1222\)
\(\beta_{10}\)\(=\)\((\)\( 428 \nu^{11} + 3500 \nu^{9} + 23625 \nu^{7} + 36694 \nu^{5} + 40250 \nu^{3} + 4375 \nu \)\()/1222\)
\(\beta_{11}\)\(=\)\( -\nu^{11} - 8 \nu^{9} - 54 \nu^{7} - 78 \nu^{5} - 92 \nu^{3} - 10 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - 3 \beta_{7}\)
\(\nu^{3}\)\(=\)\(\beta_{11} + \beta_{10} - 6 \beta_{6} - \beta_{5} - \beta_{4} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} - 7 \beta_{8} + 17 \beta_{7} + 7 \beta_{2} - 17\)
\(\nu^{5}\)\(=\)\(-7 \beta_{11} - 8 \beta_{10} + 38 \beta_{6}\)
\(\nu^{6}\)\(=\)\(-8 \beta_{3} - 46 \beta_{2} + 107\)
\(\nu^{7}\)\(=\)\(46 \beta_{5} + 54 \beta_{4} + 245 \beta_{1}\)
\(\nu^{8}\)\(=\)\(54 \beta_{9} + 299 \beta_{8} - 689 \beta_{7} + 54 \beta_{3}\)
\(\nu^{9}\)\(=\)\(299 \beta_{11} + 353 \beta_{10} - 1586 \beta_{6} - 299 \beta_{5} - 353 \beta_{4} - 1586 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-353 \beta_{9} - 1939 \beta_{8} + 4459 \beta_{7} + 1939 \beta_{2} - 4459\)
\(\nu^{11}\)\(=\)\(-1939 \beta_{11} - 2292 \beta_{10} + 10276 \beta_{6}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−1.27287 2.20467i
−0.593667 1.02826i
−0.165418 0.286513i
0.165418 + 0.286513i
0.593667 + 1.02826i
1.27287 + 2.20467i
−1.27287 + 2.20467i
−0.593667 + 1.02826i
−0.165418 + 0.286513i
0.165418 0.286513i
0.593667 1.02826i
1.27287 2.20467i
−1.27287 + 2.20467i −1.07646 + 1.86449i −2.24039 3.88048i 0 −2.74039 4.74650i −1.46928 2.54486i 6.31544 −0.817544 1.41603i 0
126.2 −0.593667 + 1.02826i −0.172555 + 0.298874i 0.295120 + 0.511162i 0 −0.204880 0.354863i −1.01478 1.75765i −3.07548 1.44045 + 2.49493i 0
126.3 −0.165418 + 0.286513i 1.34590 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i −1.67674 2.90420i −1.28714 −2.12291 3.67698i 0
126.4 0.165418 0.286513i −1.34590 + 2.33117i 0.945274 + 1.63726i 0 0.445274 + 0.771236i 1.67674 + 2.90420i 1.28714 −2.12291 3.67698i 0
126.5 0.593667 1.02826i 0.172555 0.298874i 0.295120 + 0.511162i 0 −0.204880 0.354863i 1.01478 + 1.75765i 3.07548 1.44045 + 2.49493i 0
126.6 1.27287 2.20467i 1.07646 1.86449i −2.24039 3.88048i 0 −2.74039 4.74650i 1.46928 + 2.54486i −6.31544 −0.817544 1.41603i 0
276.1 −1.27287 2.20467i −1.07646 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i −1.46928 + 2.54486i 6.31544 −0.817544 + 1.41603i 0
276.2 −0.593667 1.02826i −0.172555 0.298874i 0.295120 0.511162i 0 −0.204880 + 0.354863i −1.01478 + 1.75765i −3.07548 1.44045 2.49493i 0
276.3 −0.165418 0.286513i 1.34590 + 2.33117i 0.945274 1.63726i 0 0.445274 0.771236i −1.67674 + 2.90420i −1.28714 −2.12291 + 3.67698i 0
276.4 0.165418 + 0.286513i −1.34590 2.33117i 0.945274 1.63726i 0 0.445274 0.771236i 1.67674 2.90420i 1.28714 −2.12291 + 3.67698i 0
276.5 0.593667 + 1.02826i 0.172555 + 0.298874i 0.295120 0.511162i 0 −0.204880 + 0.354863i 1.01478 1.75765i 3.07548 1.44045 2.49493i 0
276.6 1.27287 + 2.20467i 1.07646 + 1.86449i −2.24039 + 3.88048i 0 −2.74039 + 4.74650i 1.46928 2.54486i −6.31544 −0.817544 + 1.41603i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 276.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.e 12
5.b even 2 1 inner 325.2.e.e 12
5.c odd 4 2 65.2.n.a 12
13.c even 3 1 inner 325.2.e.e 12
13.c even 3 1 4225.2.a.br 6
13.e even 6 1 4225.2.a.bq 6
15.e even 4 2 585.2.bs.a 12
20.e even 4 2 1040.2.dh.a 12
65.f even 4 2 845.2.l.f 24
65.h odd 4 2 845.2.n.e 12
65.k even 4 2 845.2.l.f 24
65.l even 6 1 4225.2.a.bq 6
65.n even 6 1 inner 325.2.e.e 12
65.n even 6 1 4225.2.a.br 6
65.o even 12 2 845.2.d.d 12
65.o even 12 2 845.2.l.f 24
65.q odd 12 2 65.2.n.a 12
65.q odd 12 2 845.2.b.d 6
65.r odd 12 2 845.2.b.e 6
65.r odd 12 2 845.2.n.e 12
65.t even 12 2 845.2.d.d 12
65.t even 12 2 845.2.l.f 24
195.bl even 12 2 585.2.bs.a 12
260.bj even 12 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 5.c odd 4 2
65.2.n.a 12 65.q odd 12 2
325.2.e.e 12 1.a even 1 1 trivial
325.2.e.e 12 5.b even 2 1 inner
325.2.e.e 12 13.c even 3 1 inner
325.2.e.e 12 65.n even 6 1 inner
585.2.bs.a 12 15.e even 4 2
585.2.bs.a 12 195.bl even 12 2
845.2.b.d 6 65.q odd 12 2
845.2.b.e 6 65.r odd 12 2
845.2.d.d 12 65.o even 12 2
845.2.d.d 12 65.t even 12 2
845.2.l.f 24 65.f even 4 2
845.2.l.f 24 65.k even 4 2
845.2.l.f 24 65.o even 12 2
845.2.l.f 24 65.t even 12 2
845.2.n.e 12 65.h odd 4 2
845.2.n.e 12 65.r odd 12 2
1040.2.dh.a 12 20.e even 4 2
1040.2.dh.a 12 260.bj even 12 2
4225.2.a.bq 6 13.e even 6 1
4225.2.a.bq 6 65.l even 6 1
4225.2.a.br 6 13.c even 3 1
4225.2.a.br 6 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8 T_{2}^{10} + 54 T_{2}^{8} + 78 T_{2}^{6} + 92 T_{2}^{4} + 10 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T^{2} + 92 T^{4} + 78 T^{6} + 54 T^{8} + 8 T^{10} + T^{12} \)
$3$ \( 16 + 140 T^{2} + 1177 T^{4} + 412 T^{6} + 109 T^{8} + 12 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( 160000 + 71600 T^{2} + 22441 T^{4} + 3496 T^{6} + 397 T^{8} + 24 T^{10} + T^{12} \)
$11$ \( ( 64 - 104 T + 169 T^{2} - 16 T^{3} + 13 T^{4} + T^{6} )^{2} \)
$13$ \( 4826809 + 428415 T^{2} + 6591 T^{4} + 322 T^{6} + 39 T^{8} + 15 T^{10} + T^{12} \)
$17$ \( 28561 + 27547 T^{2} + 20654 T^{4} + 5367 T^{6} + 1062 T^{8} + 35 T^{10} + T^{12} \)
$19$ \( ( 100 + 10 T + 61 T^{2} + 14 T^{3} + 37 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$23$ \( 16 + 140 T^{2} + 1177 T^{4} + 412 T^{6} + 109 T^{8} + 12 T^{10} + T^{12} \)
$29$ \( ( 9 + 3 T + T^{2} )^{6} \)
$31$ \( ( 40 - 40 T + 4 T^{2} + T^{3} )^{4} \)
$37$ \( 28561 + 27547 T^{2} + 20654 T^{4} + 5367 T^{6} + 1062 T^{8} + 35 T^{10} + T^{12} \)
$41$ \( ( 25 + 145 T + 806 T^{2} + 213 T^{3} + 78 T^{4} - 7 T^{5} + T^{6} )^{2} \)
$43$ \( 65536 + 72448 T^{2} + 59609 T^{4} + 22128 T^{6} + 6117 T^{8} + 80 T^{10} + T^{12} \)
$47$ \( ( -270400 + 14640 T^{2} - 236 T^{4} + T^{6} )^{2} \)
$53$ \( ( -400 + 1040 T^{2} - 171 T^{4} + T^{6} )^{2} \)
$59$ \( ( 18496 - 7480 T + 3297 T^{2} - 162 T^{3} + 59 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$61$ \( ( 13225 - 5635 T + 2746 T^{2} - 83 T^{3} + 58 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$67$ \( 406586896 + 52486892 T^{2} + 4759209 T^{4} + 219972 T^{6} + 7397 T^{8} + 100 T^{10} + T^{12} \)
$71$ \( ( 676 + 26 T + 157 T^{2} + 46 T^{3} + 37 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$73$ \( ( -250000 + 13900 T^{2} - 215 T^{4} + T^{6} )^{2} \)
$79$ \( ( -160 + 180 T - 26 T^{2} + T^{3} )^{4} \)
$83$ \( ( -640000 + 23600 T^{2} - 276 T^{4} + T^{6} )^{2} \)
$89$ \( ( 2515396 + 249002 T + 40509 T^{2} + 1602 T^{3} + 257 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$97$ \( 41740124416 + 2934418352 T^{2} + 149090649 T^{4} + 3613032 T^{6} + 64037 T^{8} + 280 T^{10} + T^{12} \)
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