Properties

Label 1334.2.a.j
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 16 x^{7} + 44 x^{6} + 87 x^{5} - 209 x^{4} - 160 x^{3} + 348 x^{2} + 12 x - 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 16 x^{7} + 44 x^{6} + 87 x^{5} - 209 x^{4} - 160 x^{3} + 348 x^{2} + 12 x - 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 13 \nu^{4} - 7 \nu^{3} - 44 \nu^{2} + 8 \nu + 22 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 13 \nu^{4} + 9 \nu^{3} + 42 \nu^{2} - 22 \nu - 18 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{8} - 3 \nu^{7} - 12 \nu^{6} + 34 \nu^{5} + 43 \nu^{4} - 105 \nu^{3} - 48 \nu^{2} + 62 \nu + 28 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{8} + 5 \nu^{7} + 6 \nu^{6} - 54 \nu^{5} + 21 \nu^{4} + 143 \nu^{3} - 134 \nu^{2} - 18 \nu + 44 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} + 9 \nu^{5} - 43 \nu^{4} - 14 \nu^{3} + 119 \nu^{2} - 26 \nu - 44 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{8} + 5 \nu^{7} + 6 \nu^{6} - 56 \nu^{5} + 23 \nu^{4} + 165 \nu^{3} - 144 \nu^{2} - 70 \nu + 52 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 14 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(-2 \beta_{8} + \beta_{7} + 3 \beta_{6} + \beta_{5} + 12 \beta_{4} + 13 \beta_{3} + 15 \beta_{2} + 71 \beta_{1} + 33\)
\(\nu^{6}\)\(=\)\(-2 \beta_{8} + 14 \beta_{7} + 16 \beta_{6} + 14 \beta_{5} + 18 \beta_{4} + 30 \beta_{3} + 81 \beta_{2} + 161 \beta_{1} + 216\)
\(\nu^{7}\)\(=\)\(-26 \beta_{8} + 20 \beta_{7} + 48 \beta_{6} + 22 \beta_{5} + 123 \beta_{4} + 137 \beta_{3} + 177 \beta_{2} + 662 \beta_{1} + 404\)
\(\nu^{8}\)\(=\)\(-34 \beta_{8} + 151 \beta_{7} + 191 \beta_{6} + 161 \beta_{5} + 239 \beta_{4} + 348 \beta_{3} + 759 \beta_{2} + 1728 \beta_{1} + 1895\)

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
3.18584
3.11529
2.04237
1.07506
0.783963
−0.540845
−2.05766
−2.10346
−2.50055
−1.00000 −3.18584 1.00000 −0.979916 3.18584 4.69416 −1.00000 7.14956 0.979916
1.2 −1.00000 −3.11529 1.00000 2.08342 3.11529 −4.29295 −1.00000 6.70504 −2.08342
1.3 −1.00000 −2.04237 1.00000 4.28519 2.04237 0.891555 −1.00000 1.17127 −4.28519
1.4 −1.00000 −1.07506 1.00000 −2.43466 1.07506 1.40780 −1.00000 −1.84425 2.43466
1.5 −1.00000 −0.783963 1.00000 −1.63370 0.783963 −4.03607 −1.00000 −2.38540 1.63370
1.6 −1.00000 0.540845 1.00000 1.92844 −0.540845 2.85963 −1.00000 −2.70749 −1.92844
1.7 −1.00000 2.05766 1.00000 3.66144 −2.05766 4.21443 −1.00000 1.23398 −3.66144
1.8 −1.00000 2.10346 1.00000 −3.58097 −2.10346 2.82290 −1.00000 1.42454 3.58097
1.9 −1.00000 2.50055 1.00000 1.67075 −2.50055 −2.56147 −1.00000 3.25275 −1.67075
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

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 1334.2.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.j 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\(T_{3}^{9} + \cdots\)
\(T_{5}^{9} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( 100 + 12 T - 348 T^{2} - 160 T^{3} + 209 T^{4} + 87 T^{5} - 44 T^{6} - 16 T^{7} + 3 T^{8} + T^{9} \)
$5$ \( -1470 - 322 T + 1912 T^{2} - 10 T^{3} - 779 T^{4} + 95 T^{5} + 114 T^{6} - 20 T^{7} - 5 T^{8} + T^{9} \)
$7$ \( 8896 - 18816 T + 9312 T^{2} + 3048 T^{3} - 2954 T^{4} + 140 T^{5} + 242 T^{6} - 32 T^{7} - 6 T^{8} + T^{9} \)
$11$ \( 240 - 1712 T + 4312 T^{2} - 4256 T^{3} + 663 T^{4} + 995 T^{5} - 138 T^{6} - 62 T^{7} + 3 T^{8} + T^{9} \)
$13$ \( 243056 - 199616 T - 9360 T^{2} + 45764 T^{3} - 9117 T^{4} - 1999 T^{5} + 702 T^{6} - 8 T^{7} - 13 T^{8} + T^{9} \)
$17$ \( 468 + 1828 T + 28 T^{2} - 2252 T^{3} - 370 T^{4} + 712 T^{5} + 98 T^{6} - 56 T^{7} - 2 T^{8} + T^{9} \)
$19$ \( -61376 - 68800 T + 24992 T^{2} + 20736 T^{3} - 5004 T^{4} - 1884 T^{5} + 480 T^{6} + 38 T^{7} - 16 T^{8} + T^{9} \)
$23$ \( ( 1 + T )^{9} \)
$29$ \( ( -1 + T )^{9} \)
$31$ \( 33088 - 13696 T - 24528 T^{2} + 17600 T^{3} - 879 T^{4} - 1885 T^{5} + 394 T^{6} + 34 T^{7} - 15 T^{8} + T^{9} \)
$37$ \( -40768 - 92736 T + 6592 T^{2} + 58920 T^{3} - 21122 T^{4} - 784 T^{5} + 1050 T^{6} - 62 T^{7} - 12 T^{8} + T^{9} \)
$41$ \( -34848 + 65888 T - 16096 T^{2} - 24264 T^{3} + 7270 T^{4} + 3248 T^{5} - 490 T^{6} - 110 T^{7} + 6 T^{8} + T^{9} \)
$43$ \( 132080 + 220464 T - 14472 T^{2} - 55512 T^{3} + 2499 T^{4} + 4183 T^{5} - 166 T^{6} - 118 T^{7} + 3 T^{8} + T^{9} \)
$47$ \( 273312 + 1468448 T + 1343984 T^{2} + 395660 T^{3} + 325 T^{4} - 18875 T^{5} - 2836 T^{6} - 32 T^{7} + 19 T^{8} + T^{9} \)
$53$ \( 3058362 + 1268518 T - 499608 T^{2} - 251838 T^{3} + 12445 T^{4} + 13247 T^{5} + 354 T^{6} - 208 T^{7} - 5 T^{8} + T^{9} \)
$59$ \( 422400 - 942592 T + 484352 T^{2} + 44800 T^{3} - 50680 T^{4} + 1576 T^{5} + 1472 T^{6} - 100 T^{7} - 12 T^{8} + T^{9} \)
$61$ \( -201344 + 620032 T + 853184 T^{2} - 215712 T^{3} - 225730 T^{4} + 21396 T^{5} + 3246 T^{6} - 282 T^{7} - 12 T^{8} + T^{9} \)
$67$ \( -675484 - 2120940 T - 865112 T^{2} + 173328 T^{3} + 117326 T^{4} + 9844 T^{5} - 1668 T^{6} - 210 T^{7} + 6 T^{8} + T^{9} \)
$71$ \( 26575104 - 24676480 T + 5451776 T^{2} + 634752 T^{3} - 293488 T^{4} + 10228 T^{5} + 3592 T^{6} - 244 T^{7} - 12 T^{8} + T^{9} \)
$73$ \( -1965088 + 9438208 T - 2798384 T^{2} - 980048 T^{3} + 139054 T^{4} + 35756 T^{5} - 956 T^{6} - 354 T^{7} + T^{9} \)
$79$ \( 16643750 - 50836250 T + 33293000 T^{2} + 2023150 T^{3} - 928905 T^{4} - 11567 T^{5} + 8868 T^{6} - 134 T^{7} - 29 T^{8} + T^{9} \)
$83$ \( 672218736 - 215070272 T - 14783832 T^{2} + 12177472 T^{3} - 1220396 T^{4} - 40310 T^{5} + 11132 T^{6} - 272 T^{7} - 24 T^{8} + T^{9} \)
$89$ \( -111132 - 324772 T - 230104 T^{2} - 20652 T^{3} + 24446 T^{4} + 5052 T^{5} - 584 T^{6} - 156 T^{7} + 2 T^{8} + T^{9} \)
$97$ \( -790352 + 1006304 T + 410936 T^{2} - 298016 T^{3} - 69252 T^{4} + 17066 T^{5} + 2260 T^{6} - 272 T^{7} - 12 T^{8} + T^{9} \)
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