Properties

Label 6042.2.a.y
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 14 x^{5} + 29 x^{4} + 48 x^{3} - 14 x^{2} - 35 x - 10\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 14 x^{5} + 29 x^{4} + 48 x^{3} - 14 x^{2} - 35 x - 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 16 \nu^{6} - 51 \nu^{5} - 216 \nu^{4} + 517 \nu^{3} + 682 \nu^{2} - 480 \nu - 445 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( 24 \nu^{6} - 89 \nu^{5} - 274 \nu^{4} + 888 \nu^{3} + 548 \nu^{2} - 695 \nu - 405 \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{6} + 108 \nu^{5} + 303 \nu^{4} - 1086 \nu^{3} - 456 \nu^{2} + 915 \nu + 335 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{6} + 34 \nu^{5} + 99 \nu^{4} - 338 \nu^{3} - 163 \nu^{2} + 260 \nu + 115 \)\()/5\)
\(\beta_{6}\)\(=\)\((\)\( -93 \nu^{6} + 348 \nu^{5} + 1043 \nu^{4} - 3466 \nu^{3} - 1886 \nu^{2} + 2665 \nu + 1285 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 11 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(16 \beta_{6} - 20 \beta_{5} - \beta_{4} + 22 \beta_{3} + 2 \beta_{2} + 21 \beta_{1} + 43\)
\(\nu^{5}\)\(=\)\(44 \beta_{6} - 60 \beta_{5} - 22 \beta_{4} + 19 \beta_{3} + 20 \beta_{2} + 147 \beta_{1} + 77\)
\(\nu^{6}\)\(=\)\(249 \beta_{6} - 354 \beta_{5} - 19 \beta_{4} + 240 \beta_{3} + 60 \beta_{2} + 384 \beta_{1} + 554\)

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
4.05485
3.00316
0.982384
−0.456607
−0.724990
−0.835135
−3.02367
−1.00000 1.00000 1.00000 −3.05485 −1.00000 −2.36098 −1.00000 1.00000 3.05485
1.2 −1.00000 1.00000 1.00000 −2.00316 −1.00000 −4.20447 −1.00000 1.00000 2.00316
1.3 −1.00000 1.00000 1.00000 0.0176159 −1.00000 0.282315 −1.00000 1.00000 −0.0176159
1.4 −1.00000 1.00000 1.00000 1.45661 −1.00000 1.71471 −1.00000 1.00000 −1.45661
1.5 −1.00000 1.00000 1.00000 1.72499 −1.00000 −0.765322 −1.00000 1.00000 −1.72499
1.6 −1.00000 1.00000 1.00000 1.83513 −1.00000 0.943730 −1.00000 1.00000 −1.83513
1.7 −1.00000 1.00000 1.00000 4.02367 −1.00000 −4.60998 −1.00000 1.00000 −4.02367
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

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 6042.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.y 7 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)
\(53\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{7} - 4 T_{5}^{6} - 11 T_{5}^{5} + 51 T_{5}^{4} - T_{5}^{3} - 140 T_{5}^{2} + 116 T_{5} - 2 \)
\( T_{7}^{7} + 9 T_{7}^{6} + 16 T_{7}^{5} - 39 T_{7}^{4} - 80 T_{7}^{3} + 56 T_{7}^{2} + 48 T_{7} - 16 \)
\( T_{11}^{7} + 2 T_{11}^{6} - 29 T_{11}^{5} - 11 T_{11}^{4} + 193 T_{11}^{3} - 14 T_{11}^{2} - 330 T_{11} + 190 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ \( ( 1 - T )^{7} \)
$5$ \( 1 - 4 T + 24 T^{2} - 69 T^{3} + 249 T^{4} - 620 T^{5} + 1726 T^{6} - 3752 T^{7} + 8630 T^{8} - 15500 T^{9} + 31125 T^{10} - 43125 T^{11} + 75000 T^{12} - 62500 T^{13} + 78125 T^{14} \)
$7$ \( 1 + 9 T + 65 T^{2} + 339 T^{3} + 1509 T^{4} + 5579 T^{5} + 18213 T^{6} + 51042 T^{7} + 127491 T^{8} + 273371 T^{9} + 517587 T^{10} + 813939 T^{11} + 1092455 T^{12} + 1058841 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 2 T + 48 T^{2} + 121 T^{3} + 1139 T^{4} + 3132 T^{5} + 17534 T^{6} + 45136 T^{7} + 192874 T^{8} + 378972 T^{9} + 1516009 T^{10} + 1771561 T^{11} + 7730448 T^{12} + 3543122 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 7 T + 87 T^{2} + 473 T^{3} + 3267 T^{4} + 14089 T^{5} + 69241 T^{6} + 237190 T^{7} + 900133 T^{8} + 2381041 T^{9} + 7177599 T^{10} + 13509353 T^{11} + 32302491 T^{12} + 33787663 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 10 T + 112 T^{2} + 789 T^{3} + 5037 T^{4} + 27556 T^{5} + 129618 T^{6} + 579162 T^{7} + 2203506 T^{8} + 7963684 T^{9} + 24746781 T^{10} + 65898069 T^{11} + 159023984 T^{12} + 241375690 T^{13} + 410338673 T^{14} \)
$19$ \( ( 1 + T )^{7} \)
$23$ \( 1 - 2 T + 78 T^{2} - 159 T^{3} + 3101 T^{4} - 5964 T^{5} + 89128 T^{6} - 153570 T^{7} + 2049944 T^{8} - 3154956 T^{9} + 37729867 T^{10} - 44494719 T^{11} + 502034754 T^{12} - 296071778 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 6 T + 110 T^{2} + 445 T^{3} + 6224 T^{4} + 20646 T^{5} + 238505 T^{6} + 650414 T^{7} + 6916645 T^{8} + 17363286 T^{9} + 151797136 T^{10} + 314740045 T^{11} + 2256226390 T^{12} + 3568939926 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 - 3 T + 159 T^{2} - 426 T^{3} + 11971 T^{4} - 27517 T^{5} + 555845 T^{6} - 1066828 T^{7} + 17231195 T^{8} - 26443837 T^{9} + 356628061 T^{10} - 393419946 T^{11} + 4552035009 T^{12} - 2662511043 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 3 T + 181 T^{2} + 587 T^{3} + 16105 T^{4} + 49785 T^{5} + 891645 T^{6} + 2387338 T^{7} + 32990865 T^{8} + 68155665 T^{9} + 815766565 T^{10} + 1100132507 T^{11} + 12551256217 T^{12} + 7697179227 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 12 T + 230 T^{2} + 1965 T^{3} + 23416 T^{4} + 160784 T^{5} + 1442345 T^{6} + 8098942 T^{7} + 59136145 T^{8} + 270277904 T^{9} + 1613854136 T^{10} + 5552620365 T^{11} + 26646926230 T^{12} + 57001250892 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 26 T + 468 T^{2} + 6125 T^{3} + 67933 T^{4} + 626432 T^{5} + 5065442 T^{6} + 35248362 T^{7} + 217814006 T^{8} + 1158272768 T^{9} + 5401149031 T^{10} + 20940156125 T^{11} + 68799951324 T^{12} + 164355439274 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 - 17 T + 256 T^{2} - 2681 T^{3} + 25732 T^{4} - 211135 T^{5} + 1637203 T^{6} - 11581674 T^{7} + 76948541 T^{8} - 466397215 T^{9} + 2671573436 T^{10} - 13082424761 T^{11} + 58712321792 T^{12} - 183246660593 T^{13} + 506623120463 T^{14} \)
$53$ \( ( 1 - T )^{7} \)
$59$ \( 1 - 7 T + 218 T^{2} - 1502 T^{3} + 28455 T^{4} - 160717 T^{5} + 2358570 T^{6} - 11767452 T^{7} + 139155630 T^{8} - 559455877 T^{9} + 5844059445 T^{10} - 18200276222 T^{11} + 155853497182 T^{12} - 295263735487 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 18 T + 232 T^{2} + 3029 T^{3} + 30886 T^{4} + 293890 T^{5} + 2645765 T^{6} + 21026718 T^{7} + 161391665 T^{8} + 1093564690 T^{9} + 7010535166 T^{10} + 41939052389 T^{11} + 195946341832 T^{12} + 927366738498 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 3 T + 119 T^{2} + 463 T^{3} + 11831 T^{4} + 35537 T^{5} + 983841 T^{6} + 3657594 T^{7} + 65917347 T^{8} + 159525593 T^{9} + 3558327053 T^{10} + 9329969023 T^{11} + 160664887733 T^{12} + 271375146507 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 2 T + 281 T^{2} - 776 T^{3} + 41133 T^{4} - 115198 T^{5} + 4161861 T^{6} - 9961008 T^{7} + 295492131 T^{8} - 580713118 T^{9} + 14721953163 T^{10} - 19719464456 T^{11} + 506988447631 T^{12} - 256200567842 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - T + 381 T^{2} + 49 T^{3} + 66847 T^{4} + 52637 T^{5} + 7206899 T^{6} + 6385238 T^{7} + 526103627 T^{8} + 280502573 T^{9} + 26004619399 T^{10} + 1391513809 T^{11} + 789840276933 T^{12} - 151334226289 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 18 T + 296 T^{2} + 3691 T^{3} + 42697 T^{4} + 438548 T^{5} + 4151902 T^{6} + 36524826 T^{7} + 328000258 T^{8} + 2736978068 T^{9} + 21051286183 T^{10} + 143764748971 T^{11} + 910808694104 T^{12} + 4375574199378 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 + 17 T + 483 T^{2} + 5993 T^{3} + 102363 T^{4} + 1005755 T^{5} + 12880873 T^{6} + 103358102 T^{7} + 1069112459 T^{8} + 6928646195 T^{9} + 58529832681 T^{10} + 284417717753 T^{11} + 1902556630569 T^{12} + 5557986347273 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 + 13 T + 378 T^{2} + 5366 T^{3} + 80635 T^{4} + 948785 T^{5} + 11394250 T^{6} + 102159976 T^{7} + 1014088250 T^{8} + 7515325985 T^{9} + 56845175315 T^{10} + 336674865206 T^{11} + 2110774471722 T^{12} + 6460756782493 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 20 T + 522 T^{2} + 6781 T^{3} + 114717 T^{4} + 1220570 T^{5} + 16554064 T^{6} + 147166538 T^{7} + 1605744208 T^{8} + 11484343130 T^{9} + 104699108541 T^{10} + 600317054461 T^{11} + 4482591614154 T^{12} + 16659440098580 T^{13} + 80798284478113 T^{14} \)
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