Properties

Label 546.8.a.p
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 302081 x^{4} - 2628147 x^{3} + 19116974952 x^{2} - 78725393748 x - 5138711063280\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -7 - \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} +O(q^{10})\) \( q -8 q^{2} + 27 q^{3} + 64 q^{4} + ( -7 - \beta_{1} ) q^{5} -216 q^{6} + 343 q^{7} -512 q^{8} + 729 q^{9} + ( 56 + 8 \beta_{1} ) q^{10} + ( 1228 + 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{11} + 1728 q^{12} -2197 q^{13} -2744 q^{14} + ( -189 - 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 1325 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - 8 \beta_{5} ) q^{17} -5832 q^{18} + ( -9516 - 43 \beta_{1} + 6 \beta_{2} - 9 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} ) q^{19} + ( -448 - 64 \beta_{1} ) q^{20} + 9261 q^{21} + ( -9824 - 16 \beta_{1} + 16 \beta_{3} + 8 \beta_{5} ) q^{22} + ( 5278 + 115 \beta_{1} - 2 \beta_{2} - 14 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{23} -13824 q^{24} + ( 22611 + 5 \beta_{1} - 26 \beta_{2} - 9 \beta_{3} + 25 \beta_{4} - 3 \beta_{5} ) q^{25} + 17576 q^{26} + 19683 q^{27} + 21952 q^{28} + ( -6098 + 111 \beta_{1} + 15 \beta_{2} - 6 \beta_{3} + 11 \beta_{4} + 23 \beta_{5} ) q^{29} + ( 1512 + 216 \beta_{1} ) q^{30} + ( 35847 + 111 \beta_{1} + 27 \beta_{2} - 13 \beta_{3} - 62 \beta_{4} + 19 \beta_{5} ) q^{31} -32768 q^{32} + ( 33156 + 54 \beta_{1} - 54 \beta_{3} - 27 \beta_{5} ) q^{33} + ( -10600 + 16 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} - 8 \beta_{4} + 64 \beta_{5} ) q^{34} + ( -2401 - 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( 22189 - 78 \beta_{1} + 98 \beta_{2} - 37 \beta_{3} + 9 \beta_{4} - 90 \beta_{5} ) q^{37} + ( 76128 + 344 \beta_{1} - 48 \beta_{2} + 72 \beta_{3} + 24 \beta_{4} + 88 \beta_{5} ) q^{38} -59319 q^{39} + ( 3584 + 512 \beta_{1} ) q^{40} + ( 86159 - 468 \beta_{1} - 110 \beta_{2} + 111 \beta_{3} - 17 \beta_{4} + 91 \beta_{5} ) q^{41} -74088 q^{42} + ( -540 - 119 \beta_{1} - 119 \beta_{2} + 24 \beta_{3} + 137 \beta_{4} - 111 \beta_{5} ) q^{43} + ( 78592 + 128 \beta_{1} - 128 \beta_{3} - 64 \beta_{5} ) q^{44} + ( -5103 - 729 \beta_{1} ) q^{45} + ( -42224 - 920 \beta_{1} + 16 \beta_{2} + 112 \beta_{3} + 56 \beta_{4} + 56 \beta_{5} ) q^{46} + ( 244136 - 939 \beta_{1} - 231 \beta_{2} - 35 \beta_{3} + 35 \beta_{4} + 38 \beta_{5} ) q^{47} + 110592 q^{48} + 117649 q^{49} + ( -180888 - 40 \beta_{1} + 208 \beta_{2} + 72 \beta_{3} - 200 \beta_{4} + 24 \beta_{5} ) q^{50} + ( 35775 - 54 \beta_{1} + 108 \beta_{2} + 54 \beta_{3} + 27 \beta_{4} - 216 \beta_{5} ) q^{51} -140608 q^{52} + ( -223711 - 2365 \beta_{1} + 67 \beta_{2} + 205 \beta_{3} + 30 \beta_{4} + 103 \beta_{5} ) q^{53} -157464 q^{54} + ( -260873 - 2562 \beta_{1} + 388 \beta_{2} + 3 \beta_{3} - 131 \beta_{4} + 20 \beta_{5} ) q^{55} -175616 q^{56} + ( -256932 - 1161 \beta_{1} + 162 \beta_{2} - 243 \beta_{3} - 81 \beta_{4} - 297 \beta_{5} ) q^{57} + ( 48784 - 888 \beta_{1} - 120 \beta_{2} + 48 \beta_{3} - 88 \beta_{4} - 184 \beta_{5} ) q^{58} + ( 302065 - 1716 \beta_{1} + 402 \beta_{2} + 194 \beta_{3} - 133 \beta_{4} + 295 \beta_{5} ) q^{59} + ( -12096 - 1728 \beta_{1} ) q^{60} + ( 675425 - 3962 \beta_{1} + 115 \beta_{2} - 196 \beta_{3} - 599 \beta_{4} + 154 \beta_{5} ) q^{61} + ( -286776 - 888 \beta_{1} - 216 \beta_{2} + 104 \beta_{3} + 496 \beta_{4} - 152 \beta_{5} ) q^{62} + 250047 q^{63} + 262144 q^{64} + ( 15379 + 2197 \beta_{1} ) q^{65} + ( -265248 - 432 \beta_{1} + 432 \beta_{3} + 216 \beta_{5} ) q^{66} + ( 399434 - 4574 \beta_{1} - 628 \beta_{2} - 322 \beta_{3} + 792 \beta_{4} + 208 \beta_{5} ) q^{67} + ( 84800 - 128 \beta_{1} + 256 \beta_{2} + 128 \beta_{3} + 64 \beta_{4} - 512 \beta_{5} ) q^{68} + ( 142506 + 3105 \beta_{1} - 54 \beta_{2} - 378 \beta_{3} - 189 \beta_{4} - 189 \beta_{5} ) q^{69} + ( 19208 + 2744 \beta_{1} ) q^{70} + ( 1722623 + 4984 \beta_{1} + 282 \beta_{2} + 357 \beta_{3} + 181 \beta_{4} + 787 \beta_{5} ) q^{71} -373248 q^{72} + ( 864632 - 6657 \beta_{1} - 815 \beta_{2} + 708 \beta_{3} - 567 \beta_{4} + 411 \beta_{5} ) q^{73} + ( -177512 + 624 \beta_{1} - 784 \beta_{2} + 296 \beta_{3} - 72 \beta_{4} + 720 \beta_{5} ) q^{74} + ( 610497 + 135 \beta_{1} - 702 \beta_{2} - 243 \beta_{3} + 675 \beta_{4} - 81 \beta_{5} ) q^{75} + ( -609024 - 2752 \beta_{1} + 384 \beta_{2} - 576 \beta_{3} - 192 \beta_{4} - 704 \beta_{5} ) q^{76} + ( 421204 + 686 \beta_{1} - 686 \beta_{3} - 343 \beta_{5} ) q^{77} + 474552 q^{78} + ( 771433 - 5799 \beta_{1} + 849 \beta_{2} + 81 \beta_{3} + 1090 \beta_{4} + 503 \beta_{5} ) q^{79} + ( -28672 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( -689272 + 3744 \beta_{1} + 880 \beta_{2} - 888 \beta_{3} + 136 \beta_{4} - 728 \beta_{5} ) q^{82} + ( 1982060 + 253 \beta_{1} - 1071 \beta_{2} + 1245 \beta_{3} + 43 \beta_{4} - 840 \beta_{5} ) q^{83} + 592704 q^{84} + ( 124289 + 3832 \beta_{1} - 3434 \beta_{2} + 583 \beta_{3} + 401 \beta_{4} + 252 \beta_{5} ) q^{85} + ( 4320 + 952 \beta_{1} + 952 \beta_{2} - 192 \beta_{3} - 1096 \beta_{4} + 888 \beta_{5} ) q^{86} + ( -164646 + 2997 \beta_{1} + 405 \beta_{2} - 162 \beta_{3} + 297 \beta_{4} + 621 \beta_{5} ) q^{87} + ( -628736 - 1024 \beta_{1} + 1024 \beta_{3} + 512 \beta_{5} ) q^{88} + ( 1630452 + 1243 \beta_{1} + 2265 \beta_{2} - 2356 \beta_{3} - 1121 \beta_{4} - 514 \beta_{5} ) q^{89} + ( 40824 + 5832 \beta_{1} ) q^{90} -753571 q^{91} + ( 337792 + 7360 \beta_{1} - 128 \beta_{2} - 896 \beta_{3} - 448 \beta_{4} - 448 \beta_{5} ) q^{92} + ( 967869 + 2997 \beta_{1} + 729 \beta_{2} - 351 \beta_{3} - 1674 \beta_{4} + 513 \beta_{5} ) q^{93} + ( -1953088 + 7512 \beta_{1} + 1848 \beta_{2} + 280 \beta_{3} - 280 \beta_{4} - 304 \beta_{5} ) q^{94} + ( 4371228 + 15525 \beta_{1} - 1798 \beta_{2} - 567 \beta_{3} + 685 \beta_{4} + 451 \beta_{5} ) q^{95} -884736 q^{96} + ( 868707 - 9905 \beta_{1} + 2907 \beta_{2} - 2345 \beta_{3} + 100 \beta_{4} - 947 \beta_{5} ) q^{97} -941192 q^{98} + ( 895212 + 1458 \beta_{1} - 1458 \beta_{3} - 729 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} + 162 q^{3} + 384 q^{4} - 43 q^{5} - 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q - 48 q^{2} + 162 q^{3} + 384 q^{4} - 43 q^{5} - 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 344 q^{10} + 7370 q^{11} + 10368 q^{12} - 13182 q^{13} - 16464 q^{14} - 1161 q^{15} + 24576 q^{16} + 7950 q^{17} - 34992 q^{18} - 57145 q^{19} - 2752 q^{20} + 55566 q^{21} - 58960 q^{22} + 31769 q^{23} - 82944 q^{24} + 135721 q^{25} + 105456 q^{26} + 118098 q^{27} + 131712 q^{28} - 36455 q^{29} + 9288 q^{30} + 215069 q^{31} - 196608 q^{32} + 198990 q^{33} - 63600 q^{34} - 14749 q^{35} + 279936 q^{36} + 133074 q^{37} + 457160 q^{38} - 355914 q^{39} + 22016 q^{40} + 516452 q^{41} - 444528 q^{42} - 3085 q^{43} + 471680 q^{44} - 31347 q^{45} - 254152 q^{46} + 1463947 q^{47} + 663552 q^{48} + 705894 q^{49} - 1085768 q^{50} + 214650 q^{51} - 843648 q^{52} - 1344571 q^{53} - 944784 q^{54} - 1568062 q^{55} - 1053696 q^{56} - 1542915 q^{57} + 291640 q^{58} + 1810408 q^{59} - 74304 q^{60} + 4047390 q^{61} - 1720552 q^{62} + 1500282 q^{63} + 1572864 q^{64} + 94471 q^{65} - 1591920 q^{66} + 2393614 q^{67} + 508800 q^{68} + 857763 q^{69} + 117992 q^{70} + 10341084 q^{71} - 2239488 q^{72} + 5180001 q^{73} - 1064592 q^{74} + 3664467 q^{75} - 3657280 q^{76} + 2527910 q^{77} + 2847312 q^{78} + 4624979 q^{79} - 176128 q^{80} + 3188646 q^{81} - 4131616 q^{82} + 11892699 q^{83} + 3556224 q^{84} + 750368 q^{85} + 24680 q^{86} - 984285 q^{87} - 3773440 q^{88} + 9781713 q^{89} + 250776 q^{90} - 4521426 q^{91} + 2033216 q^{92} + 5806863 q^{93} - 11711576 q^{94} + 26244263 q^{95} - 5308416 q^{96} + 5202537 q^{97} - 5647152 q^{98} + 5372730 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 302081 x^{4} - 2628147 x^{3} + 19116974952 x^{2} - 78725393748 x - 5138711063280\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(1010393471 \nu^{5} - 202190516283 \nu^{4} - 312298922699325 \nu^{3} + 33810283300944563 \nu^{2} + 21357081927796074006 \nu - 172283862240231957840\)\()/ 477302882530737900 \)
\(\beta_{3}\)\(=\)\((\)\(153828543 \nu^{5} + 7918632266 \nu^{4} - 50688900701525 \nu^{3} - 2357914938206446 \nu^{2} + 3634645866270427938 \nu + 50013877993819754280\)\()/ 34093063037909850 \)
\(\beta_{4}\)\(=\)\((\)\(1227084392 \nu^{5} - 92166649386 \nu^{4} - 381326633469375 \nu^{3} + 21649201441137326 \nu^{2} + 25720346060087511927 \nu - 962522047703028032730\)\()/ 238651441265368950 \)
\(\beta_{5}\)\(=\)\((\)\(2616932156 \nu^{5} - 58187785243 \nu^{4} - 759959699149025 \nu^{3} + 3864517652596993 \nu^{2} + 44745344631471185151 \nu - 315045876792375472440\)\()/ 238651441265368950 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-3 \beta_{5} + 25 \beta_{4} - 9 \beta_{3} - 26 \beta_{2} - 9 \beta_{1} + 100687\)
\(\nu^{3}\)\(=\)\(2980 \beta_{5} - 1894 \beta_{4} - 4048 \beta_{3} - 2208 \beta_{2} + 175749 \beta_{1} + 1436454\)
\(\nu^{4}\)\(=\)\(-339815 \beta_{5} + 4694485 \beta_{4} - 1193155 \beta_{3} - 7099150 \beta_{2} + 2626407 \beta_{1} + 17672944475\)
\(\nu^{5}\)\(=\)\(953464464 \beta_{5} - 482555470 \beta_{4} - 1188782538 \beta_{3} - 760660672 \beta_{2} + 34010976477 \beta_{1} + 781806983654\)

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
472.635
285.819
18.6679
−14.4733
−315.732
−445.917
−8.00000 27.0000 64.0000 −479.635 −216.000 343.000 −512.000 729.000 3837.08
1.2 −8.00000 27.0000 64.0000 −292.819 −216.000 343.000 −512.000 729.000 2342.55
1.3 −8.00000 27.0000 64.0000 −25.6679 −216.000 343.000 −512.000 729.000 205.343
1.4 −8.00000 27.0000 64.0000 7.47333 −216.000 343.000 −512.000 729.000 −59.7866
1.5 −8.00000 27.0000 64.0000 308.732 −216.000 343.000 −512.000 729.000 −2469.86
1.6 −8.00000 27.0000 64.0000 438.917 −216.000 343.000 −512.000 729.000 −3511.33
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 546.8.a.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.p 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 43 T_{5}^{5} - 301311 T_{5}^{4} - 5822771 T_{5}^{3} + 19083393670 T_{5}^{2} + 346335038400 T_{5} - \)\(36\!\cdots\!00\)\( \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T )^{6} \)
$3$ \( ( -27 + T )^{6} \)
$5$ \( -3650725242000 + 346335038400 T + 19083393670 T^{2} - 5822771 T^{3} - 301311 T^{4} + 43 T^{5} + T^{6} \)
$7$ \( ( -343 + T )^{6} \)
$11$ \( -\)\(12\!\cdots\!52\)\( - 841600968805718136 T - 726548291277356 T^{2} + 470322996274 T^{3} - 46050213 T^{4} - 7370 T^{5} + T^{6} \)
$13$ \( ( 2197 + T )^{6} \)
$17$ \( -\)\(12\!\cdots\!08\)\( - \)\(55\!\cdots\!32\)\( T + 1082656459185974284 T^{2} + 12076465707510 T^{3} - 2009054405 T^{4} - 7950 T^{5} + T^{6} \)
$19$ \( -\)\(30\!\cdots\!52\)\( + \)\(30\!\cdots\!28\)\( T - 505648971300519644 T^{2} - 125436502054109 T^{3} - 1996476801 T^{4} + 57145 T^{5} + T^{6} \)
$23$ \( \)\(40\!\cdots\!36\)\( + \)\(60\!\cdots\!76\)\( T + 23939143436689450380 T^{2} + 18770012665665 T^{3} - 9372686381 T^{4} - 31769 T^{5} + T^{6} \)
$29$ \( \)\(11\!\cdots\!68\)\( - \)\(14\!\cdots\!96\)\( T + \)\(34\!\cdots\!06\)\( T^{2} + 133297490017461 T^{3} - 37825526027 T^{4} + 36455 T^{5} + T^{6} \)
$31$ \( \)\(27\!\cdots\!56\)\( - \)\(19\!\cdots\!28\)\( T + \)\(77\!\cdots\!80\)\( T^{2} + 15838779691270288 T^{3} - 75837119980 T^{4} - 215069 T^{5} + T^{6} \)
$37$ \( -\)\(12\!\cdots\!76\)\( - \)\(39\!\cdots\!04\)\( T + \)\(12\!\cdots\!12\)\( T^{2} + 11074299633462406 T^{3} - 260806655409 T^{4} - 133074 T^{5} + T^{6} \)
$41$ \( -\)\(97\!\cdots\!28\)\( - \)\(93\!\cdots\!48\)\( T + \)\(16\!\cdots\!92\)\( T^{2} + 138912885084969504 T^{3} - 330280802084 T^{4} - 516452 T^{5} + T^{6} \)
$43$ \( -\)\(33\!\cdots\!92\)\( + \)\(24\!\cdots\!72\)\( T + \)\(11\!\cdots\!24\)\( T^{2} - 82741108357250541 T^{3} - 784842439469 T^{4} + 3085 T^{5} + T^{6} \)
$47$ \( -\)\(21\!\cdots\!32\)\( + \)\(46\!\cdots\!56\)\( T - \)\(48\!\cdots\!08\)\( T^{2} + 1273926793070384180 T^{3} - 392865394572 T^{4} - 1463947 T^{5} + T^{6} \)
$53$ \( -\)\(86\!\cdots\!84\)\( + \)\(65\!\cdots\!08\)\( T + \)\(11\!\cdots\!48\)\( T^{2} - 1056639814155127832 T^{3} - 2059790446942 T^{4} + 1344571 T^{5} + T^{6} \)
$59$ \( -\)\(18\!\cdots\!76\)\( - \)\(45\!\cdots\!04\)\( T + \)\(45\!\cdots\!44\)\( T^{2} + 9781365979757942960 T^{3} - 5717487868092 T^{4} - 1810408 T^{5} + T^{6} \)
$61$ \( \)\(26\!\cdots\!48\)\( - \)\(58\!\cdots\!08\)\( T - \)\(10\!\cdots\!32\)\( T^{2} + 42904235815826091972 T^{3} - 8202391886519 T^{4} - 4047390 T^{5} + T^{6} \)
$67$ \( -\)\(32\!\cdots\!08\)\( - \)\(18\!\cdots\!88\)\( T + \)\(17\!\cdots\!92\)\( T^{2} + 51308599043227095800 T^{3} - 26430707601412 T^{4} - 2393614 T^{5} + T^{6} \)
$71$ \( \)\(11\!\cdots\!92\)\( - \)\(29\!\cdots\!84\)\( T - \)\(34\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + 17207204909556 T^{4} - 10341084 T^{5} + T^{6} \)
$73$ \( -\)\(30\!\cdots\!88\)\( - \)\(58\!\cdots\!68\)\( T - \)\(57\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!77\)\( T^{3} - 34801659181391 T^{4} - 5180001 T^{5} + T^{6} \)
$79$ \( \)\(10\!\cdots\!56\)\( - \)\(43\!\cdots\!32\)\( T + \)\(13\!\cdots\!96\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} - 85894681250056 T^{4} - 4624979 T^{5} + T^{6} \)
$83$ \( -\)\(41\!\cdots\!80\)\( + \)\(74\!\cdots\!36\)\( T - \)\(44\!\cdots\!68\)\( T^{2} + \)\(99\!\cdots\!96\)\( T^{3} - 37284924250424 T^{4} - 11892699 T^{5} + T^{6} \)
$89$ \( -\)\(14\!\cdots\!08\)\( - \)\(10\!\cdots\!44\)\( T + \)\(28\!\cdots\!00\)\( T^{2} + \)\(84\!\cdots\!64\)\( T^{3} - 113835302988770 T^{4} - 9781713 T^{5} + T^{6} \)
$97$ \( -\)\(44\!\cdots\!88\)\( + \)\(10\!\cdots\!24\)\( T + \)\(11\!\cdots\!48\)\( T^{2} + \)\(13\!\cdots\!12\)\( T^{3} - 234391422311402 T^{4} - 5202537 T^{5} + T^{6} \)
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