Properties

Label 144.3.o.a
Level $144$
Weight $3$
Character orbit 144.o
Analytic conductor $3.924$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
Defining polynomial: \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + ( -1 + 2 \beta_{1} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + ( -1 + 2 \beta_{1} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{9} + ( 5 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} + ( 2 - 8 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{15} + ( -2 - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{17} + ( 4 - 6 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{21} + ( -4 - 8 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{23} + ( 10 - 12 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{25} + ( 15 + 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{27} + ( -12 + 23 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{29} + ( -4 + 8 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{31} + ( 18 - 9 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{33} + ( -27 + 41 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - \beta_{4} - \beta_{6} - 5 \beta_{7} ) q^{35} + ( -10 + 6 \beta_{1} - 3 \beta_{2} - 12 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} ) q^{37} + ( -30 + 24 \beta_{1} - \beta_{2} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{39} + ( 21 - 21 \beta_{1} + 12 \beta_{2} + 9 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} + 3 \beta_{7} ) q^{41} + ( -7 + 3 \beta_{1} - 9 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} + \beta_{6} + 6 \beta_{7} ) q^{43} + ( -27 + 27 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{45} + ( 33 - 13 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{47} + ( 12 - 12 \beta_{1} - 9 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{49} + ( 4 - 40 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 9 \beta_{6} ) q^{51} + ( -42 + 6 \beta_{1} - 9 \beta_{2} - 3 \beta_{4} + 12 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{53} + ( -15 + 21 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{7} ) q^{55} + ( -22 - 20 \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} - 6 \beta_{7} ) q^{57} + ( -36 - 20 \beta_{1} - 11 \beta_{2} - \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{59} + ( 8 - \beta_{1} + 3 \beta_{3} + 10 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{61} + ( 45 + 9 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{63} + ( -6 + 25 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} - 3 \beta_{7} ) q^{65} + ( -8 + 4 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 3 \beta_{6} + 6 \beta_{7} ) q^{67} + ( 9 + 15 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} - 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -24 + 62 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} - 6 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{71} + ( 14 - 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{73} + ( -74 + 62 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} - 21 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} ) q^{75} + ( 51 - 51 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{77} + ( 21 - 11 \beta_{1} + 21 \beta_{2} + 15 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + \beta_{6} - 9 \beta_{7} ) q^{79} + ( 3 + 42 \beta_{1} + 15 \beta_{2} + 21 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{81} + ( 103 - 57 \beta_{1} + 10 \beta_{2} + 14 \beta_{3} - 6 \beta_{4} + \beta_{5} + 11 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -10 + 10 \beta_{1} + 12 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} - 7 \beta_{6} - 5 \beta_{7} ) q^{85} + ( 37 - 97 \beta_{1} + 2 \beta_{2} + 13 \beta_{3} - 6 \beta_{4} + 15 \beta_{5} + 18 \beta_{6} + 9 \beta_{7} ) q^{87} + ( -10 - 6 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 10 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{89} + ( 29 - 29 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 11 \beta_{7} ) q^{91} + ( 21 - 57 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} - 12 \beta_{4} + 15 \beta_{5} + 6 \beta_{6} + 9 \beta_{7} ) q^{93} + ( -32 - 72 \beta_{1} + 28 \beta_{2} + 2 \beta_{3} - 20 \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -20 + 7 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} - 13 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 13 \beta_{7} ) q^{97} + ( 87 - 3 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} - 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{3} + 3q^{5} + 3q^{7} - 3q^{9} + O(q^{10}) \) \( 8q - 3q^{3} + 3q^{5} + 3q^{7} - 3q^{9} + 18q^{11} + 5q^{13} - 21q^{15} + 6q^{17} - 33q^{21} - 81q^{23} - 23q^{25} + 108q^{27} + 69q^{29} + 45q^{31} + 72q^{33} - 20q^{37} - 141q^{39} + 54q^{41} - 117q^{45} + 207q^{47} + 41q^{49} - 141q^{51} - 252q^{53} - 273q^{57} - 306q^{59} + 7q^{61} + 441q^{63} + 93q^{65} + 12q^{67} + 189q^{69} + 74q^{73} - 387q^{75} + 207q^{77} + 33q^{79} + 117q^{81} + 549q^{83} - 30q^{85} - 87q^{87} - 168q^{89} - 27q^{93} - 684q^{95} - 10q^{97} + 585q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} + 10 \nu + 2 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 10 \nu^{5} + 4 \nu^{4} + 31 \nu^{3} + 22 \nu^{2} + 30 \nu + 10 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 11 \nu^{5} - 2 \nu^{4} + 32 \nu^{3} - 14 \nu^{2} + 16 \nu - 16 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} + 12 \nu^{5} + 20 \nu^{4} + 39 \nu^{3} + 50 \nu^{2} + 14 \nu + 10 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 11 \nu^{5} + 34 \nu^{4} - 32 \nu^{3} + 70 \nu^{2} - 16 \nu + 16 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 11 \nu^{5} - 10 \nu^{4} + 35 \nu^{3} - 28 \nu^{2} + 22 \nu - 14 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} + 31 \nu^{5} + 88 \nu^{3} + 6 \nu^{2} + 52 \nu + 16 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{3} - \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - \beta_{2} - 2 \beta_{1} - 2\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 11 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 48\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-16 \beta_{6} - 7 \beta_{5} - 2 \beta_{4} + 25 \beta_{3} + 2 \beta_{2} + 16 \beta_{1} + 8\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(34 \beta_{7} + 8 \beta_{6} + 11 \beta_{5} - 8 \beta_{4} - 57 \beta_{3} - 42 \beta_{2} + 34 \beta_{1} - 244\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(2 \beta_{7} + 86 \beta_{6} + 43 \beta_{5} - 125 \beta_{3} + 2 \beta_{2} - 126 \beta_{1} - 20\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.07834i
0.385731i
2.06288i
2.33086i
1.07834i
0.385731i
2.06288i
2.33086i
0 −2.64956 1.40707i 0 −3.01729 + 5.22611i 0 10.2332 5.90815i 0 5.04032 + 7.45622i 0
31.2 0 −1.28651 + 2.71015i 0 −0.454613 + 0.787412i 0 −6.10709 + 3.52593i 0 −5.68980 6.97325i 0
31.3 0 −0.456412 2.96508i 0 4.61660 7.99619i 0 −5.33093 + 3.07781i 0 −8.58338 + 2.70659i 0
31.4 0 2.89248 + 0.795973i 0 0.355304 0.615405i 0 2.70480 1.56162i 0 7.73285 + 4.60467i 0
79.1 0 −2.64956 + 1.40707i 0 −3.01729 5.22611i 0 10.2332 + 5.90815i 0 5.04032 7.45622i 0
79.2 0 −1.28651 2.71015i 0 −0.454613 0.787412i 0 −6.10709 3.52593i 0 −5.68980 + 6.97325i 0
79.3 0 −0.456412 + 2.96508i 0 4.61660 + 7.99619i 0 −5.33093 3.07781i 0 −8.58338 2.70659i 0
79.4 0 2.89248 0.795973i 0 0.355304 + 0.615405i 0 2.70480 + 1.56162i 0 7.73285 4.60467i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.o.a 8
3.b odd 2 1 432.3.o.b 8
4.b odd 2 1 144.3.o.c yes 8
8.b even 2 1 576.3.o.f 8
8.d odd 2 1 576.3.o.d 8
9.c even 3 1 144.3.o.c yes 8
9.c even 3 1 1296.3.g.j 8
9.d odd 6 1 432.3.o.a 8
9.d odd 6 1 1296.3.g.k 8
12.b even 2 1 432.3.o.a 8
24.f even 2 1 1728.3.o.e 8
24.h odd 2 1 1728.3.o.f 8
36.f odd 6 1 inner 144.3.o.a 8
36.f odd 6 1 1296.3.g.j 8
36.h even 6 1 432.3.o.b 8
36.h even 6 1 1296.3.g.k 8
72.j odd 6 1 1728.3.o.e 8
72.l even 6 1 1728.3.o.f 8
72.n even 6 1 576.3.o.d 8
72.p odd 6 1 576.3.o.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 1.a even 1 1 trivial
144.3.o.a 8 36.f odd 6 1 inner
144.3.o.c yes 8 4.b odd 2 1
144.3.o.c yes 8 9.c even 3 1
432.3.o.a 8 9.d odd 6 1
432.3.o.a 8 12.b even 2 1
432.3.o.b 8 3.b odd 2 1
432.3.o.b 8 36.h even 6 1
576.3.o.d 8 8.d odd 2 1
576.3.o.d 8 72.n even 6 1
576.3.o.f 8 8.b even 2 1
576.3.o.f 8 72.p odd 6 1
1296.3.g.j 8 9.c even 3 1
1296.3.g.j 8 36.f odd 6 1
1296.3.g.k 8 9.d odd 6 1
1296.3.g.k 8 36.h even 6 1
1728.3.o.e 8 24.f even 2 1
1728.3.o.e 8 72.j odd 6 1
1728.3.o.f 8 24.h odd 2 1
1728.3.o.f 8 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 + 2187 T + 486 T^{2} - 243 T^{3} - 126 T^{4} - 27 T^{5} + 6 T^{6} + 3 T^{7} + T^{8} \)
$5$ \( 1296 - 324 T + 2133 T^{2} + 729 T^{3} + 3186 T^{4} + 189 T^{5} + 66 T^{6} - 3 T^{7} + T^{8} \)
$7$ \( 2566404 - 446958 T - 161487 T^{2} + 32643 T^{3} + 12366 T^{4} + 351 T^{5} - 114 T^{6} - 3 T^{7} + T^{8} \)
$11$ \( 12131289 + 2821230 T - 282852 T^{2} - 116640 T^{3} + 12393 T^{4} + 2592 T^{5} - 36 T^{6} - 18 T^{7} + T^{8} \)
$13$ \( 10201636 - 7515482 T + 4607155 T^{2} - 716663 T^{3} + 99640 T^{4} - 3251 T^{5} + 316 T^{6} - 5 T^{7} + T^{8} \)
$17$ \( ( 84168 + 1908 T - 822 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$19$ \( 2931572736 + 85791744 T^{2} + 739584 T^{4} + 1731 T^{6} + T^{8} \)
$23$ \( 19131876 - 3188646 T - 2302911 T^{2} + 413343 T^{3} + 345546 T^{4} + 45927 T^{5} + 2754 T^{6} + 81 T^{7} + T^{8} \)
$29$ \( 4046639163876 - 240302807082 T + 13105243395 T^{2} - 346769991 T^{3} + 10589400 T^{4} - 198963 T^{5} + 5340 T^{6} - 69 T^{7} + T^{8} \)
$31$ \( 944784 + 11573604 T + 46287855 T^{2} - 11895093 T^{3} + 818424 T^{4} + 44955 T^{5} - 324 T^{6} - 45 T^{7} + T^{8} \)
$37$ \( ( -613568 - 117320 T - 3756 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$41$ \( 5431756955769 - 445142421774 T + 30187580904 T^{2} - 767400804 T^{3} + 19934505 T^{4} - 236196 T^{5} + 5616 T^{6} - 54 T^{7} + T^{8} \)
$43$ \( 29016737649 - 20365527708 T + 5307258510 T^{2} - 380905416 T^{3} + 10320939 T^{4} - 3186 T^{6} + T^{8} \)
$47$ \( 28643839776036 - 1972043877186 T + 38368451709 T^{2} + 474219603 T^{3} - 18415998 T^{4} - 266409 T^{5} + 15570 T^{6} - 207 T^{7} + T^{8} \)
$53$ \( ( -6508512 - 274104 T + 972 T^{2} + 126 T^{3} + T^{4} )^{2} \)
$59$ \( 48359409452649 + 1123183376802 T - 44378047044 T^{2} - 1232674848 T^{3} + 48727089 T^{4} + 2335392 T^{5} + 38844 T^{6} + 306 T^{7} + T^{8} \)
$61$ \( 309954973696 - 123735132736 T + 45856336249 T^{2} - 1420643911 T^{3} + 42523942 T^{4} - 400003 T^{5} + 6406 T^{6} - 7 T^{7} + T^{8} \)
$67$ \( 68036119056801 + 520837032744 T - 49299630426 T^{2} - 387577872 T^{3} + 29174067 T^{4} + 73656 T^{5} - 6090 T^{6} - 12 T^{7} + T^{8} \)
$71$ \( 726110197530624 + 765915906816 T^{2} + 231242688 T^{4} + 26208 T^{6} + T^{8} \)
$73$ \( ( 416536 + 25628 T - 1002 T^{2} - 37 T^{3} + T^{4} )^{2} \)
$79$ \( 240627852449856 + 798427622664 T - 176002346205 T^{2} - 586923813 T^{3} + 113950044 T^{4} + 376299 T^{5} - 11040 T^{6} - 33 T^{7} + T^{8} \)
$83$ \( 1517530356962064 + 28477438539300 T - 984805785801 T^{2} - 21823289325 T^{3} + 1063934676 T^{4} - 16389297 T^{5} + 130320 T^{6} - 549 T^{7} + T^{8} \)
$89$ \( ( -1161936 - 109152 T - 984 T^{2} + 84 T^{3} + T^{4} )^{2} \)
$97$ \( 30429664983481 + 3216990050002 T + 257418140392 T^{2} + 8850998044 T^{3} + 235988233 T^{4} + 1016476 T^{5} + 15088 T^{6} + 10 T^{7} + T^{8} \)
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