Properties

Label 350.2.o.d
Level 350
Weight 2
Character orbit 350.o
Analytic conductor 2.795
Analytic rank 0
Dimension 16
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.478584585616890104119296.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{7} ) q^{2} + ( -\beta_{4} - \beta_{13} + \beta_{14} ) q^{3} + \beta_{9} q^{4} + ( -1 + \beta_{2} - 2 \beta_{6} - \beta_{8} ) q^{6} + ( -\beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{7} + ( -\beta_{4} - \beta_{13} ) q^{8} + ( 2 \beta_{3} - 3 \beta_{9} + \beta_{11} - 4 \beta_{12} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{7} ) q^{2} + ( -\beta_{4} - \beta_{13} + \beta_{14} ) q^{3} + \beta_{9} q^{4} + ( -1 + \beta_{2} - 2 \beta_{6} - \beta_{8} ) q^{6} + ( -\beta_{1} - 2 \beta_{5} + 2 \beta_{7} ) q^{7} + ( -\beta_{4} - \beta_{13} ) q^{8} + ( 2 \beta_{3} - 3 \beta_{9} + \beta_{11} - 4 \beta_{12} ) q^{9} + ( 1 + \beta_{2} - 2 \beta_{8} ) q^{11} + ( \beta_{1} + \beta_{5} ) q^{12} + ( \beta_{4} - 2 \beta_{13} - \beta_{15} ) q^{13} + ( -\beta_{3} + 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{14} -\beta_{6} q^{16} + ( 4 \beta_{5} - 2 \beta_{7} ) q^{17} + ( \beta_{4} + 4 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{18} + ( \beta_{9} - 3 \beta_{11} + 2 \beta_{12} ) q^{19} + ( 3 + 2 \beta_{2} - 3 \beta_{6} - \beta_{8} ) q^{21} + ( 2 \beta_{1} - \beta_{7} - \beta_{10} ) q^{22} + 3 \beta_{4} q^{23} + ( \beta_{3} - \beta_{9} + \beta_{11} - 2 \beta_{12} ) q^{24} + ( 3 + 2 \beta_{6} + \beta_{8} ) q^{26} + ( 5 \beta_{5} - 9 \beta_{7} - \beta_{10} ) q^{27} + ( -2 \beta_{4} - 2 \beta_{13} - \beta_{15} ) q^{28} + ( \beta_{3} - \beta_{9} + 2 \beta_{11} - 3 \beta_{12} ) q^{29} + ( -2 + 2 \beta_{6} ) q^{31} + \beta_{7} q^{32} + ( 4 \beta_{4} + 8 \beta_{13} ) q^{33} + ( -4 \beta_{9} - 2 \beta_{12} ) q^{34} + ( -2 - \beta_{2} - \beta_{8} ) q^{36} + ( -\beta_{1} - \beta_{7} + 2 \beta_{10} ) q^{37} + ( \beta_{4} + 2 \beta_{13} - 3 \beta_{14} ) q^{38} + ( -2 \beta_{3} + 6 \beta_{9} + 2 \beta_{11} ) q^{39} + ( -3 - 6 \beta_{6} ) q^{41} + ( \beta_{1} - 4 \beta_{5} + 5 \beta_{7} + \beta_{10} ) q^{42} + ( -4 \beta_{4} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{43} + ( 2 \beta_{3} + \beta_{9} + \beta_{11} ) q^{44} + ( 3 + 3 \beta_{6} ) q^{46} + ( -3 \beta_{1} + 2 \beta_{5} + 5 \beta_{7} ) q^{47} + ( \beta_{13} + \beta_{15} ) q^{48} + ( -3 \beta_{3} - \beta_{9} - 3 \beta_{11} + 3 \beta_{12} ) q^{49} + ( 2 - 4 \beta_{2} + 6 \beta_{6} + 2 \beta_{8} ) q^{51} + ( -\beta_{1} - 4 \beta_{5} + 2 \beta_{7} + \beta_{10} ) q^{52} + ( -3 \beta_{4} - 6 \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{53} + ( -4 \beta_{9} - \beta_{11} + 5 \beta_{12} ) q^{54} + ( -\beta_{6} + \beta_{8} ) q^{56} + ( -4 \beta_{1} - 18 \beta_{5} + 2 \beta_{7} + 2 \beta_{10} ) q^{57} + ( -\beta_{4} + \beta_{14} + \beta_{15} ) q^{58} + ( 6 \beta_{9} + 12 \beta_{12} ) q^{59} + ( -9 - 3 \beta_{6} + 3 \beta_{8} ) q^{61} + ( 2 \beta_{5} - 4 \beta_{7} ) q^{62} + ( -8 \beta_{4} + \beta_{14} + 3 \beta_{15} ) q^{63} -\beta_{12} q^{64} + ( -4 + 4 \beta_{6} ) q^{66} + ( \beta_{1} + 7 \beta_{7} + \beta_{10} ) q^{67} + ( 2 \beta_{4} + 4 \beta_{13} ) q^{68} + ( -3 \beta_{3} + 3 \beta_{9} + 3 \beta_{12} ) q^{69} -6 q^{71} + ( \beta_{1} + 4 \beta_{5} - 3 \beta_{7} - 2 \beta_{10} ) q^{72} + ( 6 \beta_{4} + 2 \beta_{13} + 2 \beta_{14} ) q^{73} + ( -\beta_{3} - 2 \beta_{9} + \beta_{11} ) q^{74} + ( 2 - 3 \beta_{2} + 4 \beta_{6} + 3 \beta_{8} ) q^{76} + ( 3 \beta_{1} - 4 \beta_{5} + 8 \beta_{7} - 3 \beta_{10} ) q^{77} + ( -8 \beta_{4} - 10 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{78} + ( -10 \beta_{9} - 10 \beta_{12} ) q^{79} + ( 5 - 2 \beta_{2} + 7 \beta_{6} + 4 \beta_{8} ) q^{81} + ( 3 \beta_{5} + 3 \beta_{7} ) q^{82} + ( 6 \beta_{4} - 3 \beta_{13} + 3 \beta_{15} ) q^{83} + ( \beta_{3} + 3 \beta_{9} + 2 \beta_{11} - 3 \beta_{12} ) q^{84} + ( 1 - 2 \beta_{2} - 3 \beta_{6} + \beta_{8} ) q^{86} + ( 3 \beta_{1} + 14 \beta_{5} - 7 \beta_{7} - 3 \beta_{10} ) q^{87} + ( \beta_{4} - \beta_{14} + 2 \beta_{15} ) q^{88} + ( 3 \beta_{11} - 3 \beta_{12} ) q^{89} + ( 6 - 3 \beta_{2} + 10 \beta_{6} + 3 \beta_{8} ) q^{91} -3 \beta_{5} q^{92} + ( 2 \beta_{4} - 2 \beta_{14} - 2 \beta_{15} ) q^{93} + ( -3 \beta_{3} - 2 \beta_{9} - 3 \beta_{11} - 4 \beta_{12} ) q^{94} + ( -1 - \beta_{6} - \beta_{8} ) q^{96} + ( 4 \beta_{5} - 12 \beta_{7} + 4 \beta_{10} ) q^{97} + ( \beta_{4} + \beta_{13} - 3 \beta_{15} ) q^{98} + ( -\beta_{3} + \beta_{9} - 2 \beta_{11} + 12 \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 12q^{11} + 8q^{16} + 76q^{21} + 36q^{26} - 48q^{31} - 40q^{36} + 24q^{46} - 24q^{51} + 12q^{56} - 108q^{61} - 96q^{66} - 96q^{71} + 32q^{81} + 36q^{86} + 16q^{91} - 12q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 31 x^{12} + 336 x^{8} - 19375 x^{4} + 390625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{12} + 1344 \nu^{8} + 336 \nu^{4} - 19375 \)\()/378000\)
\(\beta_{3}\)\(=\)\((\)\( 31 \nu^{14} - 336 \nu^{10} + 850416 \nu^{6} - 390625 \nu^{2} \)\()/47250000\)
\(\beta_{4}\)\(=\)\((\)\( 31 \nu^{15} - 336 \nu^{11} + 850416 \nu^{7} - 390625 \nu^{3} \)\()/ 236250000 \)
\(\beta_{5}\)\(=\)\((\)\( 71 \nu^{13} + 924 \nu^{9} + 23856 \nu^{5} - 1375625 \nu \)\()/2362500\)
\(\beta_{6}\)\(=\)\((\)\( -31 \nu^{12} + 336 \nu^{8} - 10416 \nu^{4} + 390625 \)\()/210000\)
\(\beta_{7}\)\(=\)\((\)\( -341 \nu^{13} + 3696 \nu^{9} + 95424 \nu^{5} + 4296875 \nu \)\()/9450000\)
\(\beta_{8}\)\(=\)\((\)\( -341 \nu^{12} + 3696 \nu^{8} + 95424 \nu^{4} + 4296875 \)\()/1890000\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{14} + 559 \nu^{2} \)\()/75600\)
\(\beta_{10}\)\(=\)\((\)\( -1111 \nu^{13} + 18816 \nu^{9} - 373296 \nu^{5} + 21525625 \nu \)\()/9450000\)
\(\beta_{11}\)\(=\)\((\)\( -71 \nu^{14} - 924 \nu^{10} - 23856 \nu^{6} + 1375625 \nu^{2} \)\()/2362500\)
\(\beta_{12}\)\(=\)\((\)\( 31 \nu^{14} - 336 \nu^{10} + 6666 \nu^{6} - 390625 \nu^{2} \)\()/843750\)
\(\beta_{13}\)\(=\)\((\)\( -2531 \nu^{15} + 336 \nu^{11} - 850416 \nu^{7} + 49038125 \nu^{3} \)\()/ 236250000 \)
\(\beta_{14}\)\(=\)\((\)\( \nu^{15} - 31 \nu^{11} + 336 \nu^{7} - 19375 \nu^{3} \)\()/78125\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} + 559 \nu^{3} \)\()/75600\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} + \beta_{11} - 5 \beta_{9} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{15} + 5 \beta_{13} + 5 \beta_{4}\)
\(\nu^{4}\)\(=\)\(9 \beta_{8} - 11 \beta_{6}\)
\(\nu^{5}\)\(=\)\(-11 \beta_{10} + 45 \beta_{7} + 11 \beta_{5} + 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{12} + 56 \beta_{3}\)
\(\nu^{7}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{13} + 280 \beta_{4}\)
\(\nu^{8}\)\(=\)\(5 \beta_{6} + 279 \beta_{2} + 5\)
\(\nu^{9}\)\(=\)\(284 \beta_{10} + 1111 \beta_{5}\)
\(\nu^{10}\)\(=\)\(-1420 \beta_{12} - 1111 \beta_{11} - 1420 \beta_{9}\)
\(\nu^{11}\)\(=\)\(-2531 \beta_{14} - 3024 \beta_{13}\)
\(\nu^{12}\)\(=\)\(-3024 \beta_{8} - 3024 \beta_{6} + 3024 \beta_{2} + 12655\)
\(\nu^{13}\)\(=\)\(-15120 \beta_{7} + 15120 \beta_{5} + 15679 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-559 \beta_{12} + 559 \beta_{11} - 78395 \beta_{9} + 559 \beta_{3}\)
\(\nu^{15}\)\(=\)\(-77836 \beta_{15} + 2795 \beta_{13} + 2795 \beta_{4}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{6}\) \(\beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.0811201 2.23460i
−1.04705 + 1.97578i
1.04705 1.97578i
−0.0811201 + 2.23460i
1.97578 + 1.04705i
−2.23460 0.0811201i
2.23460 + 0.0811201i
−1.97578 1.04705i
1.97578 1.04705i
−2.23460 + 0.0811201i
2.23460 0.0811201i
−1.97578 + 1.04705i
0.0811201 + 2.23460i
−1.04705 1.97578i
1.04705 + 1.97578i
−0.0811201 2.23460i
−0.965926 0.258819i −0.788227 2.94170i 0.866025 + 0.500000i 0 3.04547i −2.01297 + 1.71696i −0.707107 0.707107i −5.43424 + 3.13746i 0
143.2 −0.965926 0.258819i 0.339939 + 1.26867i 0.866025 + 0.500000i 0 1.31342i −0.884806 2.49342i −0.707107 0.707107i 1.10411 0.637459i 0
143.3 0.965926 + 0.258819i −0.339939 1.26867i 0.866025 + 0.500000i 0 1.31342i 0.884806 + 2.49342i 0.707107 + 0.707107i 1.10411 0.637459i 0
143.4 0.965926 + 0.258819i 0.788227 + 2.94170i 0.866025 + 0.500000i 0 3.04547i 2.01297 1.71696i 0.707107 + 0.707107i −5.43424 + 3.13746i 0
157.1 −0.258819 + 0.965926i −1.26867 + 0.339939i −0.866025 0.500000i 0 1.31342i −2.49342 + 0.884806i 0.707107 0.707107i −1.10411 + 0.637459i 0
157.2 −0.258819 + 0.965926i 2.94170 0.788227i −0.866025 0.500000i 0 3.04547i 1.71696 + 2.01297i 0.707107 0.707107i 5.43424 3.13746i 0
157.3 0.258819 0.965926i −2.94170 + 0.788227i −0.866025 0.500000i 0 3.04547i −1.71696 2.01297i −0.707107 + 0.707107i 5.43424 3.13746i 0
157.4 0.258819 0.965926i 1.26867 0.339939i −0.866025 0.500000i 0 1.31342i 2.49342 0.884806i −0.707107 + 0.707107i −1.10411 + 0.637459i 0
243.1 −0.258819 0.965926i −1.26867 0.339939i −0.866025 + 0.500000i 0 1.31342i −2.49342 0.884806i 0.707107 + 0.707107i −1.10411 0.637459i 0
243.2 −0.258819 0.965926i 2.94170 + 0.788227i −0.866025 + 0.500000i 0 3.04547i 1.71696 2.01297i 0.707107 + 0.707107i 5.43424 + 3.13746i 0
243.3 0.258819 + 0.965926i −2.94170 0.788227i −0.866025 + 0.500000i 0 3.04547i −1.71696 + 2.01297i −0.707107 0.707107i 5.43424 + 3.13746i 0
243.4 0.258819 + 0.965926i 1.26867 + 0.339939i −0.866025 + 0.500000i 0 1.31342i 2.49342 + 0.884806i −0.707107 0.707107i −1.10411 0.637459i 0
257.1 −0.965926 + 0.258819i −0.788227 + 2.94170i 0.866025 0.500000i 0 3.04547i −2.01297 1.71696i −0.707107 + 0.707107i −5.43424 3.13746i 0
257.2 −0.965926 + 0.258819i 0.339939 1.26867i 0.866025 0.500000i 0 1.31342i −0.884806 + 2.49342i −0.707107 + 0.707107i 1.10411 + 0.637459i 0
257.3 0.965926 0.258819i −0.339939 + 1.26867i 0.866025 0.500000i 0 1.31342i 0.884806 2.49342i 0.707107 0.707107i 1.10411 + 0.637459i 0
257.4 0.965926 0.258819i 0.788227 2.94170i 0.866025 0.500000i 0 3.04547i 2.01297 + 1.71696i 0.707107 0.707107i −5.43424 3.13746i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.o.d 16
5.b even 2 1 inner 350.2.o.d 16
5.c odd 4 2 inner 350.2.o.d 16
7.d odd 6 1 inner 350.2.o.d 16
35.i odd 6 1 inner 350.2.o.d 16
35.k even 12 2 inner 350.2.o.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.o.d 16 1.a even 1 1 trivial
350.2.o.d 16 5.b even 2 1 inner
350.2.o.d 16 5.c odd 4 2 inner
350.2.o.d 16 7.d odd 6 1 inner
350.2.o.d 16 35.i odd 6 1 inner
350.2.o.d 16 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 89 T_{3}^{12} + 7665 T_{3}^{8} - 22784 T_{3}^{4} + 65536 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( 1 + 7 T^{4} - 111 T^{8} - 14 T^{12} + 15070 T^{16} - 1134 T^{20} - 728271 T^{24} + 3720087 T^{28} + 43046721 T^{32} \)
$5$ \( \)
$7$ \( 1 + 73 T^{4} + 2928 T^{8} + 175273 T^{12} + 5764801 T^{16} \)
$11$ \( ( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 396 T^{5} - 121 T^{6} - 3993 T^{7} + 14641 T^{8} )^{4} \)
$13$ \( ( 1 + T^{4} - 50736 T^{8} + 28561 T^{12} + 815730721 T^{16} )^{2} \)
$17$ \( ( 1 + 94 T^{4} - 74685 T^{8} + 7850974 T^{12} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 + 11 T^{2} - 503 T^{4} - 1078 T^{6} + 215374 T^{8} - 389158 T^{10} - 65551463 T^{12} + 517504691 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 - 311 T^{4} - 183120 T^{8} - 87030551 T^{12} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 83 T^{2} + 3276 T^{4} - 69803 T^{6} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} )^{8} \)
$37$ \( 1 - 1393 T^{4} - 596879 T^{8} + 1686914642 T^{12} - 711159912626 T^{16} + 3161549632365362 T^{20} - 2096525223976912559 T^{24} - \)\(91\!\cdots\!33\)\( T^{28} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( ( 1 - 55 T^{2} + 1681 T^{4} )^{8} \)
$43$ \( ( 1 + 217 T^{4} - 5396064 T^{8} + 741879817 T^{12} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 4153 T^{4} + 8461249 T^{8} + 4041707906 T^{12} - 30776533112690 T^{16} + 19722245276457986 T^{20} + \)\(20\!\cdots\!89\)\( T^{24} - \)\(48\!\cdots\!73\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 7727 T^{4} + 31770289 T^{8} + 93923833106 T^{12} + 247060582275790 T^{16} + 741104220570063986 T^{20} + \)\(19\!\cdots\!29\)\( T^{24} + \)\(37\!\cdots\!07\)\( T^{28} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( ( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )^{4} \)
$61$ \( ( 1 + 27 T + 383 T^{2} + 3780 T^{3} + 30702 T^{4} + 230580 T^{5} + 1425143 T^{6} + 6128487 T^{7} + 13845841 T^{8} )^{4} \)
$67$ \( 1 + 3479 T^{4} + 20488945 T^{8} - 169384668334 T^{12} - 579648904329986 T^{16} - 3413290947143302414 T^{20} + \)\(83\!\cdots\!45\)\( T^{24} + \)\(28\!\cdots\!19\)\( T^{28} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{16} \)
$73$ \( 1 - 2788 T^{4} + 31113226 T^{8} + 223421298032 T^{12} - 482429265046061 T^{16} + 6344771866045561712 T^{20} + \)\(25\!\cdots\!06\)\( T^{24} - \)\(63\!\cdots\!48\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( ( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{4} \)
$83$ \( ( 1 + 1033 T^{4} - 67132992 T^{8} + 49024445593 T^{12} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 257 T^{2} + 34849 T^{4} - 3947006 T^{6} + 387523630 T^{8} - 31264234526 T^{10} + 2186504356609 T^{12} - 127724191776977 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 + 28228 T^{4} + 363130758 T^{8} + 2499004544068 T^{12} + 7837433594376961 T^{16} )^{2} \)
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