Properties

Label 735.3.h.a
Level 735
Weight 3
Character orbit 735.h
Analytic conductor 20.027
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 735.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.0272994305\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.523596960000.16
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{2} + \beta_{5} ) q^{6} + ( -3 + 2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{8} -3 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{2} + \beta_{5} ) q^{6} + ( -3 + 2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{8} -3 q^{9} + ( \beta_{2} - \beta_{4} + \beta_{6} ) q^{10} + ( -6 - \beta_{1} + \beta_{3} - \beta_{7} ) q^{11} + ( \beta_{2} - 2 \beta_{5} + 3 \beta_{6} ) q^{12} + ( -2 \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{13} -\beta_{1} q^{15} + ( -3 - \beta_{1} + 5 \beta_{3} - 2 \beta_{7} ) q^{16} + ( -4 \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{17} + 3 \beta_{3} q^{18} + ( 2 \beta_{2} - 4 \beta_{4} + \beta_{5} + 8 \beta_{6} ) q^{19} + ( -\beta_{2} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{20} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + \beta_{7} ) q^{22} + ( -12 + 2 \beta_{1} - 3 \beta_{3} + 4 \beta_{7} ) q^{23} + ( -\beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{24} -5 q^{25} + ( -\beta_{2} + \beta_{4} - 10 \beta_{5} + 13 \beta_{6} ) q^{26} + 3 \beta_{5} q^{27} + ( -15 + 4 \beta_{1} - 5 \beta_{7} ) q^{29} + ( 2 + \beta_{1} - 3 \beta_{3} + \beta_{7} ) q^{30} + ( \beta_{2} + 4 \beta_{4} - 13 \beta_{5} + 7 \beta_{6} ) q^{31} + ( -11 + 2 \beta_{1} + \beta_{3} - 3 \beta_{7} ) q^{32} + ( \beta_{2} - 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{33} + ( 2 \beta_{2} - 2 \beta_{4} - 20 \beta_{5} + 16 \beta_{6} ) q^{34} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{36} + ( 22 - 7 \beta_{1} - 8 \beta_{3} - 4 \beta_{7} ) q^{37} + ( -7 \beta_{2} + 8 \beta_{4} + 9 \beta_{5} - 22 \beta_{6} ) q^{38} + ( -2 + \beta_{1} + 6 \beta_{3} - 4 \beta_{7} ) q^{39} + ( 3 \beta_{2} + 2 \beta_{4} - 10 \beta_{5} + 3 \beta_{6} ) q^{40} + ( -3 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + 9 \beta_{6} ) q^{41} + ( 39 - 3 \beta_{1} + 5 \beta_{3} - 5 \beta_{7} ) q^{43} + ( 1 + \beta_{1} + 7 \beta_{3} ) q^{44} + 3 \beta_{6} q^{45} + ( 11 - 13 \beta_{1} + 17 \beta_{3} - 2 \beta_{7} ) q^{46} + ( \beta_{2} + \beta_{4} + 5 \beta_{5} + 4 \beta_{6} ) q^{47} + ( 5 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} ) q^{48} + 5 \beta_{3} q^{50} + ( -11 - 2 \beta_{1} + 12 \beta_{3} - \beta_{7} ) q^{51} + ( -15 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} - 12 \beta_{6} ) q^{52} + ( 23 + 12 \beta_{1} - 3 \beta_{3} + 5 \beta_{7} ) q^{53} + ( 3 \beta_{2} - 3 \beta_{5} ) q^{54} + ( -3 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} ) q^{55} + ( 5 + 8 \beta_{1} - 6 \beta_{3} + 4 \beta_{7} ) q^{57} + ( -8 + 6 \beta_{1} + 32 \beta_{3} - 4 \beta_{7} ) q^{58} + ( 14 \beta_{2} + \beta_{4} - 12 \beta_{5} - 14 \beta_{6} ) q^{59} + ( 13 - 2 \beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{60} + ( 4 \beta_{2} - 10 \beta_{4} + 36 \beta_{5} + 14 \beta_{6} ) q^{61} + ( -20 \beta_{2} + 7 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} ) q^{62} + ( 3 + 9 \beta_{1} - \beta_{3} + 6 \beta_{7} ) q^{64} + ( 21 - 14 \beta_{3} - 2 \beta_{7} ) q^{65} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - 12 \beta_{6} ) q^{66} + ( -49 + 9 \beta_{1} + 5 \beta_{3} - \beta_{7} ) q^{67} + ( -20 \beta_{2} + 12 \beta_{4} + 30 \beta_{5} - 18 \beta_{6} ) q^{68} + ( -3 \beta_{2} + 12 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} ) q^{69} + ( 9 + 4 \beta_{1} + 28 \beta_{3} + 5 \beta_{7} ) q^{71} + ( 9 - 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{7} ) q^{72} + ( 11 \beta_{2} - 7 \beta_{4} + 27 \beta_{5} + 23 \beta_{6} ) q^{73} + ( 54 + 7 \beta_{1} - 31 \beta_{3} + 7 \beta_{7} ) q^{74} + 5 \beta_{5} q^{75} + ( 23 \beta_{2} - 6 \beta_{4} - 48 \beta_{5} + 27 \beta_{6} ) q^{76} + ( -32 + 13 \beta_{1} + 3 \beta_{3} - \beta_{7} ) q^{78} + ( 52 + 9 \beta_{1} + 5 \beta_{3} + 6 \beta_{7} ) q^{79} + ( -9 \beta_{2} - \beta_{4} + 5 \beta_{5} ) q^{80} + 9 q^{81} + ( -5 \beta_{2} + 9 \beta_{4} - 19 \beta_{5} + 6 \beta_{6} ) q^{82} + ( 12 \beta_{2} + 5 \beta_{4} - 8 \beta_{5} + 32 \beta_{6} ) q^{83} + ( 1 - 14 \beta_{3} + 3 \beta_{7} ) q^{85} + ( -19 + 18 \beta_{1} - 48 \beta_{3} + 3 \beta_{7} ) q^{86} + ( -15 \beta_{4} + 20 \beta_{5} - 12 \beta_{6} ) q^{87} + ( -25 - 10 \beta_{1} - 17 \beta_{3} - 5 \beta_{7} ) q^{88} + ( -10 \beta_{2} + 11 \beta_{4} - 24 \beta_{5} - 22 \beta_{6} ) q^{89} + ( -3 \beta_{2} + 3 \beta_{4} - 3 \beta_{6} ) q^{90} + ( -11 + 26 \beta_{1} - 53 \beta_{3} - 3 \beta_{7} ) q^{92} + ( -32 + 7 \beta_{1} - 3 \beta_{3} - 4 \beta_{7} ) q^{93} + ( \beta_{2} + 4 \beta_{4} - 5 \beta_{6} ) q^{94} + ( 24 + \beta_{1} + 14 \beta_{3} + 2 \beta_{7} ) q^{95} + ( \beta_{2} - 9 \beta_{4} + 13 \beta_{5} - 6 \beta_{6} ) q^{96} + ( -14 \beta_{2} - 10 \beta_{4} + 30 \beta_{5} + 26 \beta_{6} ) q^{97} + ( 18 + 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} + 12q^{4} - 32q^{8} - 24q^{9} + O(q^{10}) \) \( 8q - 4q^{2} + 12q^{4} - 32q^{8} - 24q^{9} - 40q^{11} + 4q^{16} + 12q^{18} - 16q^{22} - 124q^{23} - 40q^{25} - 100q^{29} - 72q^{32} - 36q^{36} + 160q^{37} + 24q^{39} + 352q^{43} + 36q^{44} + 164q^{46} + 20q^{50} - 36q^{51} + 152q^{53} + 80q^{58} + 120q^{60} - 4q^{64} + 120q^{65} - 368q^{67} + 164q^{71} + 96q^{72} + 280q^{74} - 240q^{78} + 412q^{79} + 72q^{81} - 60q^{85} - 356q^{86} - 248q^{88} - 288q^{92} - 252q^{93} + 240q^{95} + 120q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 13 x^{6} - 2 x^{5} + 91 x^{4} - 50 x^{3} + 190 x^{2} + 100 x + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{7} - 38 \nu^{6} + 87 \nu^{5} + 132 \nu^{4} - 161 \nu^{3} + 480 \nu^{2} + 300 \nu + 12250 \)\()/3090\)
\(\beta_{2}\)\(=\)\((\)\( 87 \nu^{7} - 24 \nu^{6} + 841 \nu^{5} + 1276 \nu^{4} + 10117 \nu^{3} + 4640 \nu^{2} + 46160 \nu + 13700 \)\()/21630\)
\(\beta_{3}\)\(=\)\((\)\( 87 \nu^{7} - 24 \nu^{6} + 841 \nu^{5} + 1276 \nu^{4} + 10117 \nu^{3} + 4640 \nu^{2} + 2900 \nu + 35330 \)\()/21630\)
\(\beta_{4}\)\(=\)\((\)\( 283 \nu^{7} - 451 \nu^{6} + 5139 \nu^{5} - 656 \nu^{4} + 40658 \nu^{3} + 12690 \nu^{2} + 158440 \nu + 44150 \)\()/32445\)
\(\beta_{5}\)\(=\)\((\)\( -137 \nu^{7} + 361 \nu^{6} - 1805 \nu^{5} + 1115 \nu^{4} - 11191 \nu^{3} + 16967 \nu^{2} - 21390 \nu + 15 \)\()/10815\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{7} + 32 \nu^{6} - 153 \nu^{5} + 142 \nu^{4} - 901 \nu^{3} + 1350 \nu^{2} - 1580 \nu + 50 \)\()/630\)
\(\beta_{7}\)\(=\)\((\)\( 423 \nu^{7} - 241 \nu^{6} + 4089 \nu^{5} + 6204 \nu^{4} + 34148 \nu^{3} + 22560 \nu^{2} + 14100 \nu + 61265 \)\()/10815\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{6} + 5 \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 9 \beta_{3} + \beta_{1} - 13\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{7} + 33 \beta_{6} - 45 \beta_{5} + 6 \beta_{4} + 17 \beta_{3} - 17 \beta_{2} + 11 \beta_{1} - 60\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(13 \beta_{7} + 63 \beta_{6} - 85 \beta_{5} + 39 \beta_{4} - 95 \beta_{3} - 95 \beta_{2} - 21 \beta_{1} + 167\)\()/2\)
\(\nu^{6}\)\(=\)\(34 \beta_{7} - 243 \beta_{3} - 121 \beta_{1} + 684\)
\(\nu^{7}\)\(=\)\((\)\(155 \beta_{7} - 933 \beta_{6} + 1215 \beta_{5} - 465 \beta_{4} - 1081 \beta_{3} + 1081 \beta_{2} - 311 \beta_{1} + 2141\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
−1.26021 2.18275i
−1.26021 + 2.18275i
−0.336732 0.583237i
−0.336732 + 0.583237i
0.836732 + 1.44926i
0.836732 1.44926i
1.76021 + 3.04878i
1.76021 3.04878i
−3.52043 1.73205i 8.39341 2.23607i 6.09756i 0 −15.4667 −3.00000 7.87192i
391.2 −3.52043 1.73205i 8.39341 2.23607i 6.09756i 0 −15.4667 −3.00000 7.87192i
391.3 −1.67346 1.73205i −1.19952 2.23607i 2.89852i 0 8.70121 −3.00000 3.74198i
391.4 −1.67346 1.73205i −1.19952 2.23607i 2.89852i 0 8.70121 −3.00000 3.74198i
391.5 0.673464 1.73205i −3.54645 2.23607i 1.16647i 0 −5.08226 −3.00000 1.50591i
391.6 0.673464 1.73205i −3.54645 2.23607i 1.16647i 0 −5.08226 −3.00000 1.50591i
391.7 2.52043 1.73205i 2.35256 2.23607i 4.36551i 0 −4.15226 −3.00000 5.63585i
391.8 2.52043 1.73205i 2.35256 2.23607i 4.36551i 0 −4.15226 −3.00000 5.63585i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.3.h.a 8
7.b odd 2 1 inner 735.3.h.a 8
7.c even 3 1 105.3.n.a 8
7.d odd 6 1 105.3.n.a 8
21.g even 6 1 315.3.w.a 8
21.h odd 6 1 315.3.w.a 8
35.i odd 6 1 525.3.o.l 8
35.j even 6 1 525.3.o.l 8
35.k even 12 2 525.3.s.h 16
35.l odd 12 2 525.3.s.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 7.c even 3 1
105.3.n.a 8 7.d odd 6 1
315.3.w.a 8 21.g even 6 1
315.3.w.a 8 21.h odd 6 1
525.3.o.l 8 35.i odd 6 1
525.3.o.l 8 35.j even 6 1
525.3.s.h 16 35.k even 12 2
525.3.s.h 16 35.l odd 12 2
735.3.h.a 8 1.a even 1 1 trivial
735.3.h.a 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2 T_{2}^{3} - 9 T_{2}^{2} - 10 T_{2} + 10 \) acting on \(S_{3}^{\mathrm{new}}(735, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 7 T^{2} + 14 T^{3} + 34 T^{4} + 56 T^{5} + 112 T^{6} + 128 T^{7} + 256 T^{8} )^{2} \)
$3$ \( ( 1 + 3 T^{2} )^{4} \)
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ 1
$11$ \( ( 1 + 20 T + 547 T^{2} + 6950 T^{3} + 103078 T^{4} + 840950 T^{5} + 8008627 T^{6} + 35431220 T^{7} + 214358881 T^{8} )^{2} \)
$13$ \( 1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 251281048928 T^{10} + 32487291694546 T^{12} - 4380040003026428 T^{14} + 665416609183179841 T^{16} \)
$17$ \( 1 - 1208 T^{2} + 774016 T^{4} - 348305048 T^{6} + 116530286494 T^{8} - 29090785914008 T^{10} + 5399347871453056 T^{12} - 703807662573551288 T^{14} + 48661191875666868481 T^{16} \)
$19$ \( 1 - 1196 T^{2} + 705850 T^{4} - 361350224 T^{6} + 155496381379 T^{8} - 47091522541904 T^{10} + 11987847972489850 T^{12} - 2647124643203128556 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( ( 1 + 62 T + 2347 T^{2} + 75764 T^{3} + 2007934 T^{4} + 40079156 T^{5} + 656786827 T^{6} + 9178225118 T^{7} + 78310985281 T^{8} )^{2} \)
$29$ \( ( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 13329850 T^{5} + 872784754 T^{6} + 29741166050 T^{7} + 500246412961 T^{8} )^{2} \)
$31$ \( 1 - 3890 T^{2} + 8819209 T^{4} - 13390045370 T^{6} + 15030304083796 T^{8} - 12365988090147770 T^{10} + 7521824313419004169 T^{12} - \)\(30\!\cdots\!90\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( ( 1 - 80 T + 5206 T^{2} - 193760 T^{3} + 8139931 T^{4} - 265257440 T^{5} + 9756882166 T^{6} - 205258112720 T^{7} + 3512479453921 T^{8} )^{2} \)
$41$ \( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} + \)\(39\!\cdots\!45\)\( T^{12} - \)\(22\!\cdots\!86\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 2107763852 T^{5} + 58177736617 T^{6} - 1112559896624 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 - 16718 T^{2} + 124152241 T^{4} - 535266039518 T^{6} + 1465949995987204 T^{8} - 2611927522981233758 T^{10} + \)\(29\!\cdots\!01\)\( T^{12} - \)\(19\!\cdots\!38\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 - 76 T + 8845 T^{2} - 557992 T^{3} + 33596848 T^{4} - 1567399528 T^{5} + 69791304445 T^{6} - 1684491445804 T^{7} + 62259690411361 T^{8} )^{2} \)
$59$ \( 1 - 11480 T^{2} + 76423744 T^{4} - 376265621240 T^{6} + 1444840128344926 T^{8} - 4559346364454347640 T^{10} + \)\(11\!\cdots\!24\)\( T^{12} - \)\(20\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( 1 - 9512 T^{2} + 35469436 T^{4} - 58631449112 T^{6} + 50920349318854 T^{8} - 811801722004343192 T^{10} + \)\(67\!\cdots\!16\)\( T^{12} - \)\(25\!\cdots\!52\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( ( 1 + 184 T + 27625 T^{2} + 2611588 T^{3} + 208237768 T^{4} + 11723418532 T^{5} + 556674717625 T^{6} + 16644342319096 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 4266006742 T^{5} + 309158511046 T^{6} - 10504223281522 T^{7} + 645753531245761 T^{8} )^{2} \)
$73$ \( 1 - 15422 T^{2} + 189396025 T^{4} - 1427786526086 T^{6} + 9122100606631828 T^{8} - 40546625864343014726 T^{10} + \)\(15\!\cdots\!25\)\( T^{12} - \)\(35\!\cdots\!62\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( ( 1 - 206 T + 36853 T^{2} - 4125578 T^{3} + 384207724 T^{4} - 25747732298 T^{5} + 1435427335093 T^{6} - 50076015837326 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} - \)\(10\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!80\)\( T^{12} - \)\(22\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( 1 - 39848 T^{2} + 820746880 T^{4} - 10893774296744 T^{6} + 101902892477590558 T^{8} - \)\(68\!\cdots\!04\)\( T^{10} + \)\(32\!\cdots\!80\)\( T^{12} - \)\(98\!\cdots\!08\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!16\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \)
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