Properties

Label 605.6.a.p
Level $605$
Weight $6$
Character orbit 605.a
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,6,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + 14112735 x^{13} + 987264735 x^{12} - 1043264932 x^{11} + \cdots - 32708279373824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 20) q^{4} - 25 q^{5} + ( - \beta_{9} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 13) q^{6} + (\beta_{8} + \beta_{4} - \beta_1 - 8) q^{7} + (\beta_{6} + 2 \beta_{4} - \beta_{3} + 19 \beta_1 - 4) q^{8} + (\beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 78) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 20) q^{4} - 25 q^{5} + ( - \beta_{9} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 13) q^{6} + (\beta_{8} + \beta_{4} - \beta_1 - 8) q^{7} + (\beta_{6} + 2 \beta_{4} - \beta_{3} + 19 \beta_1 - 4) q^{8} + (\beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 78) q^{9} - 25 \beta_1 q^{10} + (\beta_{18} - \beta_{17} - \beta_{14} + \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{6} - \beta_{5} - 32 \beta_{4} + \cdots - 9) q^{12}+ \cdots + (65 \beta_{19} - 107 \beta_{18} - 58 \beta_{17} - 45 \beta_{16} - 37 \beta_{15} + \cdots + 20168) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 407 q^{4} - 500 q^{5} - 264 q^{6} - 167 q^{7} - 57 q^{8} + 1598 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{2} + 407 q^{4} - 500 q^{5} - 264 q^{6} - 167 q^{7} - 57 q^{8} + 1598 q^{9} - 25 q^{10} - 253 q^{12} - 769 q^{13} - 1045 q^{14} + 6963 q^{16} + 2989 q^{17} - 3775 q^{18} - 5828 q^{19} - 10175 q^{20} - 3310 q^{21} - 695 q^{23} - 16724 q^{24} + 12500 q^{25} - 7384 q^{26} + 5925 q^{27} + 3508 q^{28} - 11268 q^{29} + 6600 q^{30} - 11465 q^{31} + 9062 q^{32} + 1217 q^{34} + 4175 q^{35} + 112083 q^{36} - 3057 q^{37} - 13510 q^{38} - 13459 q^{39} + 1425 q^{40} + 839 q^{41} - 14772 q^{42} - 43671 q^{43} - 39950 q^{45} - 81471 q^{46} + 32245 q^{47} - 104315 q^{48} + 2959 q^{49} + 625 q^{50} - 69047 q^{51} - 42696 q^{52} + 27981 q^{53} - 61212 q^{54} - 28294 q^{56} - 79425 q^{57} + 37274 q^{58} - 56847 q^{59} + 6325 q^{60} - 85616 q^{61} - 38095 q^{62} - 100055 q^{63} - 18233 q^{64} + 19225 q^{65} - 31091 q^{67} + 83972 q^{68} - 48708 q^{69} + 26125 q^{70} - 106431 q^{71} - 350510 q^{72} - 117959 q^{73} - 154757 q^{74} - 451972 q^{76} + 348898 q^{78} - 215138 q^{79} - 174075 q^{80} + 75516 q^{81} - 127864 q^{82} - 66761 q^{83} - 521275 q^{84} - 74725 q^{85} - 32222 q^{86} + 5311 q^{87} + 270560 q^{89} + 94375 q^{90} - 269192 q^{91} - 461663 q^{92} + 9345 q^{93} - 479494 q^{94} + 145700 q^{95} - 1247523 q^{96} + 45338 q^{97} + 420757 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + 14112735 x^{13} + 987264735 x^{12} - 1043264932 x^{11} + \cdots - 32708279373824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!77 \nu^{19} + \cdots - 27\!\cdots\!12 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!37 \nu^{19} + \cdots + 66\!\cdots\!72 ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 93\!\cdots\!31 \nu^{19} + \cdots - 11\!\cdots\!36 ) / 39\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!17 \nu^{19} + \cdots - 85\!\cdots\!28 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!31 \nu^{19} + \cdots - 65\!\cdots\!08 ) / 98\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50\!\cdots\!33 \nu^{19} + \cdots - 28\!\cdots\!28 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 64\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71\!\cdots\!39 \nu^{19} + \cdots - 50\!\cdots\!68 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 94\!\cdots\!03 \nu^{19} + \cdots - 75\!\cdots\!16 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!35 \nu^{19} + \cdots + 86\!\cdots\!00 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!31 \nu^{19} + \cdots + 15\!\cdots\!36 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!85 \nu^{19} + \cdots - 30\!\cdots\!56 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 72\!\cdots\!51 \nu^{19} + \cdots - 66\!\cdots\!12 ) / 39\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!07 \nu^{19} + \cdots - 42\!\cdots\!24 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 68\!\cdots\!59 \nu^{19} + \cdots + 94\!\cdots\!48 ) / 22\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 80\!\cdots\!03 \nu^{19} + \cdots + 11\!\cdots\!12 ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 28\!\cdots\!09 \nu^{19} + \cdots + 20\!\cdots\!92 ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{4} - \beta_{3} + 83\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - 2 \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{11} + \beta_{10} + 5 \beta_{9} - 4 \beta_{8} - 2 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 116 \beta_{2} + 4 \beta _1 + 4310 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 8 \beta_{18} + 2 \beta_{17} + 5 \beta_{16} - 5 \beta_{15} + 9 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 26 \beta_{9} - 15 \beta_{8} - 4 \beta_{7} + 135 \beta_{6} + 10 \beta_{5} + 411 \beta_{4} - 283 \beta_{3} + 17 \beta_{2} + \cdots - 157 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 134 \beta_{19} - 361 \beta_{18} + 170 \beta_{17} - 140 \beta_{16} - 147 \beta_{15} + 29 \beta_{14} + 29 \beta_{13} - 4 \beta_{12} - 764 \beta_{11} + 133 \beta_{10} + 843 \beta_{9} - 643 \beta_{8} + \beta_{7} - 212 \beta_{6} + 453 \beta_{5} + \cdots + 401873 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 1616 \beta_{18} + 536 \beta_{17} + 1130 \beta_{16} - 892 \beta_{15} + 1670 \beta_{14} + 630 \beta_{13} + 1038 \beta_{12} + 502 \beta_{11} - 256 \beta_{10} + 5516 \beta_{9} - 2894 \beta_{8} - 696 \beta_{7} + \cdots + 7452 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13892 \beta_{19} - 49832 \beta_{18} + 22680 \beta_{17} - 15070 \beta_{16} - 17058 \beta_{15} + 5432 \beta_{14} + 5716 \beta_{13} - 2036 \beta_{12} - 127946 \beta_{11} + 15524 \beta_{10} + 110520 \beta_{9} + \cdots + 39541608 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12481 \beta_{19} - 249102 \beta_{18} + 93275 \beta_{17} + 181555 \beta_{16} - 125929 \beta_{15} + 231272 \beta_{14} + 103252 \beta_{13} + 191668 \beta_{12} + 42809 \beta_{11} - 20929 \beta_{10} + \cdots + 5447998 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1337413 \beta_{19} - 6321964 \beta_{18} + 2797532 \beta_{17} - 1442413 \beta_{16} - 1864295 \beta_{15} + 736899 \beta_{14} + 840533 \beta_{13} - 470075 \beta_{12} - 18279883 \beta_{11} + \cdots + 4011058321 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2692252 \beta_{19} - 34661771 \beta_{18} + 13649034 \beta_{17} + 25644280 \beta_{16} - 16443653 \beta_{15} + 28798645 \beta_{14} + 15100477 \beta_{13} + 28510850 \beta_{12} + \cdots + 1351708001 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 125800466 \beta_{19} - 774466126 \beta_{18} + 333510036 \beta_{17} - 125448632 \beta_{16} - 201681424 \beta_{15} + 88838464 \beta_{14} + 112673860 \beta_{13} + \cdots + 415378775286 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 405101796 \beta_{19} - 4579359616 \beta_{18} + 1827374616 \beta_{17} + 3397046160 \beta_{16} - 2068533572 \beta_{15} + 3410940172 \beta_{14} + \cdots + 253404682644 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11781256913 \beta_{19} - 93275343778 \beta_{18} + 39078781481 \beta_{17} - 9572480593 \beta_{16} - 21992374457 \beta_{15} + 10172693176 \beta_{14} + \cdots + 43701921820630 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 52873075427 \beta_{19} - 587133723344 \beta_{18} + 232550480486 \beta_{17} + 433532994865 \beta_{16} - 254684368521 \beta_{15} + 393431647329 \beta_{14} + \cdots + 41223854994291 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1107156456910 \beta_{19} - 11130736807385 \beta_{18} + 4536158683702 \beta_{17} - 547555567636 \beta_{16} - 2433121227275 \beta_{15} + \cdots + 46\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6397086175894 \beta_{19} - 73874705672452 \beta_{18} + 28704694377240 \beta_{17} + 54024106275830 \beta_{16} - 30932585818436 \beta_{15} + \cdots + 61\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 104749406552836 \beta_{19} + \cdots + 50\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 738711912072257 \beta_{19} + \cdots + 86\!\cdots\!62 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.4229
−10.1575
−9.60667
−9.34075
−6.92807
−5.73355
−5.37088
−4.73032
−1.05592
0.238351
1.08038
1.84684
3.12060
5.27089
5.77720
8.03981
8.98229
8.99127
10.1332
10.8658
−10.4229 −11.0208 76.6377 −25.0000 114.870 −4.64894 −465.256 −121.541 260.573
1.2 −10.1575 30.4037 71.1747 −25.0000 −308.825 10.6636 −397.917 681.384 253.937
1.3 −9.60667 21.6000 60.2880 −25.0000 −207.504 −51.1649 −271.754 223.560 240.167
1.4 −9.34075 −29.7904 55.2496 −25.0000 278.265 104.862 −217.169 644.469 233.519
1.5 −6.92807 −8.76999 15.9982 −25.0000 60.7591 −97.5351 110.862 −166.087 173.202
1.6 −5.73355 3.90196 0.873636 −25.0000 −22.3721 −214.940 178.465 −227.775 143.339
1.7 −5.37088 −0.268292 −3.15366 −25.0000 1.44096 221.735 188.806 −242.928 134.272
1.8 −4.73032 8.08839 −9.62406 −25.0000 −38.2607 159.979 196.895 −177.578 118.258
1.9 −1.05592 22.6862 −30.8850 −25.0000 −23.9549 12.2465 66.4016 271.665 26.3980
1.10 0.238351 −19.5755 −31.9432 −25.0000 −4.66584 −208.743 −15.2409 140.200 −5.95878
1.11 1.08038 9.44854 −30.8328 −25.0000 10.2080 −16.3713 −67.8833 −153.725 −27.0095
1.12 1.84684 −21.5804 −28.5892 −25.0000 −39.8555 −85.4988 −111.898 222.716 −46.1709
1.13 3.12060 −6.47958 −22.2618 −25.0000 −20.2202 −46.8054 −169.330 −201.015 −78.0151
1.14 5.27089 −17.5243 −4.21770 −25.0000 −92.3688 132.134 −190.900 64.1015 −131.772
1.15 5.77720 22.4828 1.37599 −25.0000 129.888 −56.5871 −176.921 262.477 −144.430
1.16 8.03981 5.09012 32.6385 −25.0000 40.9236 87.3874 5.13363 −217.091 −200.995
1.17 8.98229 23.5092 48.6815 −25.0000 211.166 −232.955 149.838 309.680 −224.557
1.18 8.99127 10.3783 48.8429 −25.0000 93.3145 132.461 151.440 −135.290 −224.782
1.19 10.1332 −21.6453 70.6821 −25.0000 −219.336 136.893 391.974 225.517 −253.330
1.20 10.8658 −20.9346 86.0646 −25.0000 −227.471 −150.113 587.453 195.259 −271.644
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)

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 605.6.a.p 20
11.b odd 2 1 605.6.a.o 20
11.c even 5 2 55.6.g.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.g.b 40 11.c even 5 2
605.6.a.o 20 11.b odd 2 1
605.6.a.p 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - T_{2}^{19} - 523 T_{2}^{18} + 521 T_{2}^{17} + 115018 T_{2}^{16} - 115347 T_{2}^{15} - 13821739 T_{2}^{14} + 14112735 T_{2}^{13} + 987264735 T_{2}^{12} - 1043264932 T_{2}^{11} + \cdots - 32708279373824 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots - 32708279373824 \) Copy content Toggle raw display
$3$ \( T^{20} - 3229 T^{18} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( (T + 25)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 167 T^{19} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 769 T^{19} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} - 2989 T^{19} + \cdots - 17\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + 5828 T^{19} + \cdots + 49\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{20} + 695 T^{19} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + 11268 T^{19} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + 11465 T^{19} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + 3057 T^{19} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{20} - 839 T^{19} + \cdots - 35\!\cdots\!59 \) Copy content Toggle raw display
$43$ \( T^{20} + 43671 T^{19} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} - 32245 T^{19} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{20} - 27981 T^{19} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + 56847 T^{19} + \cdots - 32\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + 85616 T^{19} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{20} + 31091 T^{19} + \cdots - 27\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + 106431 T^{19} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{20} + 117959 T^{19} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + 215138 T^{19} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + 66761 T^{19} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} - 270560 T^{19} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} - 45338 T^{19} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
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