Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,2,Mod(131,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.l (of order \(5\), degree \(4\), minimal) |
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: | no |
| Analytic conductor: | \(5.19027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 9 x^{19} + 43 x^{18} - 132 x^{17} + 291 x^{16} - 480 x^{15} + 624 x^{14} - 653 x^{13} + \cdots + 31 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} - 9 x^{19} + 43 x^{18} - 132 x^{17} + 291 x^{16} - 480 x^{15} + 624 x^{14} - 653 x^{13} + \cdots + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 41602842506613 \nu^{19} + 310044321766184 \nu^{18} + \cdots - 23\!\cdots\!85 ) / 12\!\cdots\!75 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19\!\cdots\!17 \nu^{19} + \cdots + 62\!\cdots\!62 ) / 36\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11\!\cdots\!06 \nu^{19} + \cdots + 12\!\cdots\!78 ) / 36\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17\!\cdots\!19 \nu^{19} + \cdots - 60\!\cdots\!03 ) / 36\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 18\!\cdots\!64 \nu^{19} + \cdots - 23\!\cdots\!10 ) / 36\!\cdots\!75 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 32\!\cdots\!76 \nu^{19} + \cdots + 20\!\cdots\!67 ) / 36\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 32\!\cdots\!58 \nu^{19} + \cdots + 98\!\cdots\!48 ) / 36\!\cdots\!75 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 56\!\cdots\!52 \nu^{19} + \cdots + 28\!\cdots\!12 ) / 60\!\cdots\!75 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 35\!\cdots\!83 \nu^{19} + \cdots + 13\!\cdots\!94 ) / 36\!\cdots\!75 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 39\!\cdots\!01 \nu^{19} + \cdots + 35\!\cdots\!68 ) / 36\!\cdots\!75 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 40\!\cdots\!11 \nu^{19} + \cdots + 23\!\cdots\!92 ) / 36\!\cdots\!75 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 62\!\cdots\!48 \nu^{19} + \cdots + 75\!\cdots\!37 ) / 36\!\cdots\!75 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 65\!\cdots\!33 \nu^{19} + \cdots - 20\!\cdots\!74 ) / 36\!\cdots\!75 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 66\!\cdots\!95 \nu^{19} + \cdots + 12\!\cdots\!63 ) / 36\!\cdots\!75 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 84\!\cdots\!50 \nu^{19} + \cdots + 28\!\cdots\!73 ) / 36\!\cdots\!75 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 12\!\cdots\!88 \nu^{19} + \cdots + 19\!\cdots\!80 ) / 36\!\cdots\!75 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 12\!\cdots\!05 \nu^{19} + \cdots + 32\!\cdots\!78 ) / 36\!\cdots\!75 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 13\!\cdots\!75 \nu^{19} + \cdots + 47\!\cdots\!12 ) / 36\!\cdots\!75 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 20\!\cdots\!76 \nu^{19} + \cdots - 60\!\cdots\!65 ) / 36\!\cdots\!75 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} + 3 \beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{13} + 2 \beta_{11} + \cdots - 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} + 4 \beta_{18} - 6 \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + 6 \beta_{13} + \cdots - 5 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4 \beta_{18} - 15 \beta_{17} - 3 \beta_{16} - 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} + \cdots - 11 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7 \beta_{19} + 3 \beta_{18} - 17 \beta_{17} + 4 \beta_{16} - 8 \beta_{15} - 16 \beta_{14} + 17 \beta_{13} + \cdots - 30 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 54 \beta_{19} + 26 \beta_{17} + 15 \beta_{16} - 36 \beta_{15} - 24 \beta_{14} + 15 \beta_{13} + \cdots - 51 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 158 \beta_{19} - 48 \beta_{18} + 157 \beta_{17} - 29 \beta_{16} - 62 \beta_{15} + \beta_{14} - 17 \beta_{13} + \cdots + 10 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 223 \beta_{19} - 224 \beta_{18} + 362 \beta_{17} - 232 \beta_{16} + 63 \beta_{15} + 119 \beta_{14} + \cdots + 329 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 186 \beta_{19} - 449 \beta_{18} + 351 \beta_{17} - 557 \beta_{16} + 734 \beta_{15} + 443 \beta_{14} + \cdots + 1190 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 1828 \beta_{19} + 52 \beta_{18} - 667 \beta_{17} - 434 \beta_{16} + 2272 \beta_{15} + 1007 \beta_{14} + \cdots + 2144 ) / 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 948 \beta_{19} + 583 \beta_{18} - 747 \beta_{17} + 297 \beta_{16} + 671 \beta_{15} + 258 \beta_{14} + \cdots - 12 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 5169 \beta_{19} + 7923 \beta_{18} - 7946 \beta_{17} + 6234 \beta_{16} - 1904 \beta_{15} + \cdots - 12246 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 8941 \beta_{19} + 5134 \beta_{18} - 3671 \beta_{17} + 10732 \beta_{16} - 24589 \beta_{15} - 9478 \beta_{14} + \cdots - 35965 ) / 5 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 56200 \beta_{19} - 31776 \beta_{18} + 37730 \beta_{17} + 2157 \beta_{16} - 67675 \beta_{15} + \cdots - 38451 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 128682 \beta_{19} - 126572 \beta_{18} + 155413 \beta_{17} - 43481 \beta_{16} - 80008 \beta_{15} + \cdots + 87835 ) / 5 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 86219 \beta_{19} - 216840 \beta_{18} + 301811 \beta_{17} - 140695 \beta_{16} + 120284 \beta_{15} + \cdots + 501014 ) / 5 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 506172 \beta_{19} + 11457 \beta_{18} + 65602 \beta_{17} - 204689 \beta_{16} + 838468 \beta_{15} + \cdots + 1116355 ) / 5 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 2274617 \beta_{19} + 1250846 \beta_{18} - 1652148 \beta_{17} + 172408 \beta_{16} + 2025878 \beta_{15} + \cdots + 755289 ) / 5 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 4661261 \beta_{19} + 4121881 \beta_{18} - 6034594 \beta_{17} + 1944033 \beta_{16} + 1685959 \beta_{15} + \cdots - 4084120 ) / 5 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 1916403 \beta_{19} + 6797252 \beta_{18} - 10457397 \beta_{17} + 5838451 \beta_{16} - 6671978 \beta_{15} + \cdots - 18425246 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-\beta_{13}\) | \(1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 131.1 |
|
−0.309017 | − | 0.951057i | −1.69698 | + | 1.23293i | −0.809017 | + | 0.587785i | −1.53931 | + | 1.62189i | 1.69698 | + | 1.23293i | −3.44863 | 0.809017 | + | 0.587785i | 0.432586 | − | 1.33136i | 2.01818 | + | 0.962774i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.2 | −0.309017 | − | 0.951057i | −1.47354 | + | 1.07059i | −0.809017 | + | 0.587785i | 1.58287 | + | 1.57940i | 1.47354 | + | 1.07059i | 2.37656 | 0.809017 | + | 0.587785i | 0.0981028 | − | 0.301929i | 1.01296 | − | 1.99347i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.3 | −0.309017 | − | 0.951057i | −0.286240 | + | 0.207966i | −0.809017 | + | 0.587785i | −2.22306 | + | 0.240824i | 0.286240 | + | 0.207966i | 2.80842 | 0.809017 | + | 0.587785i | −0.888367 | + | 2.73411i | 0.916001 | + | 2.03984i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.4 | −0.309017 | − | 0.951057i | −0.126245 | + | 0.0917222i | −0.809017 | + | 0.587785i | −1.23420 | − | 1.86460i | 0.126245 | + | 0.0917222i | −2.41858 | 0.809017 | + | 0.587785i | −0.919526 | + | 2.83001i | −1.39195 | + | 1.74999i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 131.5 | −0.309017 | − | 0.951057i | 1.46497 | − | 1.06437i | −0.809017 | + | 0.587785i | −0.204340 | + | 2.22671i | −1.46497 | − | 1.06437i | 2.68223 | 0.809017 | + | 0.587785i | 0.0862215 | − | 0.265362i | 2.18087 | − | 0.493753i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 261.1 | 0.809017 | − | 0.587785i | −0.777911 | + | 2.39416i | 0.309017 | − | 0.951057i | 2.08891 | + | 0.797773i | 0.777911 | + | 2.39416i | 1.47068 | −0.309017 | − | 0.951057i | −2.69982 | − | 1.96153i | 2.15889 | − | 0.582420i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 261.2 | 0.809017 | − | 0.587785i | −0.311946 | + | 0.960072i | 0.309017 | − | 0.951057i | −2.22385 | + | 0.233444i | 0.311946 | + | 0.960072i | −0.400732 | −0.309017 | − | 0.951057i | 1.60262 | + | 1.16437i | −1.66192 | + | 1.49601i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 261.3 | 0.809017 | − | 0.587785i | −0.0885960 | + | 0.272670i | 0.309017 | − | 0.951057i | −0.642903 | − | 2.14165i | 0.0885960 | + | 0.272670i | 3.26367 | −0.309017 | − | 0.951057i | 2.36055 | + | 1.71504i | −1.77895 | − | 1.35474i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 261.4 | 0.809017 | − | 0.587785i | 0.370600 | − | 1.14059i | 0.309017 | − | 0.951057i | 1.06050 | + | 1.96859i | −0.370600 | − | 1.14059i | 1.46282 | −0.309017 | − | 0.951057i | 1.26345 | + | 0.917949i | 2.01507 | + | 0.969274i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 261.5 | 0.809017 | − | 0.587785i | 0.925886 | − | 2.84959i | 0.309017 | − | 0.951057i | −1.66463 | + | 1.49299i | −0.925886 | − | 2.84959i | −3.79644 | −0.309017 | − | 0.951057i | −4.83582 | − | 3.51343i | −0.469156 | + | 2.18630i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 391.1 | 0.809017 | + | 0.587785i | −0.777911 | − | 2.39416i | 0.309017 | + | 0.951057i | 2.08891 | − | 0.797773i | 0.777911 | − | 2.39416i | 1.47068 | −0.309017 | + | 0.951057i | −2.69982 | + | 1.96153i | 2.15889 | + | 0.582420i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 391.2 | 0.809017 | + | 0.587785i | −0.311946 | − | 0.960072i | 0.309017 | + | 0.951057i | −2.22385 | − | 0.233444i | 0.311946 | − | 0.960072i | −0.400732 | −0.309017 | + | 0.951057i | 1.60262 | − | 1.16437i | −1.66192 | − | 1.49601i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 391.3 | 0.809017 | + | 0.587785i | −0.0885960 | − | 0.272670i | 0.309017 | + | 0.951057i | −0.642903 | + | 2.14165i | 0.0885960 | − | 0.272670i | 3.26367 | −0.309017 | + | 0.951057i | 2.36055 | − | 1.71504i | −1.77895 | + | 1.35474i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 391.4 | 0.809017 | + | 0.587785i | 0.370600 | + | 1.14059i | 0.309017 | + | 0.951057i | 1.06050 | − | 1.96859i | −0.370600 | + | 1.14059i | 1.46282 | −0.309017 | + | 0.951057i | 1.26345 | − | 0.917949i | 2.01507 | − | 0.969274i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 391.5 | 0.809017 | + | 0.587785i | 0.925886 | + | 2.84959i | 0.309017 | + | 0.951057i | −1.66463 | − | 1.49299i | −0.925886 | + | 2.84959i | −3.79644 | −0.309017 | + | 0.951057i | −4.83582 | + | 3.51343i | −0.469156 | − | 2.18630i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.1 | −0.309017 | + | 0.951057i | −1.69698 | − | 1.23293i | −0.809017 | − | 0.587785i | −1.53931 | − | 1.62189i | 1.69698 | − | 1.23293i | −3.44863 | 0.809017 | − | 0.587785i | 0.432586 | + | 1.33136i | 2.01818 | − | 0.962774i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.2 | −0.309017 | + | 0.951057i | −1.47354 | − | 1.07059i | −0.809017 | − | 0.587785i | 1.58287 | − | 1.57940i | 1.47354 | − | 1.07059i | 2.37656 | 0.809017 | − | 0.587785i | 0.0981028 | + | 0.301929i | 1.01296 | + | 1.99347i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.3 | −0.309017 | + | 0.951057i | −0.286240 | − | 0.207966i | −0.809017 | − | 0.587785i | −2.22306 | − | 0.240824i | 0.286240 | − | 0.207966i | 2.80842 | 0.809017 | − | 0.587785i | −0.888367 | − | 2.73411i | 0.916001 | − | 2.03984i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.4 | −0.309017 | + | 0.951057i | −0.126245 | − | 0.0917222i | −0.809017 | − | 0.587785i | −1.23420 | + | 1.86460i | 0.126245 | − | 0.0917222i | −2.41858 | 0.809017 | − | 0.587785i | −0.919526 | − | 2.83001i | −1.39195 | − | 1.74999i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 521.5 | −0.309017 | + | 0.951057i | 1.46497 | + | 1.06437i | −0.809017 | − | 0.587785i | −0.204340 | − | 2.22671i | −1.46497 | + | 1.06437i | 2.68223 | 0.809017 | − | 0.587785i | 0.0862215 | + | 0.265362i | 2.18087 | + | 0.493753i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 650.2.l.b | ✓ | 20 |
| 25.d | even | 5 | 1 | inner | 650.2.l.b | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.2.l.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 650.2.l.b | ✓ | 20 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 4 T_{3}^{19} + 19 T_{3}^{18} + 59 T_{3}^{17} + 174 T_{3}^{16} + 367 T_{3}^{15} + 676 T_{3}^{14} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \)
|
| $3$ |
\( T^{20} + 4 T^{19} + \cdots + 1 \)
|
| $5$ |
\( T^{20} + 10 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( (T^{10} - 4 T^{9} + \cdots + 1595)^{2} \)
|
| $11$ |
\( T^{20} + 10 T^{18} + \cdots + 121 \)
|
| $13$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \)
|
| $17$ |
\( T^{20} + \cdots + 100180081 \)
|
| $19$ |
\( T^{20} + 4 T^{19} + \cdots + 297025 \)
|
| $23$ |
\( T^{20} + 17 T^{19} + \cdots + 4116841 \)
|
| $29$ |
\( T^{20} + \cdots + 32042790025 \)
|
| $31$ |
\( T^{20} + 2 T^{19} + \cdots + 3222025 \)
|
| $37$ |
\( T^{20} + \cdots + 20992822321 \)
|
| $41$ |
\( T^{20} + \cdots + 34279795461025 \)
|
| $43$ |
\( (T^{10} - T^{9} + \cdots + 3448001)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 2715556321 \)
|
| $53$ |
\( T^{20} + \cdots + 10\!\cdots\!41 \)
|
| $59$ |
\( T^{20} + \cdots + 2003937516025 \)
|
| $61$ |
\( T^{20} + \cdots + 99\!\cdots\!61 \)
|
| $67$ |
\( T^{20} + \cdots + 57274699680025 \)
|
| $71$ |
\( T^{20} + \cdots + 45\!\cdots\!25 \)
|
| $73$ |
\( T^{20} + \cdots + 83\!\cdots\!21 \)
|
| $79$ |
\( T^{20} + \cdots + 44\!\cdots\!25 \)
|
| $83$ |
\( T^{20} + \cdots + 10538049025 \)
|
| $89$ |
\( T^{20} + \cdots + 46\!\cdots\!25 \)
|
| $97$ |
\( T^{20} + \cdots + 16832713467361 \)
|
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