Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [704,2,Mod(63,704)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("704.63");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 704 = 2^{6} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 704.u (of order \(10\), degree \(4\), not 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: | \(5.62146830230\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 44) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 73 \nu^{15} - 155 \nu^{14} + 343 \nu^{13} - 483 \nu^{12} + 373 \nu^{11} + 28 \nu^{10} - 1002 \nu^{9} + \cdots + 768 ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{15} + 58 \nu^{14} - 126 \nu^{13} + 270 \nu^{12} - 376 \nu^{11} + 295 \nu^{10} + 32 \nu^{9} + \cdots + 6656 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 113 \nu^{15} - 299 \nu^{14} + 663 \nu^{13} - 1043 \nu^{12} + 1013 \nu^{11} - 316 \nu^{10} + \cdots - 6784 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37 \nu^{15} - 152 \nu^{14} + 334 \nu^{13} - 602 \nu^{12} + 704 \nu^{11} - 405 \nu^{10} - 498 \nu^{9} + \cdots - 9088 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 147 \nu^{15} + 497 \nu^{14} - 1101 \nu^{13} + 1881 \nu^{12} - 2063 \nu^{11} + 1012 \nu^{10} + \cdots + 22400 ) / 128 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} + \cdots + 6464 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 115 \nu^{15} + 541 \nu^{14} - 1193 \nu^{13} + 2213 \nu^{12} - 2683 \nu^{11} + 1672 \nu^{10} + \cdots + 37376 ) / 128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 167 \nu^{15} + 617 \nu^{14} - 1361 \nu^{13} + 2389 \nu^{12} - 2707 \nu^{11} + 1452 \nu^{10} + \cdots + 32384 ) / 128 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 225 \nu^{15} + 821 \nu^{14} - 1813 \nu^{13} + 3169 \nu^{12} - 3579 \nu^{11} + 1894 \nu^{10} + \cdots + 42240 ) / 128 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + \cdots + 56320 ) / 128 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 311 \nu^{15} + 1185 \nu^{14} - 2617 \nu^{13} + 4629 \nu^{12} - 5299 \nu^{11} + 2908 \nu^{10} + \cdots + 64896 ) / 128 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 263 \nu^{15} - 951 \nu^{14} + 2099 \nu^{13} - 3663 \nu^{12} + 4117 \nu^{11} - 2162 \nu^{10} + \cdots - 48192 ) / 64 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 607 \nu^{15} + 2231 \nu^{14} - 4923 \nu^{13} + 8623 \nu^{12} - 9749 \nu^{11} + 5186 \nu^{10} + \cdots + 115840 ) / 128 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 819 \nu^{15} - 2871 \nu^{14} + 6339 \nu^{13} - 10967 \nu^{12} + 12197 \nu^{11} - 6230 \nu^{10} + \cdots - 138880 ) / 128 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 935 \nu^{15} - 3323 \nu^{14} + 7335 \nu^{13} - 12731 \nu^{12} + 14225 \nu^{11} - 7342 \nu^{10} + \cdots - 163968 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{15} - 2\beta_{13} + \beta_{9} + \beta_{8} + \beta_{3} - 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots - \beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{14} + \beta_{13} - \beta_{12} - 4 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - 3 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + \cdots - 1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{15} - 4 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + 8 \beta_{8} + \cdots + 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 3 \beta_{15} + 6 \beta_{14} - 8 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 5 \beta_{8} + \cdots + 15 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 8 \beta_{15} + 3 \beta_{14} - 8 \beta_{13} - 8 \beta_{12} - 16 \beta_{11} + 5 \beta_{10} - 8 \beta_{9} + \cdots + 4 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 4 \beta_{15} - 5 \beta_{14} - 11 \beta_{13} + \beta_{12} + 2 \beta_{11} - 4 \beta_{10} + \beta_{8} + \cdots + 4 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 11 \beta_{15} - 8 \beta_{14} - 15 \beta_{13} + 2 \beta_{12} + \beta_{11} + 20 \beta_{9} + \beta_{8} + \cdots - 7 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 2 \beta_{14} + 6 \beta_{13} - 21 \beta_{12} - 15 \beta_{11} - 23 \beta_{10} - 19 \beta_{9} + \cdots + 14 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 15 \beta_{15} - 48 \beta_{14} - 18 \beta_{13} - 32 \beta_{12} - 48 \beta_{11} - 12 \beta_{10} - 23 \beta_{9} + \cdots - 9 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 44 \beta_{15} + 15 \beta_{14} - 10 \beta_{13} - 44 \beta_{12} - 22 \beta_{11} + 73 \beta_{10} + \cdots + 100 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 4 \beta_{15} + 49 \beta_{14} + 17 \beta_{13} - 45 \beta_{12} - 112 \beta_{11} - 48 \beta_{10} + \cdots - 48 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 99 \beta_{15} + 98 \beta_{14} - 45 \beta_{13} - 43 \beta_{11} + 80 \beta_{10} - 190 \beta_{9} + \cdots - 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).
| \(n\) | \(133\) | \(321\) | \(639\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{10}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | −1.70537 | + | 0.554109i | 0 | 2.39991 | + | 1.74363i | 0 | −0.815620 | + | 2.51022i | 0 | 0.174207 | − | 0.126569i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.2 | 0 | −0.704424 | + | 0.228881i | 0 | −1.09089 | − | 0.792578i | 0 | 0.503194 | − | 1.54867i | 0 | −1.98322 | + | 1.44090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.3 | 0 | 0.704424 | − | 0.228881i | 0 | −1.09089 | − | 0.792578i | 0 | −0.503194 | + | 1.54867i | 0 | −1.98322 | + | 1.44090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 63.4 | 0 | 1.70537 | − | 0.554109i | 0 | 2.39991 | + | 1.74363i | 0 | 0.815620 | − | 2.51022i | 0 | 0.174207 | − | 0.126569i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.1 | 0 | −1.59814 | + | 2.19965i | 0 | 0.720859 | − | 2.21858i | 0 | −1.04462 | + | 0.758960i | 0 | −1.35736 | − | 4.17752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.2 | 0 | −0.539857 | + | 0.743049i | 0 | −0.529876 | + | 1.63079i | 0 | −1.93399 | + | 1.40513i | 0 | 0.666375 | + | 2.05089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.3 | 0 | 0.539857 | − | 0.743049i | 0 | −0.529876 | + | 1.63079i | 0 | 1.93399 | − | 1.40513i | 0 | 0.666375 | + | 2.05089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.4 | 0 | 1.59814 | − | 2.19965i | 0 | 0.720859 | − | 2.21858i | 0 | 1.04462 | − | 0.758960i | 0 | −1.35736 | − | 4.17752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.1 | 0 | −1.59814 | − | 2.19965i | 0 | 0.720859 | + | 2.21858i | 0 | −1.04462 | − | 0.758960i | 0 | −1.35736 | + | 4.17752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.2 | 0 | −0.539857 | − | 0.743049i | 0 | −0.529876 | − | 1.63079i | 0 | −1.93399 | − | 1.40513i | 0 | 0.666375 | − | 2.05089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.3 | 0 | 0.539857 | + | 0.743049i | 0 | −0.529876 | − | 1.63079i | 0 | 1.93399 | + | 1.40513i | 0 | 0.666375 | − | 2.05089i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.4 | 0 | 1.59814 | + | 2.19965i | 0 | 0.720859 | + | 2.21858i | 0 | 1.04462 | + | 0.758960i | 0 | −1.35736 | + | 4.17752i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.1 | 0 | −1.70537 | − | 0.554109i | 0 | 2.39991 | − | 1.74363i | 0 | −0.815620 | − | 2.51022i | 0 | 0.174207 | + | 0.126569i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.2 | 0 | −0.704424 | − | 0.228881i | 0 | −1.09089 | + | 0.792578i | 0 | 0.503194 | + | 1.54867i | 0 | −1.98322 | − | 1.44090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.3 | 0 | 0.704424 | + | 0.228881i | 0 | −1.09089 | + | 0.792578i | 0 | −0.503194 | − | 1.54867i | 0 | −1.98322 | − | 1.44090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 447.4 | 0 | 1.70537 | + | 0.554109i | 0 | 2.39991 | − | 1.74363i | 0 | 0.815620 | + | 2.51022i | 0 | 0.174207 | + | 0.126569i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 11.d | odd | 10 | 1 | inner |
| 44.g | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 704.2.u.c | 16 | |
| 4.b | odd | 2 | 1 | inner | 704.2.u.c | 16 | |
| 8.b | even | 2 | 1 | 44.2.g.a | ✓ | 16 | |
| 8.d | odd | 2 | 1 | 44.2.g.a | ✓ | 16 | |
| 11.d | odd | 10 | 1 | inner | 704.2.u.c | 16 | |
| 24.f | even | 2 | 1 | 396.2.r.a | 16 | ||
| 24.h | odd | 2 | 1 | 396.2.r.a | 16 | ||
| 44.g | even | 10 | 1 | inner | 704.2.u.c | 16 | |
| 88.b | odd | 2 | 1 | 484.2.g.i | 16 | ||
| 88.g | even | 2 | 1 | 484.2.g.i | 16 | ||
| 88.k | even | 10 | 1 | 44.2.g.a | ✓ | 16 | |
| 88.k | even | 10 | 1 | 484.2.c.d | 16 | ||
| 88.k | even | 10 | 1 | 484.2.g.f | 16 | ||
| 88.k | even | 10 | 1 | 484.2.g.j | 16 | ||
| 88.l | odd | 10 | 1 | 484.2.c.d | 16 | ||
| 88.l | odd | 10 | 1 | 484.2.g.f | 16 | ||
| 88.l | odd | 10 | 1 | 484.2.g.i | 16 | ||
| 88.l | odd | 10 | 1 | 484.2.g.j | 16 | ||
| 88.o | even | 10 | 1 | 484.2.c.d | 16 | ||
| 88.o | even | 10 | 1 | 484.2.g.f | 16 | ||
| 88.o | even | 10 | 1 | 484.2.g.i | 16 | ||
| 88.o | even | 10 | 1 | 484.2.g.j | 16 | ||
| 88.p | odd | 10 | 1 | 44.2.g.a | ✓ | 16 | |
| 88.p | odd | 10 | 1 | 484.2.c.d | 16 | ||
| 88.p | odd | 10 | 1 | 484.2.g.f | 16 | ||
| 88.p | odd | 10 | 1 | 484.2.g.j | 16 | ||
| 264.r | odd | 10 | 1 | 396.2.r.a | 16 | ||
| 264.u | even | 10 | 1 | 396.2.r.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.2.g.a | ✓ | 16 | 8.b | even | 2 | 1 | |
| 44.2.g.a | ✓ | 16 | 8.d | odd | 2 | 1 | |
| 44.2.g.a | ✓ | 16 | 88.k | even | 10 | 1 | |
| 44.2.g.a | ✓ | 16 | 88.p | odd | 10 | 1 | |
| 396.2.r.a | 16 | 24.f | even | 2 | 1 | ||
| 396.2.r.a | 16 | 24.h | odd | 2 | 1 | ||
| 396.2.r.a | 16 | 264.r | odd | 10 | 1 | ||
| 396.2.r.a | 16 | 264.u | even | 10 | 1 | ||
| 484.2.c.d | 16 | 88.k | even | 10 | 1 | ||
| 484.2.c.d | 16 | 88.l | odd | 10 | 1 | ||
| 484.2.c.d | 16 | 88.o | even | 10 | 1 | ||
| 484.2.c.d | 16 | 88.p | odd | 10 | 1 | ||
| 484.2.g.f | 16 | 88.k | even | 10 | 1 | ||
| 484.2.g.f | 16 | 88.l | odd | 10 | 1 | ||
| 484.2.g.f | 16 | 88.o | even | 10 | 1 | ||
| 484.2.g.f | 16 | 88.p | odd | 10 | 1 | ||
| 484.2.g.i | 16 | 88.b | odd | 2 | 1 | ||
| 484.2.g.i | 16 | 88.g | even | 2 | 1 | ||
| 484.2.g.i | 16 | 88.l | odd | 10 | 1 | ||
| 484.2.g.i | 16 | 88.o | even | 10 | 1 | ||
| 484.2.g.j | 16 | 88.k | even | 10 | 1 | ||
| 484.2.g.j | 16 | 88.l | odd | 10 | 1 | ||
| 484.2.g.j | 16 | 88.o | even | 10 | 1 | ||
| 484.2.g.j | 16 | 88.p | odd | 10 | 1 | ||
| 704.2.u.c | 16 | 1.a | even | 1 | 1 | trivial | |
| 704.2.u.c | 16 | 4.b | odd | 2 | 1 | inner | |
| 704.2.u.c | 16 | 11.d | odd | 10 | 1 | inner | |
| 704.2.u.c | 16 | 44.g | even | 10 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - T_{3}^{14} + 42T_{3}^{12} - 253T_{3}^{10} + 675T_{3}^{8} - 357T_{3}^{6} + 432T_{3}^{4} - 319T_{3}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(704, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - T^{14} + \cdots + 121 \)
|
| $5$ |
\( (T^{8} - 3 T^{7} + \cdots + 256)^{2} \)
|
| $7$ |
\( T^{16} + 11 T^{14} + \cdots + 30976 \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( (T^{8} - 5 T^{7} + 16 T^{6} + \cdots + 16)^{2} \)
|
| $17$ |
\( (T^{8} + 5 T^{7} - 24 T^{6} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{16} + 106 T^{14} + \cdots + 1771561 \)
|
| $23$ |
\( (T^{8} + 88 T^{6} + \cdots + 2816)^{2} \)
|
| $29$ |
\( (T^{8} - 5 T^{7} + \cdots + 400)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 428888770816 \)
|
| $37$ |
\( (T^{8} + 9 T^{7} + \cdots + 13456)^{2} \)
|
| $41$ |
\( (T^{8} - 5 T^{7} + \cdots + 2588881)^{2} \)
|
| $43$ |
\( (T^{8} - 143 T^{6} + \cdots + 148016)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 1206517709056 \)
|
| $53$ |
\( (T^{8} + 19 T^{7} + \cdots + 15376)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 17080137481 \)
|
| $61$ |
\( (T^{8} - 5 T^{7} + \cdots + 633616)^{2} \)
|
| $67$ |
\( (T^{8} + 185 T^{6} + \cdots + 1760000)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 21908736256 \)
|
| $73$ |
\( (T^{8} + 15 T^{7} + \cdots + 6561)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 28606986496 \)
|
| $83$ |
\( T^{16} + \cdots + 435963075625 \)
|
| $89$ |
\( (T^{4} + 9 T^{3} + \cdots + 116)^{4} \)
|
| $97$ |
\( (T^{4} + 17 T^{3} + \cdots + 3721)^{4} \)
|
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