Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3584,2,Mod(1793,3584)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3584.1793");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3584 = 2^{9} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3584.b (of order \(2\), degree \(1\), 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: | \(28.6183840844\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.46091337728.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14x^{6} + 35x^{4} + 16x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 14x^{6} + 35x^{4} + 16x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{6} + 14\nu^{4} + 35\nu^{2} + 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} + 14\nu^{5} + 35\nu^{3} + 16\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{6} - 41\nu^{4} - 92\nu^{2} - 22 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} + 41\nu^{5} + 92\nu^{3} + 26\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} - 41\nu^{5} - 92\nu^{3} - 22\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( 5\nu^{6} + 69\nu^{4} + 160\nu^{2} + 39 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{7} - 151\nu^{5} - 344\nu^{3} - 84\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} - \beta_{3} + 2\beta _1 - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{7} - 13\beta_{5} - 7\beta_{4} + 2\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 13\beta_{6} + 15\beta_{3} - 20\beta _1 + 63 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -26\beta_{7} + 147\beta_{5} + 65\beta_{4} - 20\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -147\beta_{6} - 175\beta_{3} + 212\beta _1 - 661 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 294\beta_{7} - 1619\beta_{5} - 681\beta_{4} + 212\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).
| \(n\) | \(1023\) | \(1025\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1793.1 |
|
0 | − | 2.93637i | 0 | 2.25526i | 0 | −1.00000 | 0 | −5.62226 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1793.2 | 0 | − | 2.53038i | 0 | 1.73998i | 0 | −1.00000 | 0 | −3.40281 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1793.3 | 0 | − | 0.889918i | 0 | − | 1.35748i | 0 | −1.00000 | 0 | 2.20805 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1793.4 | 0 | − | 0.427759i | 0 | − | 4.24777i | 0 | −1.00000 | 0 | 2.81702 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1793.5 | 0 | 0.427759i | 0 | 4.24777i | 0 | −1.00000 | 0 | 2.81702 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1793.6 | 0 | 0.889918i | 0 | 1.35748i | 0 | −1.00000 | 0 | 2.20805 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1793.7 | 0 | 2.53038i | 0 | − | 1.73998i | 0 | −1.00000 | 0 | −3.40281 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1793.8 | 0 | 2.93637i | 0 | − | 2.25526i | 0 | −1.00000 | 0 | −5.62226 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3584.2.b.e | 8 | |
| 4.b | odd | 2 | 1 | 3584.2.b.f | 8 | ||
| 8.b | even | 2 | 1 | inner | 3584.2.b.e | 8 | |
| 8.d | odd | 2 | 1 | 3584.2.b.f | 8 | ||
| 16.e | even | 4 | 2 | 3584.2.a.n | yes | 8 | |
| 16.f | odd | 4 | 2 | 3584.2.a.m | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3584.2.a.m | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 3584.2.a.n | yes | 8 | 16.e | even | 4 | 2 | |
| 3584.2.b.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 3584.2.b.e | 8 | 8.b | even | 2 | 1 | inner | |
| 3584.2.b.f | 8 | 4.b | odd | 2 | 1 | ||
| 3584.2.b.f | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):
|
\( T_{3}^{8} + 16T_{3}^{6} + 70T_{3}^{4} + 56T_{3}^{2} + 8 \)
|
|
\( T_{23}^{4} + 4T_{23}^{3} - 54T_{23}^{2} - 192T_{23} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 16 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{8} + 28 T^{6} + \cdots + 512 \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} + 60 T^{6} + \cdots + 6272 \)
|
| $13$ |
\( T^{8} + 44 T^{6} + \cdots + 512 \)
|
| $17$ |
\( (T^{4} + 4 T^{3} + \cdots + 544)^{2} \)
|
| $19$ |
\( T^{8} + 112 T^{6} + \cdots + 75272 \)
|
| $23$ |
\( (T^{4} + 4 T^{3} - 54 T^{2} + \cdots - 16)^{2} \)
|
| $29$ |
\( T^{8} + 72 T^{6} + \cdots + 512 \)
|
| $31$ |
\( (T^{2} - 8 T + 8)^{4} \)
|
| $37$ |
\( T^{8} + 264 T^{6} + \cdots + 5431808 \)
|
| $41$ |
\( (T^{4} - 8 T^{3} + \cdots + 1904)^{2} \)
|
| $43$ |
\( T^{8} + 252 T^{6} + \cdots + 9400448 \)
|
| $47$ |
\( (T^{4} - 104 T^{2} + \cdots + 2176)^{2} \)
|
| $53$ |
\( T^{8} + 216 T^{6} + \cdots + 8192 \)
|
| $59$ |
\( T^{8} + 112 T^{6} + \cdots + 40328 \)
|
| $61$ |
\( T^{8} + 188 T^{6} + \cdots + 32768 \)
|
| $67$ |
\( T^{8} + 140 T^{6} + \cdots + 147968 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 8704)^{2} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots + 544)^{2} \)
|
| $79$ |
\( (T^{4} - 36 T^{3} + \cdots + 3104)^{2} \)
|
| $83$ |
\( T^{8} + 464 T^{6} + \cdots + 15523592 \)
|
| $89$ |
\( (T^{4} - 20 T^{3} + \cdots - 19232)^{2} \)
|
| $97$ |
\( (T^{4} + 20 T^{3} + \cdots - 14624)^{2} \)
|
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