Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,2,Mod(1,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("322.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.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: | \(2.57118294509\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.316.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 6 ) / 2 \)
|
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 |
|
1.00000 | −2.24914 | 1.00000 | 4.24914 | −2.24914 | −1.00000 | 1.00000 | 2.05863 | 4.24914 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.14637 | 1.00000 | 0.853635 | 1.14637 | −1.00000 | 1.00000 | −1.68585 | 0.853635 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 3.10278 | 1.00000 | −1.10278 | 3.10278 | −1.00000 | 1.00000 | 6.62721 | −1.10278 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 322.2.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2898.2.a.be | 3 | ||
| 4.b | odd | 2 | 1 | 2576.2.a.w | 3 | ||
| 5.b | even | 2 | 1 | 8050.2.a.bh | 3 | ||
| 7.b | odd | 2 | 1 | 2254.2.a.p | 3 | ||
| 23.b | odd | 2 | 1 | 7406.2.a.x | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2254.2.a.p | 3 | 7.b | odd | 2 | 1 | ||
| 2576.2.a.w | 3 | 4.b | odd | 2 | 1 | ||
| 2898.2.a.be | 3 | 3.b | odd | 2 | 1 | ||
| 7406.2.a.x | 3 | 23.b | odd | 2 | 1 | ||
| 8050.2.a.bh | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\):
|
\( T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 8 \)
|
|
\( T_{5}^{3} - 4T_{5}^{2} - 2T_{5} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
|
| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
|
| $5$ |
\( T^{3} - 4 T^{2} + \cdots + 4 \)
|
| $7$ |
\( (T + 1)^{3} \)
|
| $11$ |
\( T^{3} + 4 T^{2} + \cdots - 16 \)
|
| $13$ |
\( T^{3} - 2 T^{2} + \cdots + 16 \)
|
| $17$ |
\( T^{3} - 8 T^{2} + \cdots + 44 \)
|
| $19$ |
\( T^{3} - 28T - 16 \)
|
| $23$ |
\( (T + 1)^{3} \)
|
| $29$ |
\( T^{3} + 10 T^{2} + \cdots - 352 \)
|
| $31$ |
\( T^{3} + 2 T^{2} + \cdots - 32 \)
|
| $37$ |
\( T^{3} + 10 T^{2} + \cdots - 344 \)
|
| $41$ |
\( T^{3} - 6 T^{2} + \cdots + 344 \)
|
| $43$ |
\( T^{3} + 4 T^{2} + \cdots - 128 \)
|
| $47$ |
\( T^{3} - 2 T^{2} + \cdots + 32 \)
|
| $53$ |
\( T^{3} + 2 T^{2} + \cdots + 136 \)
|
| $59$ |
\( T^{3} - 6 T^{2} + \cdots + 216 \)
|
| $61$ |
\( T^{3} + 8 T^{2} + \cdots - 524 \)
|
| $67$ |
\( T^{3} - 4 T^{2} + \cdots + 16 \)
|
| $71$ |
\( T^{3} + 28 T^{2} + \cdots + 64 \)
|
| $73$ |
\( T^{3} + 6 T^{2} + \cdots - 1448 \)
|
| $79$ |
\( T^{3} + 20 T^{2} + \cdots - 2432 \)
|
| $83$ |
\( T^{3} - 28 T^{2} + \cdots - 656 \)
|
| $89$ |
\( T^{3} - 34T + 76 \)
|
| $97$ |
\( T^{3} - 16 T^{2} + \cdots + 172 \)
|
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