Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,4,Mod(1,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.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: | \(19.1166188419\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| 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)
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 |
|
0 | 0 | 0 | −3.00000 | 0 | −4.00000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
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 | 324.4.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 324.4.a.b | yes | 1 | |
| 4.b | odd | 2 | 1 | 1296.4.a.d | 1 | ||
| 9.c | even | 3 | 2 | 324.4.e.f | 2 | ||
| 9.d | odd | 6 | 2 | 324.4.e.c | 2 | ||
| 12.b | even | 2 | 1 | 1296.4.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.4.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 324.4.a.b | yes | 1 | 3.b | odd | 2 | 1 | |
| 324.4.e.c | 2 | 9.d | odd | 6 | 2 | ||
| 324.4.e.f | 2 | 9.c | even | 3 | 2 | ||
| 1296.4.a.d | 1 | 4.b | odd | 2 | 1 | ||
| 1296.4.a.e | 1 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 3 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 3 \)
|
| $7$ |
\( T + 4 \)
|
| $11$ |
\( T - 24 \)
|
| $13$ |
\( T + 25 \)
|
| $17$ |
\( T - 21 \)
|
| $19$ |
\( T + 52 \)
|
| $23$ |
\( T + 168 \)
|
| $29$ |
\( T - 177 \)
|
| $31$ |
\( T + 124 \)
|
| $37$ |
\( T + 265 \)
|
| $41$ |
\( T + 426 \)
|
| $43$ |
\( T + 160 \)
|
| $47$ |
\( T - 540 \)
|
| $53$ |
\( T - 258 \)
|
| $59$ |
\( T + 528 \)
|
| $61$ |
\( T + 505 \)
|
| $67$ |
\( T + 244 \)
|
| $71$ |
\( T + 204 \)
|
| $73$ |
\( T + 397 \)
|
| $79$ |
\( T - 200 \)
|
| $83$ |
\( T - 540 \)
|
| $89$ |
\( T - 453 \)
|
| $97$ |
\( T - 290 \)
|
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