Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3640,1,Mod(909,3640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3640, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3640.909");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3640.n (of order \(2\), degree \(1\), 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: | yes |
| Analytic conductor: | \(1.81659664598\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-14}, \sqrt{65})\) |
| Artin image: | $D_4$ |
| Artin field: | Galois closure of 4.0.203840.2 |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times\).
| \(n\) | \(521\) | \(561\) | \(911\) | \(1457\) | \(1821\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 909.1 |
|
1.00000 | 0 | 1.00000 | 1.00000 | 0 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 56.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-14}) \) |
| 65.d | even | 2 | 1 | RM by \(\Q(\sqrt{65}) \) |
| 3640.n | odd | 2 | 1 | CM by \(\Q(\sqrt{-910}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3640.1.n.d | yes | 1 |
| 5.b | even | 2 | 1 | 3640.1.n.a | ✓ | 1 | |
| 7.b | odd | 2 | 1 | 3640.1.n.c | yes | 1 | |
| 8.b | even | 2 | 1 | 3640.1.n.c | yes | 1 | |
| 13.b | even | 2 | 1 | 3640.1.n.a | ✓ | 1 | |
| 35.c | odd | 2 | 1 | 3640.1.n.b | yes | 1 | |
| 40.f | even | 2 | 1 | 3640.1.n.b | yes | 1 | |
| 56.h | odd | 2 | 1 | CM | 3640.1.n.d | yes | 1 |
| 65.d | even | 2 | 1 | RM | 3640.1.n.d | yes | 1 |
| 91.b | odd | 2 | 1 | 3640.1.n.b | yes | 1 | |
| 104.e | even | 2 | 1 | 3640.1.n.b | yes | 1 | |
| 280.c | odd | 2 | 1 | 3640.1.n.a | ✓ | 1 | |
| 455.h | odd | 2 | 1 | 3640.1.n.c | yes | 1 | |
| 520.p | even | 2 | 1 | 3640.1.n.c | yes | 1 | |
| 728.l | odd | 2 | 1 | 3640.1.n.a | ✓ | 1 | |
| 3640.n | odd | 2 | 1 | CM | 3640.1.n.d | yes | 1 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3640.1.n.a | ✓ | 1 | 5.b | even | 2 | 1 | |
| 3640.1.n.a | ✓ | 1 | 13.b | even | 2 | 1 | |
| 3640.1.n.a | ✓ | 1 | 280.c | odd | 2 | 1 | |
| 3640.1.n.a | ✓ | 1 | 728.l | odd | 2 | 1 | |
| 3640.1.n.b | yes | 1 | 35.c | odd | 2 | 1 | |
| 3640.1.n.b | yes | 1 | 40.f | even | 2 | 1 | |
| 3640.1.n.b | yes | 1 | 91.b | odd | 2 | 1 | |
| 3640.1.n.b | yes | 1 | 104.e | even | 2 | 1 | |
| 3640.1.n.c | yes | 1 | 7.b | odd | 2 | 1 | |
| 3640.1.n.c | yes | 1 | 8.b | even | 2 | 1 | |
| 3640.1.n.c | yes | 1 | 455.h | odd | 2 | 1 | |
| 3640.1.n.c | yes | 1 | 520.p | even | 2 | 1 | |
| 3640.1.n.d | yes | 1 | 1.a | even | 1 | 1 | trivial |
| 3640.1.n.d | yes | 1 | 56.h | odd | 2 | 1 | CM |
| 3640.1.n.d | yes | 1 | 65.d | even | 2 | 1 | RM |
| 3640.1.n.d | yes | 1 | 3640.n | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3640, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{61} + 2 \)
|
|
\( T_{83} + 2 \)
|
|
\( T_{137} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 1 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 1 \)
|
| $7$ |
\( T + 1 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T + 1 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T + 2 \)
|
| $67$ |
\( T \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T + 2 \)
|
| $83$ |
\( T + 2 \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T \)
|
| show more | |
| show less |