Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3248,1,Mod(1217,3248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3248, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("3248.1217");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3248 = 2^{4} \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3248.k (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: | yes |
| Analytic conductor: | \(1.62096316103\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 203) |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{29})\) |
| Artin image: | $D_4$ |
| Artin field: | Galois closure of 4.0.22736.2 |
| Stark unit: | Root of $x^{4} - 30332x^{3} + 41734x^{2} - 30332x + 1$ |
$q$-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 \(\left(\mathbb{Z}/3248\mathbb{Z}\right)^\times\).
| \(n\) | \(465\) | \(785\) | \(2031\) | \(2437\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(0\) | \(0\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1217.1 |
|
0 | 0 | 0 | 0 | 0 | 1.00000 | 0 | −1.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 29.b | even | 2 | 1 | RM by \(\Q(\sqrt{29}) \) |
| 203.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-203}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3248.1.k.b | 1 | |
| 4.b | odd | 2 | 1 | 203.1.c.a | ✓ | 1 | |
| 7.b | odd | 2 | 1 | CM | 3248.1.k.b | 1 | |
| 12.b | even | 2 | 1 | 1827.1.b.a | 1 | ||
| 28.d | even | 2 | 1 | 203.1.c.a | ✓ | 1 | |
| 28.f | even | 6 | 2 | 1421.1.i.a | 2 | ||
| 28.g | odd | 6 | 2 | 1421.1.i.a | 2 | ||
| 29.b | even | 2 | 1 | RM | 3248.1.k.b | 1 | |
| 84.h | odd | 2 | 1 | 1827.1.b.a | 1 | ||
| 116.d | odd | 2 | 1 | 203.1.c.a | ✓ | 1 | |
| 203.c | odd | 2 | 1 | CM | 3248.1.k.b | 1 | |
| 348.b | even | 2 | 1 | 1827.1.b.a | 1 | ||
| 812.c | even | 2 | 1 | 203.1.c.a | ✓ | 1 | |
| 812.n | odd | 6 | 2 | 1421.1.i.a | 2 | ||
| 812.s | even | 6 | 2 | 1421.1.i.a | 2 | ||
| 2436.j | odd | 2 | 1 | 1827.1.b.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 203.1.c.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
| 203.1.c.a | ✓ | 1 | 28.d | even | 2 | 1 | |
| 203.1.c.a | ✓ | 1 | 116.d | odd | 2 | 1 | |
| 203.1.c.a | ✓ | 1 | 812.c | even | 2 | 1 | |
| 1421.1.i.a | 2 | 28.f | even | 6 | 2 | ||
| 1421.1.i.a | 2 | 28.g | odd | 6 | 2 | ||
| 1421.1.i.a | 2 | 812.n | odd | 6 | 2 | ||
| 1421.1.i.a | 2 | 812.s | even | 6 | 2 | ||
| 1827.1.b.a | 1 | 12.b | even | 2 | 1 | ||
| 1827.1.b.a | 1 | 84.h | odd | 2 | 1 | ||
| 1827.1.b.a | 1 | 348.b | even | 2 | 1 | ||
| 1827.1.b.a | 1 | 2436.j | odd | 2 | 1 | ||
| 3248.1.k.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 3248.1.k.b | 1 | 7.b | odd | 2 | 1 | CM | |
| 3248.1.k.b | 1 | 29.b | even | 2 | 1 | RM | |
| 3248.1.k.b | 1 | 203.c | odd | 2 | 1 | CM | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{1}^{\mathrm{new}}(3248, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 1 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T - 2 \)
|
| $29$ |
\( T + 1 \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T - 2 \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T \)
|
| $67$ |
\( T - 2 \)
|
| $71$ |
\( T + 2 \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T \)
|
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