Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [588,4,Mod(361,588)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.i (of order \(3\), degree \(2\), 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: | \(34.6931230834\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(493\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
0 | −1.50000 | + | 2.59808i | 0 | 2.00000 | + | 3.46410i | 0 | 0 | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||||||||||
| 373.1 | 0 | −1.50000 | − | 2.59808i | 0 | 2.00000 | − | 3.46410i | 0 | 0 | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 588.4.i.b | 2 | |
| 3.b | odd | 2 | 1 | 1764.4.k.g | 2 | ||
| 7.b | odd | 2 | 1 | 588.4.i.g | 2 | ||
| 7.c | even | 3 | 1 | 588.4.a.e | yes | 1 | |
| 7.c | even | 3 | 1 | inner | 588.4.i.b | 2 | |
| 7.d | odd | 6 | 1 | 588.4.a.b | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 588.4.i.g | 2 | ||
| 21.c | even | 2 | 1 | 1764.4.k.j | 2 | ||
| 21.g | even | 6 | 1 | 1764.4.a.d | 1 | ||
| 21.g | even | 6 | 1 | 1764.4.k.j | 2 | ||
| 21.h | odd | 6 | 1 | 1764.4.a.i | 1 | ||
| 21.h | odd | 6 | 1 | 1764.4.k.g | 2 | ||
| 28.f | even | 6 | 1 | 2352.4.a.be | 1 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 588.4.a.b | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 588.4.a.e | yes | 1 | 7.c | even | 3 | 1 | |
| 588.4.i.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.4.i.b | 2 | 7.c | even | 3 | 1 | inner | |
| 588.4.i.g | 2 | 7.b | odd | 2 | 1 | ||
| 588.4.i.g | 2 | 7.d | odd | 6 | 1 | ||
| 1764.4.a.d | 1 | 21.g | even | 6 | 1 | ||
| 1764.4.a.i | 1 | 21.h | odd | 6 | 1 | ||
| 1764.4.k.g | 2 | 3.b | odd | 2 | 1 | ||
| 1764.4.k.g | 2 | 21.h | odd | 6 | 1 | ||
| 1764.4.k.j | 2 | 21.c | even | 2 | 1 | ||
| 1764.4.k.j | 2 | 21.g | even | 6 | 1 | ||
| 2352.4.a.g | 1 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.be | 1 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 4T_{5} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 3T + 9 \)
|
| $5$ |
\( T^{2} - 4T + 16 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 20T + 400 \)
|
| $13$ |
\( (T - 4)^{2} \)
|
| $17$ |
\( T^{2} - 24T + 576 \)
|
| $19$ |
\( T^{2} - 44T + 1936 \)
|
| $23$ |
\( T^{2} + 72T + 5184 \)
|
| $29$ |
\( (T + 38)^{2} \)
|
| $31$ |
\( T^{2} - 184T + 33856 \)
|
| $37$ |
\( T^{2} - 30T + 900 \)
|
| $41$ |
\( (T - 216)^{2} \)
|
| $43$ |
\( (T + 164)^{2} \)
|
| $47$ |
\( T^{2} - 520T + 270400 \)
|
| $53$ |
\( T^{2} - 146T + 21316 \)
|
| $59$ |
\( T^{2} - 460T + 211600 \)
|
| $61$ |
\( T^{2} - 628T + 394384 \)
|
| $67$ |
\( T^{2} + 556T + 309136 \)
|
| $71$ |
\( (T - 592)^{2} \)
|
| $73$ |
\( T^{2} - 1024 T + 1048576 \)
|
| $79$ |
\( T^{2} - 104T + 10816 \)
|
| $83$ |
\( (T - 324)^{2} \)
|
| $89$ |
\( T^{2} - 896T + 802816 \)
|
| $97$ |
\( (T - 920)^{2} \)
|
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