Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [588,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.361");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(183.682394985\) |
| 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: | no (minimal twist has level 84) |
| 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 | 13.5000 | − | 23.3827i | 0 | −50.0000 | − | 86.6025i | 0 | 0 | 0 | −364.500 | − | 631.333i | 0 | ||||||||||||||||||
| 373.1 | 0 | 13.5000 | + | 23.3827i | 0 | −50.0000 | + | 86.6025i | 0 | 0 | 0 | −364.500 | + | 631.333i | 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 | 588.8.i.f | 2 | |
| 7.b | odd | 2 | 1 | 588.8.i.c | 2 | ||
| 7.c | even | 3 | 1 | 84.8.a.a | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 588.8.i.f | 2 | |
| 7.d | odd | 6 | 1 | 588.8.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 588.8.i.c | 2 | ||
| 21.h | odd | 6 | 1 | 252.8.a.a | 1 | ||
| 28.g | odd | 6 | 1 | 336.8.a.j | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.8.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
| 252.8.a.a | 1 | 21.h | odd | 6 | 1 | ||
| 336.8.a.j | 1 | 28.g | odd | 6 | 1 | ||
| 588.8.a.c | 1 | 7.d | odd | 6 | 1 | ||
| 588.8.i.c | 2 | 7.b | odd | 2 | 1 | ||
| 588.8.i.c | 2 | 7.d | odd | 6 | 1 | ||
| 588.8.i.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.8.i.f | 2 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 100T_{5} + 10000 \)
acting on \(S_{8}^{\mathrm{new}}(588, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 27T + 729 \)
|
| $5$ |
\( T^{2} + 100T + 10000 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 2774 T + 7695076 \)
|
| $13$ |
\( (T + 3294)^{2} \)
|
| $17$ |
\( T^{2} + 5900 T + 34810000 \)
|
| $19$ |
\( T^{2} + 6644 T + 44142736 \)
|
| $23$ |
\( T^{2} + 1982 T + 3928324 \)
|
| $29$ |
\( (T + 208106)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 13874955264 \)
|
| $37$ |
\( T^{2} + \cdots + 112685090596 \)
|
| $41$ |
\( (T + 265488)^{2} \)
|
| $43$ |
\( (T + 93292)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 432327290256 \)
|
| $53$ |
\( T^{2} + \cdots + 370537603524 \)
|
| $59$ |
\( T^{2} + \cdots + 287424654400 \)
|
| $61$ |
\( T^{2} + \cdots + 3229532468100 \)
|
| $67$ |
\( T^{2} + \cdots + 4507876326976 \)
|
| $71$ |
\( (T + 1191214)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 1116044344900 \)
|
| $79$ |
\( T^{2} + \cdots + 996970298256 \)
|
| $83$ |
\( (T - 3898004)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 21366138011904 \)
|
| $97$ |
\( (T - 15287710)^{2} \)
|
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