Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,Mod(177,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.177");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.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: | \(23.1287487223\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 177.1 |
|
0 | −2.82843 | − | 4.89898i | 0 | −2.82843 | + | 4.89898i | 0 | 0 | 0 | −2.50000 | + | 4.33013i | 0 | ||||||||||||||||||||||||
| 177.2 | 0 | 2.82843 | + | 4.89898i | 0 | 2.82843 | − | 4.89898i | 0 | 0 | 0 | −2.50000 | + | 4.33013i | 0 | |||||||||||||||||||||||||
| 361.1 | 0 | −2.82843 | + | 4.89898i | 0 | −2.82843 | − | 4.89898i | 0 | 0 | 0 | −2.50000 | − | 4.33013i | 0 | |||||||||||||||||||||||||
| 361.2 | 0 | 2.82843 | − | 4.89898i | 0 | 2.82843 | + | 4.89898i | 0 | 0 | 0 | −2.50000 | − | 4.33013i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.4.i.j | 4 | |
| 7.b | odd | 2 | 1 | inner | 392.4.i.j | 4 | |
| 7.c | even | 3 | 1 | 392.4.a.g | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 392.4.i.j | 4 | |
| 7.d | odd | 6 | 1 | 392.4.a.g | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 392.4.i.j | 4 | |
| 28.f | even | 6 | 1 | 784.4.a.x | 2 | ||
| 28.g | odd | 6 | 1 | 784.4.a.x | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 392.4.a.g | ✓ | 2 | 7.c | even | 3 | 1 | |
| 392.4.a.g | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 392.4.i.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 392.4.i.j | 4 | 7.b | odd | 2 | 1 | inner | |
| 392.4.i.j | 4 | 7.c | even | 3 | 1 | inner | |
| 392.4.i.j | 4 | 7.d | odd | 6 | 1 | inner | |
| 784.4.a.x | 2 | 28.f | even | 6 | 1 | ||
| 784.4.a.x | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3}^{4} + 32T_{3}^{2} + 1024 \)
|
|
\( T_{5}^{4} + 32T_{5}^{2} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 32T^{2} + 1024 \)
|
| $5$ |
\( T^{4} + 32T^{2} + 1024 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $13$ |
\( (T^{2} - 32)^{2} \)
|
| $17$ |
\( T^{4} + 15488 T^{2} + 239878144 \)
|
| $19$ |
\( T^{4} + 16928 T^{2} + 286557184 \)
|
| $23$ |
\( (T^{2} + 120 T + 14400)^{2} \)
|
| $29$ |
\( (T - 218)^{4} \)
|
| $31$ |
\( T^{4} + 21632 T^{2} + 467943424 \)
|
| $37$ |
\( (T^{2} + 130 T + 16900)^{2} \)
|
| $41$ |
\( (T^{2} - 21632)^{2} \)
|
| $43$ |
\( (T - 332)^{4} \)
|
| $47$ |
\( T^{4} + 15488 T^{2} + 239878144 \)
|
| $53$ |
\( (T^{2} - 498 T + 248004)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 90653983744 \)
|
| $61$ |
\( T^{4} + \cdots + 179098240000 \)
|
| $67$ |
\( (T^{2} + 156 T + 24336)^{2} \)
|
| $71$ |
\( (T - 240)^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} + 1112 T + 1236544)^{2} \)
|
| $83$ |
\( (T^{2} - 800)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 3397386240000 \)
|
| $97$ |
\( (T^{2} - 2196608)^{2} \)
|
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