Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,2,Mod(19,392)]
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([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.m (of order \(6\), 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: | \(3.13013575923\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.39564 | − | 0.228425i | 0 | 1.89564 | + | 0.637600i | 0 | 0 | 0 | −2.50000 | − | 1.32288i | −1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||
| 19.2 | 0.895644 | + | 1.09445i | 0 | −0.395644 | + | 1.96048i | 0 | 0 | 0 | −2.50000 | + | 1.32288i | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||
| 227.1 | −1.39564 | + | 0.228425i | 0 | 1.89564 | − | 0.637600i | 0 | 0 | 0 | −2.50000 | + | 1.32288i | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||
| 227.2 | 0.895644 | − | 1.09445i | 0 | −0.395644 | − | 1.96048i | 0 | 0 | 0 | −2.50000 | − | 1.32288i | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 56.e | even | 2 | 1 | inner |
| 56.k | odd | 6 | 1 | inner |
| 56.m | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{5} \)
|
|
\( T_{23}^{4} - 28T_{23}^{2} + 784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} - T^{2} + \cdots + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} - 28T^{2} + 784 \)
|
| $29$ |
\( (T^{2} + 112)^{2} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 112 T^{2} + 12544 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( (T - 12)^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} - 112 T^{2} + 12544 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $71$ |
\( (T^{2} + 28)^{2} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} - 252 T^{2} + 63504 \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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