Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,3,Mod(99,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.99");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.g (of order \(2\), degree \(1\), 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: | \(10.6812263629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
2.00000 | 2.58579 | 4.00000 | 0 | 5.17157 | 0 | 8.00000 | −2.31371 | 0 | ||||||||||||||||||||||||
| 99.2 | 2.00000 | 5.41421 | 4.00000 | 0 | 10.8284 | 0 | 8.00000 | 20.3137 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.3.g.g | yes | 2 |
| 4.b | odd | 2 | 1 | 1568.3.g.b | 2 | ||
| 7.b | odd | 2 | 1 | 392.3.g.f | ✓ | 2 | |
| 7.c | even | 3 | 2 | 392.3.k.e | 4 | ||
| 7.d | odd | 6 | 2 | 392.3.k.f | 4 | ||
| 8.b | even | 2 | 1 | 1568.3.g.b | 2 | ||
| 8.d | odd | 2 | 1 | CM | 392.3.g.g | yes | 2 |
| 28.d | even | 2 | 1 | 1568.3.g.g | 2 | ||
| 56.e | even | 2 | 1 | 392.3.g.f | ✓ | 2 | |
| 56.h | odd | 2 | 1 | 1568.3.g.g | 2 | ||
| 56.k | odd | 6 | 2 | 392.3.k.e | 4 | ||
| 56.m | even | 6 | 2 | 392.3.k.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 392.3.g.f | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 392.3.g.f | ✓ | 2 | 56.e | even | 2 | 1 | |
| 392.3.g.g | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 392.3.g.g | yes | 2 | 8.d | odd | 2 | 1 | CM |
| 392.3.k.e | 4 | 7.c | even | 3 | 2 | ||
| 392.3.k.e | 4 | 56.k | odd | 6 | 2 | ||
| 392.3.k.f | 4 | 7.d | odd | 6 | 2 | ||
| 392.3.k.f | 4 | 56.m | even | 6 | 2 | ||
| 1568.3.g.b | 2 | 4.b | odd | 2 | 1 | ||
| 1568.3.g.b | 2 | 8.b | even | 2 | 1 | ||
| 1568.3.g.g | 2 | 28.d | even | 2 | 1 | ||
| 1568.3.g.g | 2 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8T_{3} + 14 \)
acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} - 8T + 14 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 288 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 48T + 574 \)
|
| $19$ |
\( T^{2} + 24T - 434 \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( T^{2} + 96T + 1246 \)
|
| $43$ |
\( T^{2} - 7200 \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} - 120T + 238 \)
|
| $61$ |
\( T^{2} \)
|
| $67$ |
\( (T + 62)^{2} \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} + 48T - 9506 \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} - 72T - 11186 \)
|
| $89$ |
\( T^{2} - 144T - 5474 \)
|
| $97$ |
\( T^{2} - 240T + 9982 \)
|
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