Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,Mod(1,392)]
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([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.a (trivial) |
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: | \(23.1287487223\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 8) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 4.00000 | 0 | 2.00000 | 0 | 0 | 0 | −11.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.4.a.e | 1 | |
| 4.b | odd | 2 | 1 | 784.4.a.e | 1 | ||
| 7.b | odd | 2 | 1 | 8.4.a.a | ✓ | 1 | |
| 7.c | even | 3 | 2 | 392.4.i.b | 2 | ||
| 7.d | odd | 6 | 2 | 392.4.i.g | 2 | ||
| 21.c | even | 2 | 1 | 72.4.a.c | 1 | ||
| 28.d | even | 2 | 1 | 16.4.a.a | 1 | ||
| 35.c | odd | 2 | 1 | 200.4.a.g | 1 | ||
| 35.f | even | 4 | 2 | 200.4.c.e | 2 | ||
| 56.e | even | 2 | 1 | 64.4.a.b | 1 | ||
| 56.h | odd | 2 | 1 | 64.4.a.d | 1 | ||
| 63.l | odd | 6 | 2 | 648.4.i.h | 2 | ||
| 63.o | even | 6 | 2 | 648.4.i.e | 2 | ||
| 77.b | even | 2 | 1 | 968.4.a.a | 1 | ||
| 84.h | odd | 2 | 1 | 144.4.a.e | 1 | ||
| 91.b | odd | 2 | 1 | 1352.4.a.a | 1 | ||
| 105.g | even | 2 | 1 | 1800.4.a.d | 1 | ||
| 105.k | odd | 4 | 2 | 1800.4.f.u | 2 | ||
| 112.j | even | 4 | 2 | 256.4.b.g | 2 | ||
| 112.l | odd | 4 | 2 | 256.4.b.a | 2 | ||
| 119.d | odd | 2 | 1 | 2312.4.a.a | 1 | ||
| 140.c | even | 2 | 1 | 400.4.a.g | 1 | ||
| 140.j | odd | 4 | 2 | 400.4.c.i | 2 | ||
| 168.e | odd | 2 | 1 | 576.4.a.j | 1 | ||
| 168.i | even | 2 | 1 | 576.4.a.k | 1 | ||
| 280.c | odd | 2 | 1 | 1600.4.a.o | 1 | ||
| 280.n | even | 2 | 1 | 1600.4.a.bm | 1 | ||
| 308.g | odd | 2 | 1 | 1936.4.a.l | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.4.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
| 16.4.a.a | 1 | 28.d | even | 2 | 1 | ||
| 64.4.a.b | 1 | 56.e | even | 2 | 1 | ||
| 64.4.a.d | 1 | 56.h | odd | 2 | 1 | ||
| 72.4.a.c | 1 | 21.c | even | 2 | 1 | ||
| 144.4.a.e | 1 | 84.h | odd | 2 | 1 | ||
| 200.4.a.g | 1 | 35.c | odd | 2 | 1 | ||
| 200.4.c.e | 2 | 35.f | even | 4 | 2 | ||
| 256.4.b.a | 2 | 112.l | odd | 4 | 2 | ||
| 256.4.b.g | 2 | 112.j | even | 4 | 2 | ||
| 392.4.a.e | 1 | 1.a | even | 1 | 1 | trivial | |
| 392.4.i.b | 2 | 7.c | even | 3 | 2 | ||
| 392.4.i.g | 2 | 7.d | odd | 6 | 2 | ||
| 400.4.a.g | 1 | 140.c | even | 2 | 1 | ||
| 400.4.c.i | 2 | 140.j | odd | 4 | 2 | ||
| 576.4.a.j | 1 | 168.e | odd | 2 | 1 | ||
| 576.4.a.k | 1 | 168.i | even | 2 | 1 | ||
| 648.4.i.e | 2 | 63.o | even | 6 | 2 | ||
| 648.4.i.h | 2 | 63.l | odd | 6 | 2 | ||
| 784.4.a.e | 1 | 4.b | odd | 2 | 1 | ||
| 968.4.a.a | 1 | 77.b | even | 2 | 1 | ||
| 1352.4.a.a | 1 | 91.b | odd | 2 | 1 | ||
| 1600.4.a.o | 1 | 280.c | odd | 2 | 1 | ||
| 1600.4.a.bm | 1 | 280.n | even | 2 | 1 | ||
| 1800.4.a.d | 1 | 105.g | even | 2 | 1 | ||
| 1800.4.f.u | 2 | 105.k | odd | 4 | 2 | ||
| 1936.4.a.l | 1 | 308.g | odd | 2 | 1 | ||
| 2312.4.a.a | 1 | 119.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\):
|
\( T_{3} - 4 \)
|
|
\( T_{5} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 4 \)
|
| $5$ |
\( T - 2 \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T + 44 \)
|
| $13$ |
\( T + 22 \)
|
| $17$ |
\( T + 50 \)
|
| $19$ |
\( T + 44 \)
|
| $23$ |
\( T + 56 \)
|
| $29$ |
\( T - 198 \)
|
| $31$ |
\( T - 160 \)
|
| $37$ |
\( T + 162 \)
|
| $41$ |
\( T - 198 \)
|
| $43$ |
\( T - 52 \)
|
| $47$ |
\( T + 528 \)
|
| $53$ |
\( T + 242 \)
|
| $59$ |
\( T - 668 \)
|
| $61$ |
\( T + 550 \)
|
| $67$ |
\( T - 188 \)
|
| $71$ |
\( T - 728 \)
|
| $73$ |
\( T + 154 \)
|
| $79$ |
\( T + 656 \)
|
| $83$ |
\( T + 236 \)
|
| $89$ |
\( T + 714 \)
|
| $97$ |
\( T - 478 \)
|
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