Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,2,Mod(1,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 168.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: | \(1.34148675396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 | −1.00000 | 0 | 2.00000 | 0 | 1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 168.2.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 504.2.a.e | 1 | ||
| 4.b | odd | 2 | 1 | 336.2.a.e | 1 | ||
| 5.b | even | 2 | 1 | 4200.2.a.t | 1 | ||
| 5.c | odd | 4 | 2 | 4200.2.t.j | 2 | ||
| 7.b | odd | 2 | 1 | 1176.2.a.f | 1 | ||
| 7.c | even | 3 | 2 | 1176.2.q.f | 2 | ||
| 7.d | odd | 6 | 2 | 1176.2.q.d | 2 | ||
| 8.b | even | 2 | 1 | 1344.2.a.m | 1 | ||
| 8.d | odd | 2 | 1 | 1344.2.a.b | 1 | ||
| 12.b | even | 2 | 1 | 1008.2.a.b | 1 | ||
| 16.e | even | 4 | 2 | 5376.2.c.d | 2 | ||
| 16.f | odd | 4 | 2 | 5376.2.c.bb | 2 | ||
| 20.d | odd | 2 | 1 | 8400.2.a.y | 1 | ||
| 21.c | even | 2 | 1 | 3528.2.a.v | 1 | ||
| 21.g | even | 6 | 2 | 3528.2.s.g | 2 | ||
| 21.h | odd | 6 | 2 | 3528.2.s.w | 2 | ||
| 24.f | even | 2 | 1 | 4032.2.a.bc | 1 | ||
| 24.h | odd | 2 | 1 | 4032.2.a.bh | 1 | ||
| 28.d | even | 2 | 1 | 2352.2.a.c | 1 | ||
| 28.f | even | 6 | 2 | 2352.2.q.w | 2 | ||
| 28.g | odd | 6 | 2 | 2352.2.q.d | 2 | ||
| 56.e | even | 2 | 1 | 9408.2.a.da | 1 | ||
| 56.h | odd | 2 | 1 | 9408.2.a.be | 1 | ||
| 84.h | odd | 2 | 1 | 7056.2.a.bq | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.2.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 336.2.a.e | 1 | 4.b | odd | 2 | 1 | ||
| 504.2.a.e | 1 | 3.b | odd | 2 | 1 | ||
| 1008.2.a.b | 1 | 12.b | even | 2 | 1 | ||
| 1176.2.a.f | 1 | 7.b | odd | 2 | 1 | ||
| 1176.2.q.d | 2 | 7.d | odd | 6 | 2 | ||
| 1176.2.q.f | 2 | 7.c | even | 3 | 2 | ||
| 1344.2.a.b | 1 | 8.d | odd | 2 | 1 | ||
| 1344.2.a.m | 1 | 8.b | even | 2 | 1 | ||
| 2352.2.a.c | 1 | 28.d | even | 2 | 1 | ||
| 2352.2.q.d | 2 | 28.g | odd | 6 | 2 | ||
| 2352.2.q.w | 2 | 28.f | even | 6 | 2 | ||
| 3528.2.a.v | 1 | 21.c | even | 2 | 1 | ||
| 3528.2.s.g | 2 | 21.g | even | 6 | 2 | ||
| 3528.2.s.w | 2 | 21.h | odd | 6 | 2 | ||
| 4032.2.a.bc | 1 | 24.f | even | 2 | 1 | ||
| 4032.2.a.bh | 1 | 24.h | odd | 2 | 1 | ||
| 4200.2.a.t | 1 | 5.b | even | 2 | 1 | ||
| 4200.2.t.j | 2 | 5.c | odd | 4 | 2 | ||
| 5376.2.c.d | 2 | 16.e | even | 4 | 2 | ||
| 5376.2.c.bb | 2 | 16.f | odd | 4 | 2 | ||
| 7056.2.a.bq | 1 | 84.h | odd | 2 | 1 | ||
| 8400.2.a.y | 1 | 20.d | odd | 2 | 1 | ||
| 9408.2.a.be | 1 | 56.h | odd | 2 | 1 | ||
| 9408.2.a.da | 1 | 56.e | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(168))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T + 1 \)
|
| $5$ |
\( T - 2 \)
|
| $7$ |
\( T - 1 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T - 6 \)
|
| $17$ |
\( T + 2 \)
|
| $19$ |
\( T - 4 \)
|
| $23$ |
\( T + 4 \)
|
| $29$ |
\( T + 10 \)
|
| $31$ |
\( T + 8 \)
|
| $37$ |
\( T - 6 \)
|
| $41$ |
\( T + 2 \)
|
| $43$ |
\( T + 4 \)
|
| $47$ |
\( T - 8 \)
|
| $53$ |
\( T + 10 \)
|
| $59$ |
\( T - 12 \)
|
| $61$ |
\( T + 2 \)
|
| $67$ |
\( T - 12 \)
|
| $71$ |
\( T + 12 \)
|
| $73$ |
\( T + 14 \)
|
| $79$ |
\( T + 8 \)
|
| $83$ |
\( T - 12 \)
|
| $89$ |
\( T + 2 \)
|
| $97$ |
\( T - 10 \)
|
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