Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,4,Mod(1,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.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: | \(0.354011460034\) |
| 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)
Expression as an eta quotient
\(f(z) = \eta(z)^{2}\eta(2z)^{2}\eta(3z)^{2}\eta(6z)^{2}=q\prod_{n=1}^\infty(1 - q^{n})^{2}(1 - q^{2n})^{2}(1 - q^{3n})^{2}(1 - q^{6n})^{2}\)
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | −3.00000 | 4.00000 | 6.00000 | 6.00000 | −16.0000 | −8.00000 | 9.00000 | −12.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
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 | 6.4.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 18.4.a.a | 1 | ||
| 4.b | odd | 2 | 1 | 48.4.a.c | 1 | ||
| 5.b | even | 2 | 1 | 150.4.a.i | 1 | ||
| 5.c | odd | 4 | 2 | 150.4.c.d | 2 | ||
| 7.b | odd | 2 | 1 | 294.4.a.e | 1 | ||
| 7.c | even | 3 | 2 | 294.4.e.h | 2 | ||
| 7.d | odd | 6 | 2 | 294.4.e.g | 2 | ||
| 8.b | even | 2 | 1 | 192.4.a.i | 1 | ||
| 8.d | odd | 2 | 1 | 192.4.a.c | 1 | ||
| 9.c | even | 3 | 2 | 162.4.c.f | 2 | ||
| 9.d | odd | 6 | 2 | 162.4.c.c | 2 | ||
| 11.b | odd | 2 | 1 | 726.4.a.f | 1 | ||
| 12.b | even | 2 | 1 | 144.4.a.c | 1 | ||
| 13.b | even | 2 | 1 | 1014.4.a.g | 1 | ||
| 13.d | odd | 4 | 2 | 1014.4.b.d | 2 | ||
| 15.d | odd | 2 | 1 | 450.4.a.h | 1 | ||
| 15.e | even | 4 | 2 | 450.4.c.e | 2 | ||
| 16.e | even | 4 | 2 | 768.4.d.n | 2 | ||
| 16.f | odd | 4 | 2 | 768.4.d.c | 2 | ||
| 17.b | even | 2 | 1 | 1734.4.a.d | 1 | ||
| 19.b | odd | 2 | 1 | 2166.4.a.i | 1 | ||
| 20.d | odd | 2 | 1 | 1200.4.a.b | 1 | ||
| 20.e | even | 4 | 2 | 1200.4.f.j | 2 | ||
| 21.c | even | 2 | 1 | 882.4.a.n | 1 | ||
| 21.g | even | 6 | 2 | 882.4.g.f | 2 | ||
| 21.h | odd | 6 | 2 | 882.4.g.i | 2 | ||
| 24.f | even | 2 | 1 | 576.4.a.r | 1 | ||
| 24.h | odd | 2 | 1 | 576.4.a.q | 1 | ||
| 28.d | even | 2 | 1 | 2352.4.a.e | 1 | ||
| 33.d | even | 2 | 1 | 2178.4.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.4.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 18.4.a.a | 1 | 3.b | odd | 2 | 1 | ||
| 48.4.a.c | 1 | 4.b | odd | 2 | 1 | ||
| 144.4.a.c | 1 | 12.b | even | 2 | 1 | ||
| 150.4.a.i | 1 | 5.b | even | 2 | 1 | ||
| 150.4.c.d | 2 | 5.c | odd | 4 | 2 | ||
| 162.4.c.c | 2 | 9.d | odd | 6 | 2 | ||
| 162.4.c.f | 2 | 9.c | even | 3 | 2 | ||
| 192.4.a.c | 1 | 8.d | odd | 2 | 1 | ||
| 192.4.a.i | 1 | 8.b | even | 2 | 1 | ||
| 294.4.a.e | 1 | 7.b | odd | 2 | 1 | ||
| 294.4.e.g | 2 | 7.d | odd | 6 | 2 | ||
| 294.4.e.h | 2 | 7.c | even | 3 | 2 | ||
| 450.4.a.h | 1 | 15.d | odd | 2 | 1 | ||
| 450.4.c.e | 2 | 15.e | even | 4 | 2 | ||
| 576.4.a.q | 1 | 24.h | odd | 2 | 1 | ||
| 576.4.a.r | 1 | 24.f | even | 2 | 1 | ||
| 726.4.a.f | 1 | 11.b | odd | 2 | 1 | ||
| 768.4.d.c | 2 | 16.f | odd | 4 | 2 | ||
| 768.4.d.n | 2 | 16.e | even | 4 | 2 | ||
| 882.4.a.n | 1 | 21.c | even | 2 | 1 | ||
| 882.4.g.f | 2 | 21.g | even | 6 | 2 | ||
| 882.4.g.i | 2 | 21.h | odd | 6 | 2 | ||
| 1014.4.a.g | 1 | 13.b | even | 2 | 1 | ||
| 1014.4.b.d | 2 | 13.d | odd | 4 | 2 | ||
| 1200.4.a.b | 1 | 20.d | odd | 2 | 1 | ||
| 1200.4.f.j | 2 | 20.e | even | 4 | 2 | ||
| 1734.4.a.d | 1 | 17.b | even | 2 | 1 | ||
| 2166.4.a.i | 1 | 19.b | odd | 2 | 1 | ||
| 2178.4.a.e | 1 | 33.d | even | 2 | 1 | ||
| 2352.4.a.e | 1 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 2 \)
|
| $3$ |
\( T + 3 \)
|
| $5$ |
\( T - 6 \)
|
| $7$ |
\( T + 16 \)
|
| $11$ |
\( T - 12 \)
|
| $13$ |
\( T - 38 \)
|
| $17$ |
\( T + 126 \)
|
| $19$ |
\( T - 20 \)
|
| $23$ |
\( T - 168 \)
|
| $29$ |
\( T - 30 \)
|
| $31$ |
\( T + 88 \)
|
| $37$ |
\( T - 254 \)
|
| $41$ |
\( T - 42 \)
|
| $43$ |
\( T + 52 \)
|
| $47$ |
\( T + 96 \)
|
| $53$ |
\( T - 198 \)
|
| $59$ |
\( T + 660 \)
|
| $61$ |
\( T + 538 \)
|
| $67$ |
\( T - 884 \)
|
| $71$ |
\( T - 792 \)
|
| $73$ |
\( T - 218 \)
|
| $79$ |
\( T + 520 \)
|
| $83$ |
\( T + 492 \)
|
| $89$ |
\( T - 810 \)
|
| $97$ |
\( T - 1154 \)
|
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