Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,6,Mod(1,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(1.44345437832\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 |
|
6.00000 | 0 | 4.00000 | −6.00000 | 0 | −40.0000 | −168.000 | 0 | −36.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 9.6.a.a | 1 | |
| 3.b | odd | 2 | 1 | 3.6.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 144.6.a.f | 1 | ||
| 5.b | even | 2 | 1 | 225.6.a.a | 1 | ||
| 5.c | odd | 4 | 2 | 225.6.b.b | 2 | ||
| 7.b | odd | 2 | 1 | 441.6.a.i | 1 | ||
| 8.b | even | 2 | 1 | 576.6.a.s | 1 | ||
| 8.d | odd | 2 | 1 | 576.6.a.t | 1 | ||
| 9.c | even | 3 | 2 | 81.6.c.a | 2 | ||
| 9.d | odd | 6 | 2 | 81.6.c.c | 2 | ||
| 11.b | odd | 2 | 1 | 1089.6.a.b | 1 | ||
| 12.b | even | 2 | 1 | 48.6.a.a | 1 | ||
| 15.d | odd | 2 | 1 | 75.6.a.e | 1 | ||
| 15.e | even | 4 | 2 | 75.6.b.b | 2 | ||
| 21.c | even | 2 | 1 | 147.6.a.a | 1 | ||
| 21.g | even | 6 | 2 | 147.6.e.k | 2 | ||
| 21.h | odd | 6 | 2 | 147.6.e.h | 2 | ||
| 24.f | even | 2 | 1 | 192.6.a.l | 1 | ||
| 24.h | odd | 2 | 1 | 192.6.a.d | 1 | ||
| 33.d | even | 2 | 1 | 363.6.a.d | 1 | ||
| 39.d | odd | 2 | 1 | 507.6.a.b | 1 | ||
| 48.i | odd | 4 | 2 | 768.6.d.k | 2 | ||
| 48.k | even | 4 | 2 | 768.6.d.h | 2 | ||
| 51.c | odd | 2 | 1 | 867.6.a.a | 1 | ||
| 57.d | even | 2 | 1 | 1083.6.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.6.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 9.6.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 48.6.a.a | 1 | 12.b | even | 2 | 1 | ||
| 75.6.a.e | 1 | 15.d | odd | 2 | 1 | ||
| 75.6.b.b | 2 | 15.e | even | 4 | 2 | ||
| 81.6.c.a | 2 | 9.c | even | 3 | 2 | ||
| 81.6.c.c | 2 | 9.d | odd | 6 | 2 | ||
| 144.6.a.f | 1 | 4.b | odd | 2 | 1 | ||
| 147.6.a.a | 1 | 21.c | even | 2 | 1 | ||
| 147.6.e.h | 2 | 21.h | odd | 6 | 2 | ||
| 147.6.e.k | 2 | 21.g | even | 6 | 2 | ||
| 192.6.a.d | 1 | 24.h | odd | 2 | 1 | ||
| 192.6.a.l | 1 | 24.f | even | 2 | 1 | ||
| 225.6.a.a | 1 | 5.b | even | 2 | 1 | ||
| 225.6.b.b | 2 | 5.c | odd | 4 | 2 | ||
| 363.6.a.d | 1 | 33.d | even | 2 | 1 | ||
| 441.6.a.i | 1 | 7.b | odd | 2 | 1 | ||
| 507.6.a.b | 1 | 39.d | odd | 2 | 1 | ||
| 576.6.a.s | 1 | 8.b | even | 2 | 1 | ||
| 576.6.a.t | 1 | 8.d | odd | 2 | 1 | ||
| 768.6.d.h | 2 | 48.k | even | 4 | 2 | ||
| 768.6.d.k | 2 | 48.i | odd | 4 | 2 | ||
| 867.6.a.a | 1 | 51.c | odd | 2 | 1 | ||
| 1083.6.a.c | 1 | 57.d | even | 2 | 1 | ||
| 1089.6.a.b | 1 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 6 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 6 \)
|
| $7$ |
\( T + 40 \)
|
| $11$ |
\( T - 564 \)
|
| $13$ |
\( T - 638 \)
|
| $17$ |
\( T + 882 \)
|
| $19$ |
\( T + 556 \)
|
| $23$ |
\( T - 840 \)
|
| $29$ |
\( T + 4638 \)
|
| $31$ |
\( T - 4400 \)
|
| $37$ |
\( T + 2410 \)
|
| $41$ |
\( T - 6870 \)
|
| $43$ |
\( T - 9644 \)
|
| $47$ |
\( T - 18672 \)
|
| $53$ |
\( T + 33750 \)
|
| $59$ |
\( T - 18084 \)
|
| $61$ |
\( T - 39758 \)
|
| $67$ |
\( T + 23068 \)
|
| $71$ |
\( T - 4248 \)
|
| $73$ |
\( T + 41110 \)
|
| $79$ |
\( T - 21920 \)
|
| $83$ |
\( T + 82452 \)
|
| $89$ |
\( T - 94086 \)
|
| $97$ |
\( T - 49442 \)
|
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