Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(2.81146522936\) |
| Analytic rank: | \(1\) |
| 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 | −92.0000 | −390.000 | 0 | −64.0000 | 1320.00 | 0 | 2340.00 | |||||||||||||||||||||
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 | 9.8.a.a | 1 | |
| 3.b | odd | 2 | 1 | 3.8.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 144.8.a.b | 1 | ||
| 5.b | even | 2 | 1 | 225.8.a.i | 1 | ||
| 5.c | odd | 4 | 2 | 225.8.b.f | 2 | ||
| 7.b | odd | 2 | 1 | 441.8.a.a | 1 | ||
| 8.b | even | 2 | 1 | 576.8.a.w | 1 | ||
| 8.d | odd | 2 | 1 | 576.8.a.x | 1 | ||
| 9.c | even | 3 | 2 | 81.8.c.c | 2 | ||
| 9.d | odd | 6 | 2 | 81.8.c.a | 2 | ||
| 12.b | even | 2 | 1 | 48.8.a.g | 1 | ||
| 15.d | odd | 2 | 1 | 75.8.a.a | 1 | ||
| 15.e | even | 4 | 2 | 75.8.b.c | 2 | ||
| 21.c | even | 2 | 1 | 147.8.a.b | 1 | ||
| 21.g | even | 6 | 2 | 147.8.e.a | 2 | ||
| 21.h | odd | 6 | 2 | 147.8.e.b | 2 | ||
| 24.f | even | 2 | 1 | 192.8.a.a | 1 | ||
| 24.h | odd | 2 | 1 | 192.8.a.i | 1 | ||
| 33.d | even | 2 | 1 | 363.8.a.b | 1 | ||
| 39.d | odd | 2 | 1 | 507.8.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.8.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 9.8.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 48.8.a.g | 1 | 12.b | even | 2 | 1 | ||
| 75.8.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.8.b.c | 2 | 15.e | even | 4 | 2 | ||
| 81.8.c.a | 2 | 9.d | odd | 6 | 2 | ||
| 81.8.c.c | 2 | 9.c | even | 3 | 2 | ||
| 144.8.a.b | 1 | 4.b | odd | 2 | 1 | ||
| 147.8.a.b | 1 | 21.c | even | 2 | 1 | ||
| 147.8.e.a | 2 | 21.g | even | 6 | 2 | ||
| 147.8.e.b | 2 | 21.h | odd | 6 | 2 | ||
| 192.8.a.a | 1 | 24.f | even | 2 | 1 | ||
| 192.8.a.i | 1 | 24.h | odd | 2 | 1 | ||
| 225.8.a.i | 1 | 5.b | even | 2 | 1 | ||
| 225.8.b.f | 2 | 5.c | odd | 4 | 2 | ||
| 363.8.a.b | 1 | 33.d | even | 2 | 1 | ||
| 441.8.a.a | 1 | 7.b | odd | 2 | 1 | ||
| 507.8.a.a | 1 | 39.d | odd | 2 | 1 | ||
| 576.8.a.w | 1 | 8.b | even | 2 | 1 | ||
| 576.8.a.x | 1 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 6 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 6 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 390 \)
|
| $7$ |
\( T + 64 \)
|
| $11$ |
\( T - 948 \)
|
| $13$ |
\( T + 5098 \)
|
| $17$ |
\( T + 28386 \)
|
| $19$ |
\( T + 8620 \)
|
| $23$ |
\( T - 15288 \)
|
| $29$ |
\( T + 36510 \)
|
| $31$ |
\( T + 276808 \)
|
| $37$ |
\( T - 268526 \)
|
| $41$ |
\( T - 629718 \)
|
| $43$ |
\( T - 685772 \)
|
| $47$ |
\( T + 583296 \)
|
| $53$ |
\( T - 428058 \)
|
| $59$ |
\( T + 1306380 \)
|
| $61$ |
\( T - 300662 \)
|
| $67$ |
\( T + 507244 \)
|
| $71$ |
\( T + 5560632 \)
|
| $73$ |
\( T - 1369082 \)
|
| $79$ |
\( T + 6913720 \)
|
| $83$ |
\( T - 4376748 \)
|
| $89$ |
\( T - 8528310 \)
|
| $97$ |
\( T + 8826814 \)
|
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