Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(6.91508862504\) |
| 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 |
|
−78.0000 | 0 | 4036.00 | 5370.00 | 0 | −27760.0 | −155064. | 0 | −418860. | |||||||||||||||||||||
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 | 9.12.a.a | 1 | |
| 3.b | odd | 2 | 1 | 3.12.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 144.12.a.l | 1 | ||
| 5.b | even | 2 | 1 | 225.12.a.f | 1 | ||
| 5.c | odd | 4 | 2 | 225.12.b.a | 2 | ||
| 9.c | even | 3 | 2 | 81.12.c.e | 2 | ||
| 9.d | odd | 6 | 2 | 81.12.c.a | 2 | ||
| 12.b | even | 2 | 1 | 48.12.a.f | 1 | ||
| 15.d | odd | 2 | 1 | 75.12.a.a | 1 | ||
| 15.e | even | 4 | 2 | 75.12.b.a | 2 | ||
| 21.c | even | 2 | 1 | 147.12.a.c | 1 | ||
| 24.f | even | 2 | 1 | 192.12.a.g | 1 | ||
| 24.h | odd | 2 | 1 | 192.12.a.q | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.12.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 9.12.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 48.12.a.f | 1 | 12.b | even | 2 | 1 | ||
| 75.12.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.12.b.a | 2 | 15.e | even | 4 | 2 | ||
| 81.12.c.a | 2 | 9.d | odd | 6 | 2 | ||
| 81.12.c.e | 2 | 9.c | even | 3 | 2 | ||
| 144.12.a.l | 1 | 4.b | odd | 2 | 1 | ||
| 147.12.a.c | 1 | 21.c | even | 2 | 1 | ||
| 192.12.a.g | 1 | 24.f | even | 2 | 1 | ||
| 192.12.a.q | 1 | 24.h | odd | 2 | 1 | ||
| 225.12.a.f | 1 | 5.b | even | 2 | 1 | ||
| 225.12.b.a | 2 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 78 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 78 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 5370 \)
|
| $7$ |
\( T + 27760 \)
|
| $11$ |
\( T + 637836 \)
|
| $13$ |
\( T - 766214 \)
|
| $17$ |
\( T + 3084354 \)
|
| $19$ |
\( T + 19511404 \)
|
| $23$ |
\( T + 15312360 \)
|
| $29$ |
\( T + 10751262 \)
|
| $31$ |
\( T + 50937400 \)
|
| $37$ |
\( T - 664740830 \)
|
| $41$ |
\( T + 898833450 \)
|
| $43$ |
\( T + 957947188 \)
|
| $47$ |
\( T - 1555741344 \)
|
| $53$ |
\( T + 3792417030 \)
|
| $59$ |
\( T + 555306924 \)
|
| $61$ |
\( T - 4950420998 \)
|
| $67$ |
\( T - 5292399284 \)
|
| $71$ |
\( T - 14831086248 \)
|
| $73$ |
\( T - 13971005210 \)
|
| $79$ |
\( T - 3720542360 \)
|
| $83$ |
\( T + 8768454036 \)
|
| $89$ |
\( T - 25472769174 \)
|
| $97$ |
\( T + 39092494846 \)
|
| show more | |
| show less |