Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,9,Mod(5,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.5");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.c (of order \(2\), degree \(1\), minimal) |
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: | \(4.88854332073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | 81.0000 | 0 | 0 | 0 | 4034.00 | 0 | 6561.00 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.9.c.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | CM | 12.9.c.a | ✓ | 1 |
| 4.b | odd | 2 | 1 | 48.9.e.a | 1 | ||
| 5.b | even | 2 | 1 | 300.9.g.a | 1 | ||
| 5.c | odd | 4 | 2 | 300.9.b.b | 2 | ||
| 8.b | even | 2 | 1 | 192.9.e.a | 1 | ||
| 8.d | odd | 2 | 1 | 192.9.e.b | 1 | ||
| 9.c | even | 3 | 2 | 324.9.g.a | 2 | ||
| 9.d | odd | 6 | 2 | 324.9.g.a | 2 | ||
| 12.b | even | 2 | 1 | 48.9.e.a | 1 | ||
| 15.d | odd | 2 | 1 | 300.9.g.a | 1 | ||
| 15.e | even | 4 | 2 | 300.9.b.b | 2 | ||
| 24.f | even | 2 | 1 | 192.9.e.b | 1 | ||
| 24.h | odd | 2 | 1 | 192.9.e.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.9.c.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 12.9.c.a | ✓ | 1 | 3.b | odd | 2 | 1 | CM |
| 48.9.e.a | 1 | 4.b | odd | 2 | 1 | ||
| 48.9.e.a | 1 | 12.b | even | 2 | 1 | ||
| 192.9.e.a | 1 | 8.b | even | 2 | 1 | ||
| 192.9.e.a | 1 | 24.h | odd | 2 | 1 | ||
| 192.9.e.b | 1 | 8.d | odd | 2 | 1 | ||
| 192.9.e.b | 1 | 24.f | even | 2 | 1 | ||
| 300.9.b.b | 2 | 5.c | odd | 4 | 2 | ||
| 300.9.b.b | 2 | 15.e | even | 4 | 2 | ||
| 300.9.g.a | 1 | 5.b | even | 2 | 1 | ||
| 300.9.g.a | 1 | 15.d | odd | 2 | 1 | ||
| 324.9.g.a | 2 | 9.c | even | 3 | 2 | ||
| 324.9.g.a | 2 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{9}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 81 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 4034 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T + 35806 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T + 258526 \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T + 1809406 \)
|
| $37$ |
\( T - 503522 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T - 3492194 \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T + 23826526 \)
|
| $67$ |
\( T + 5421406 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T - 16169282 \)
|
| $79$ |
\( T + 18887038 \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T - 176908034 \)
|
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