Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,13,Mod(2,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.b (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: | \(2.74198145183\) |
| 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 \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
0 | 729.000 | 4096.00 | 0 | 0 | −153502. | 0 | 531441. | 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 | 3.13.b.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | CM | 3.13.b.a | ✓ | 1 |
| 4.b | odd | 2 | 1 | 48.13.e.a | 1 | ||
| 5.b | even | 2 | 1 | 75.13.c.a | 1 | ||
| 5.c | odd | 4 | 2 | 75.13.d.a | 2 | ||
| 8.b | even | 2 | 1 | 192.13.e.a | 1 | ||
| 8.d | odd | 2 | 1 | 192.13.e.b | 1 | ||
| 9.c | even | 3 | 2 | 81.13.d.a | 2 | ||
| 9.d | odd | 6 | 2 | 81.13.d.a | 2 | ||
| 12.b | even | 2 | 1 | 48.13.e.a | 1 | ||
| 15.d | odd | 2 | 1 | 75.13.c.a | 1 | ||
| 15.e | even | 4 | 2 | 75.13.d.a | 2 | ||
| 24.f | even | 2 | 1 | 192.13.e.b | 1 | ||
| 24.h | odd | 2 | 1 | 192.13.e.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.13.b.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 3.13.b.a | ✓ | 1 | 3.b | odd | 2 | 1 | CM |
| 48.13.e.a | 1 | 4.b | odd | 2 | 1 | ||
| 48.13.e.a | 1 | 12.b | even | 2 | 1 | ||
| 75.13.c.a | 1 | 5.b | even | 2 | 1 | ||
| 75.13.c.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.13.d.a | 2 | 5.c | odd | 4 | 2 | ||
| 75.13.d.a | 2 | 15.e | even | 4 | 2 | ||
| 81.13.d.a | 2 | 9.c | even | 3 | 2 | ||
| 81.13.d.a | 2 | 9.d | odd | 6 | 2 | ||
| 192.13.e.a | 1 | 8.b | even | 2 | 1 | ||
| 192.13.e.a | 1 | 24.h | odd | 2 | 1 | ||
| 192.13.e.b | 1 | 8.d | odd | 2 | 1 | ||
| 192.13.e.b | 1 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{13}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 729 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T + 153502 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T + 9397582 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T - 17886962 \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T + 530187838 \)
|
| $37$ |
\( T - 2826257618 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T + 235885102 \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T - 74063873522 \)
|
| $67$ |
\( T + 151031344462 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T - 104459767778 \)
|
| $79$ |
\( T + 444304748158 \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T + 1662757858942 \)
|
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