Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,16,Mod(1,3)]
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([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(4.28080515300\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 |
|
−72.0000 | 2187.00 | −27584.0 | −221490. | −157464. | −2.14900e6 | 4.34534e6 | 4.78297e6 | 1.59473e7 | |||||||||||||||||||||
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 | 3.16.a.b | ✓ | 1 |
| 3.b | odd | 2 | 1 | 9.16.a.c | 1 | ||
| 4.b | odd | 2 | 1 | 48.16.a.a | 1 | ||
| 5.b | even | 2 | 1 | 75.16.a.a | 1 | ||
| 5.c | odd | 4 | 2 | 75.16.b.b | 2 | ||
| 7.b | odd | 2 | 1 | 147.16.a.b | 1 | ||
| 12.b | even | 2 | 1 | 144.16.a.l | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.16.a.b | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 9.16.a.c | 1 | 3.b | odd | 2 | 1 | ||
| 48.16.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 75.16.a.a | 1 | 5.b | even | 2 | 1 | ||
| 75.16.b.b | 2 | 5.c | odd | 4 | 2 | ||
| 144.16.a.l | 1 | 12.b | even | 2 | 1 | ||
| 147.16.a.b | 1 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 72 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 72 \)
|
| $3$ |
\( T - 2187 \)
|
| $5$ |
\( T + 221490 \)
|
| $7$ |
\( T + 2149000 \)
|
| $11$ |
\( T - 37169316 \)
|
| $13$ |
\( T + 279974266 \)
|
| $17$ |
\( T - 2492912754 \)
|
| $19$ |
\( T + 4669782244 \)
|
| $23$ |
\( T + 18467933400 \)
|
| $29$ |
\( T + 115953449418 \)
|
| $31$ |
\( T + 56187023200 \)
|
| $37$ |
\( T - 614764926830 \)
|
| $41$ |
\( T - 549859792410 \)
|
| $43$ |
\( T + 982884444028 \)
|
| $47$ |
\( T - 2076144322896 \)
|
| $53$ |
\( T + 12048378188130 \)
|
| $59$ |
\( T - 23087905758324 \)
|
| $61$ |
\( T + 8505809142442 \)
|
| $67$ |
\( T + 12331010771476 \)
|
| $71$ |
\( T - 58989192692472 \)
|
| $73$ |
\( T + 5609828808070 \)
|
| $79$ |
\( T - 159918683826800 \)
|
| $83$ |
\( T - 57675894342876 \)
|
| $89$ |
\( T + 362287610413974 \)
|
| $97$ |
\( T + 539786645144926 \)
|
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