Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,14,Mod(1,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.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: | \(135.110970479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 |
|
64.0000 | 0 | 4096.00 | −34758.0 | 0 | 117649. | 262144. | 0 | −2.22451e6 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 126.14.a.d | 1 | |
| 3.b | odd | 2 | 1 | 42.14.a.b | ✓ | 1 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.14.a.b | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 126.14.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 34758 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(126))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 64 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 34758 \)
|
| $7$ |
\( T - 117649 \)
|
| $11$ |
\( T - 9574884 \)
|
| $13$ |
\( T + 4781074 \)
|
| $17$ |
\( T - 141817998 \)
|
| $19$ |
\( T + 223101484 \)
|
| $23$ |
\( T + 194421528 \)
|
| $29$ |
\( T + 2816756238 \)
|
| $31$ |
\( T + 253184800 \)
|
| $37$ |
\( T + 3670329130 \)
|
| $41$ |
\( T + 14980174314 \)
|
| $43$ |
\( T + 19796769604 \)
|
| $47$ |
\( T - 79684803120 \)
|
| $53$ |
\( T - 172785794394 \)
|
| $59$ |
\( T - 535162833636 \)
|
| $61$ |
\( T - 9514350206 \)
|
| $67$ |
\( T - 1193767377044 \)
|
| $71$ |
\( T - 2053930205592 \)
|
| $73$ |
\( T + 291597653590 \)
|
| $79$ |
\( T - 867536684720 \)
|
| $83$ |
\( T + 3566430862356 \)
|
| $89$ |
\( T + 811579559034 \)
|
| $97$ |
\( T - 10732888990466 \)
|
| show more | |
| show less |