Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,6,Mod(1,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.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: | \(41.0582578721\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{10}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 128) |
| 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)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0 | −25.2982 | 0 | 84.0000 | 0 | 101.193 | 0 | 397.000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 25.2982 | 0 | 84.0000 | 0 | −101.193 | 0 | 397.000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.6.a.i | 2 | |
| 4.b | odd | 2 | 1 | inner | 256.6.a.i | 2 | |
| 8.b | even | 2 | 1 | 256.6.a.e | 2 | ||
| 8.d | odd | 2 | 1 | 256.6.a.e | 2 | ||
| 16.e | even | 4 | 2 | 128.6.b.c | ✓ | 4 | |
| 16.f | odd | 4 | 2 | 128.6.b.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.6.b.c | ✓ | 4 | 16.e | even | 4 | 2 | |
| 128.6.b.c | ✓ | 4 | 16.f | odd | 4 | 2 | |
| 256.6.a.e | 2 | 8.b | even | 2 | 1 | ||
| 256.6.a.e | 2 | 8.d | odd | 2 | 1 | ||
| 256.6.a.i | 2 | 1.a | even | 1 | 1 | trivial | |
| 256.6.a.i | 2 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{2} - 640 \)
|
|
\( T_{5} - 84 \)
|
|
\( T_{7}^{2} - 10240 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 640 \)
|
| $5$ |
\( (T - 84)^{2} \)
|
| $7$ |
\( T^{2} - 10240 \)
|
| $11$ |
\( T^{2} - 400000 \)
|
| $13$ |
\( (T - 404)^{2} \)
|
| $17$ |
\( (T - 958)^{2} \)
|
| $19$ |
\( T^{2} - 876160 \)
|
| $23$ |
\( T^{2} - 24586240 \)
|
| $29$ |
\( (T - 6164)^{2} \)
|
| $31$ |
\( T^{2} - 655360 \)
|
| $37$ |
\( (T - 6148)^{2} \)
|
| $41$ |
\( (T + 810)^{2} \)
|
| $43$ |
\( T^{2} - 2381440 \)
|
| $47$ |
\( T^{2} - 34447360 \)
|
| $53$ |
\( (T + 23596)^{2} \)
|
| $59$ |
\( T^{2} - 1551519360 \)
|
| $61$ |
\( (T - 8388)^{2} \)
|
| $67$ |
\( T^{2} - 750648960 \)
|
| $71$ |
\( T^{2} - 1613916160 \)
|
| $73$ |
\( (T - 24474)^{2} \)
|
| $79$ |
\( T^{2} - 2833162240 \)
|
| $83$ |
\( T^{2} - 3122995840 \)
|
| $89$ |
\( (T + 132630)^{2} \)
|
| $97$ |
\( (T - 115822)^{2} \)
|
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