Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,5,Mod(255,256)]
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([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.255");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.c (of order \(2\), degree \(1\), not 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: | no |
| Analytic conductor: | \(26.4627105495\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 64) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( -4\zeta_{12}^{3} + 8\zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\zeta_{12}^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\zeta_{12}^{3} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + 2\beta_1 ) / 16 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 2 ) / 4 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(255\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 255.1 |
|
0 | − | 3.46410i | 0 | −20.7846 | 0 | − | 40.0000i | 0 | 69.0000 | 0 | ||||||||||||||||||||||||||||
| 255.2 | 0 | − | 3.46410i | 0 | 20.7846 | 0 | 40.0000i | 0 | 69.0000 | 0 | ||||||||||||||||||||||||||||||
| 255.3 | 0 | 3.46410i | 0 | −20.7846 | 0 | 40.0000i | 0 | 69.0000 | 0 | |||||||||||||||||||||||||||||||
| 255.4 | 0 | 3.46410i | 0 | 20.7846 | 0 | − | 40.0000i | 0 | 69.0000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.5.c.j | 4 | |
| 4.b | odd | 2 | 1 | inner | 256.5.c.j | 4 | |
| 8.b | even | 2 | 1 | inner | 256.5.c.j | 4 | |
| 8.d | odd | 2 | 1 | inner | 256.5.c.j | 4 | |
| 16.e | even | 4 | 2 | 64.5.d.a | ✓ | 4 | |
| 16.f | odd | 4 | 2 | 64.5.d.a | ✓ | 4 | |
| 48.i | odd | 4 | 2 | 576.5.b.e | 4 | ||
| 48.k | even | 4 | 2 | 576.5.b.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.5.d.a | ✓ | 4 | 16.e | even | 4 | 2 | |
| 64.5.d.a | ✓ | 4 | 16.f | odd | 4 | 2 | |
| 256.5.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 256.5.c.j | 4 | 4.b | odd | 2 | 1 | inner | |
| 256.5.c.j | 4 | 8.b | even | 2 | 1 | inner | |
| 256.5.c.j | 4 | 8.d | odd | 2 | 1 | inner | |
| 576.5.b.e | 4 | 48.i | odd | 4 | 2 | ||
| 576.5.b.e | 4 | 48.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(256, [\chi])\):
|
\( T_{3}^{2} + 12 \)
|
|
\( T_{5}^{2} - 432 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 12)^{2} \)
|
| $5$ |
\( (T^{2} - 432)^{2} \)
|
| $7$ |
\( (T^{2} + 1600)^{2} \)
|
| $11$ |
\( (T^{2} + 38988)^{2} \)
|
| $13$ |
\( (T^{2} - 65712)^{2} \)
|
| $17$ |
\( (T + 198)^{4} \)
|
| $19$ |
\( (T^{2} + 63948)^{2} \)
|
| $23$ |
\( (T^{2} + 254016)^{2} \)
|
| $29$ |
\( (T^{2} - 190512)^{2} \)
|
| $31$ |
\( (T^{2} + 2876416)^{2} \)
|
| $37$ |
\( (T^{2} - 3072432)^{2} \)
|
| $41$ |
\( (T + 18)^{4} \)
|
| $43$ |
\( (T^{2} + 2397708)^{2} \)
|
| $47$ |
\( (T^{2} + 5992704)^{2} \)
|
| $53$ |
\( (T^{2} - 10648368)^{2} \)
|
| $59$ |
\( (T^{2} + 27761292)^{2} \)
|
| $61$ |
\( (T^{2} - 1009200)^{2} \)
|
| $67$ |
\( (T^{2} + 3257292)^{2} \)
|
| $71$ |
\( (T^{2} + 61591104)^{2} \)
|
| $73$ |
\( (T - 4310)^{4} \)
|
| $79$ |
\( (T^{2} + 17172736)^{2} \)
|
| $83$ |
\( (T^{2} + 82687500)^{2} \)
|
| $89$ |
\( (T + 3114)^{4} \)
|
| $97$ |
\( (T - 4730)^{4} \)
|
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