Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,8,Mod(129,256)]
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, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.129");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.b (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: | \(79.9705665239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 129.1 |
|
0 | − | 84.0000i | 0 | − | 82.0000i | 0 | −456.000 | 0 | −4869.00 | 0 | ||||||||||||||||||||||
| 129.2 | 0 | 84.0000i | 0 | 82.0000i | 0 | −456.000 | 0 | −4869.00 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.8.b.c | 2 | |
| 4.b | odd | 2 | 1 | 256.8.b.e | 2 | ||
| 8.b | even | 2 | 1 | inner | 256.8.b.c | 2 | |
| 8.d | odd | 2 | 1 | 256.8.b.e | 2 | ||
| 16.e | even | 4 | 1 | 16.8.a.c | 1 | ||
| 16.e | even | 4 | 1 | 64.8.a.a | 1 | ||
| 16.f | odd | 4 | 1 | 8.8.a.a | ✓ | 1 | |
| 16.f | odd | 4 | 1 | 64.8.a.g | 1 | ||
| 48.i | odd | 4 | 1 | 144.8.a.g | 1 | ||
| 48.i | odd | 4 | 1 | 576.8.a.k | 1 | ||
| 48.k | even | 4 | 1 | 72.8.a.d | 1 | ||
| 48.k | even | 4 | 1 | 576.8.a.j | 1 | ||
| 80.i | odd | 4 | 1 | 400.8.c.b | 2 | ||
| 80.j | even | 4 | 1 | 200.8.c.a | 2 | ||
| 80.k | odd | 4 | 1 | 200.8.a.i | 1 | ||
| 80.q | even | 4 | 1 | 400.8.a.b | 1 | ||
| 80.s | even | 4 | 1 | 200.8.c.a | 2 | ||
| 80.t | odd | 4 | 1 | 400.8.c.b | 2 | ||
| 112.j | even | 4 | 1 | 392.8.a.d | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.8.a.a | ✓ | 1 | 16.f | odd | 4 | 1 | |
| 16.8.a.c | 1 | 16.e | even | 4 | 1 | ||
| 64.8.a.a | 1 | 16.e | even | 4 | 1 | ||
| 64.8.a.g | 1 | 16.f | odd | 4 | 1 | ||
| 72.8.a.d | 1 | 48.k | even | 4 | 1 | ||
| 144.8.a.g | 1 | 48.i | odd | 4 | 1 | ||
| 200.8.a.i | 1 | 80.k | odd | 4 | 1 | ||
| 200.8.c.a | 2 | 80.j | even | 4 | 1 | ||
| 200.8.c.a | 2 | 80.s | even | 4 | 1 | ||
| 256.8.b.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 256.8.b.c | 2 | 8.b | even | 2 | 1 | inner | |
| 256.8.b.e | 2 | 4.b | odd | 2 | 1 | ||
| 256.8.b.e | 2 | 8.d | odd | 2 | 1 | ||
| 392.8.a.d | 1 | 112.j | even | 4 | 1 | ||
| 400.8.a.b | 1 | 80.q | even | 4 | 1 | ||
| 400.8.c.b | 2 | 80.i | odd | 4 | 1 | ||
| 400.8.c.b | 2 | 80.t | odd | 4 | 1 | ||
| 576.8.a.j | 1 | 48.k | even | 4 | 1 | ||
| 576.8.a.k | 1 | 48.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(256, [\chi])\):
|
\( T_{3}^{2} + 7056 \)
|
|
\( T_{7} + 456 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 7056 \)
|
| $5$ |
\( T^{2} + 6724 \)
|
| $7$ |
\( (T + 456)^{2} \)
|
| $11$ |
\( T^{2} + 6370576 \)
|
| $13$ |
\( T^{2} + 116165284 \)
|
| $17$ |
\( (T + 11150)^{2} \)
|
| $19$ |
\( T^{2} + 17007376 \)
|
| $23$ |
\( (T - 81704)^{2} \)
|
| $29$ |
\( T^{2} + 9959640804 \)
|
| $31$ |
\( (T - 40480)^{2} \)
|
| $37$ |
\( T^{2} + 175931591364 \)
|
| $41$ |
\( (T + 141402)^{2} \)
|
| $43$ |
\( T^{2} + 476690823184 \)
|
| $47$ |
\( (T - 682032)^{2} \)
|
| $53$ |
\( T^{2} + 3287396881924 \)
|
| $59$ |
\( T^{2} + 933210096784 \)
|
| $61$ |
\( T^{2} + 3563298028900 \)
|
| $67$ |
\( T^{2} + 8796372993424 \)
|
| $71$ |
\( (T + 2548232)^{2} \)
|
| $73$ |
\( (T - 1680326)^{2} \)
|
| $79$ |
\( (T + 4038064)^{2} \)
|
| $83$ |
\( T^{2} + 29006453863696 \)
|
| $89$ |
\( (T - 6473046)^{2} \)
|
| $97$ |
\( (T + 6065758)^{2} \)
|
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