Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,6,Mod(65,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.65");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 128.b (of order \(2\), degree \(1\), 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: | \(20.5291289361\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-29})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 28x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | yes |
| 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 in terms of a root \(\nu\) of
\( x^{4} + 28x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 86\nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -16\nu^{3} - 208\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{2} + 112 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 8\beta_1 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 112 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -43\beta_{2} - 104\beta_1 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | − | 15.2315i | 0 | − | 43.0813i | 0 | −22.6274 | 0 | 11.0000 | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | − | 15.2315i | 0 | 43.0813i | 0 | 22.6274 | 0 | 11.0000 | 0 | ||||||||||||||||||||||||||||||
| 65.3 | 0 | 15.2315i | 0 | − | 43.0813i | 0 | 22.6274 | 0 | 11.0000 | 0 | ||||||||||||||||||||||||||||||
| 65.4 | 0 | 15.2315i | 0 | 43.0813i | 0 | −22.6274 | 0 | 11.0000 | 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 | 128.6.b.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 128.6.b.d | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 128.6.b.d | ✓ | 4 |
| 8.d | odd | 2 | 1 | inner | 128.6.b.d | ✓ | 4 |
| 16.e | even | 4 | 2 | 256.6.a.m | 4 | ||
| 16.f | odd | 4 | 2 | 256.6.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.6.b.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 128.6.b.d | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 128.6.b.d | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 128.6.b.d | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 256.6.a.m | 4 | 16.e | even | 4 | 2 | ||
| 256.6.a.m | 4 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 232 \)
acting on \(S_{6}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 232)^{2} \)
|
| $5$ |
\( (T^{2} + 1856)^{2} \)
|
| $7$ |
\( (T^{2} - 512)^{2} \)
|
| $11$ |
\( (T^{2} + 67048)^{2} \)
|
| $13$ |
\( (T^{2} + 981824)^{2} \)
|
| $17$ |
\( (T + 218)^{4} \)
|
| $19$ |
\( (T^{2} + 3981352)^{2} \)
|
| $23$ |
\( (T^{2} - 11674112)^{2} \)
|
| $29$ |
\( (T^{2} + 23699264)^{2} \)
|
| $31$ |
\( (T^{2} - 75497472)^{2} \)
|
| $37$ |
\( (T^{2} + 10440000)^{2} \)
|
| $41$ |
\( (T - 8410)^{4} \)
|
| $43$ |
\( (T^{2} + 469800)^{2} \)
|
| $47$ |
\( (T^{2} - 329320448)^{2} \)
|
| $53$ |
\( (T^{2} + 706558784)^{2} \)
|
| $59$ |
\( (T^{2} + 1142358952)^{2} \)
|
| $61$ |
\( (T^{2} + 748385600)^{2} \)
|
| $67$ |
\( (T^{2} + 4452813352)^{2} \)
|
| $71$ |
\( (T^{2} - 466956800)^{2} \)
|
| $73$ |
\( (T - 43006)^{4} \)
|
| $79$ |
\( (T^{2} - 4193814528)^{2} \)
|
| $83$ |
\( (T^{2} + 4992714472)^{2} \)
|
| $89$ |
\( (T - 93870)^{4} \)
|
| $97$ |
\( (T + 81706)^{4} \)
|
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