Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [512,4,Mod(129,512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("512.129");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 512 = 2^{9} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 512.e (of order \(4\), degree \(2\), 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: | \(30.2089779229\) |
| 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: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(511\) |
| \(\chi(n)\) | \(i\) | \(1\) |
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 | 4.00000 | + | 4.00000i | 0 | −1.00000 | + | 1.00000i | 0 | 8.00000i | 0 | 5.00000i | 0 | ||||||||||||||||||||
| 385.1 | 0 | 4.00000 | − | 4.00000i | 0 | −1.00000 | − | 1.00000i | 0 | − | 8.00000i | 0 | − | 5.00000i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 512.4.e.g | yes | 2 |
| 4.b | odd | 2 | 1 | 512.4.e.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 512.4.e.b | yes | 2 | |
| 8.d | odd | 2 | 1 | 512.4.e.h | yes | 2 | |
| 16.e | even | 4 | 1 | 512.4.e.b | yes | 2 | |
| 16.e | even | 4 | 1 | inner | 512.4.e.g | yes | 2 |
| 16.f | odd | 4 | 1 | 512.4.e.a | ✓ | 2 | |
| 16.f | odd | 4 | 1 | 512.4.e.h | yes | 2 | |
| 32.g | even | 8 | 2 | 1024.4.a.a | 2 | ||
| 32.g | even | 8 | 2 | 1024.4.b.d | 2 | ||
| 32.h | odd | 8 | 2 | 1024.4.a.d | 2 | ||
| 32.h | odd | 8 | 2 | 1024.4.b.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 512.4.e.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 512.4.e.a | ✓ | 2 | 16.f | odd | 4 | 1 | |
| 512.4.e.b | yes | 2 | 8.b | even | 2 | 1 | |
| 512.4.e.b | yes | 2 | 16.e | even | 4 | 1 | |
| 512.4.e.g | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 512.4.e.g | yes | 2 | 16.e | even | 4 | 1 | inner |
| 512.4.e.h | yes | 2 | 8.d | odd | 2 | 1 | |
| 512.4.e.h | yes | 2 | 16.f | odd | 4 | 1 | |
| 1024.4.a.a | 2 | 32.g | even | 8 | 2 | ||
| 1024.4.a.d | 2 | 32.h | odd | 8 | 2 | ||
| 1024.4.b.a | 2 | 32.h | odd | 8 | 2 | ||
| 1024.4.b.d | 2 | 32.g | even | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(512, [\chi])\):
|
\( T_{3}^{2} - 8T_{3} + 32 \)
|
|
\( T_{5}^{2} + 2T_{5} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 8T + 32 \)
|
| $5$ |
\( T^{2} + 2T + 2 \)
|
| $7$ |
\( T^{2} + 64 \)
|
| $11$ |
\( T^{2} - 8T + 32 \)
|
| $13$ |
\( T^{2} + 6T + 18 \)
|
| $17$ |
\( (T - 24)^{2} \)
|
| $19$ |
\( T^{2} - 184T + 16928 \)
|
| $23$ |
\( T^{2} + 33856 \)
|
| $29$ |
\( T^{2} + 138T + 9522 \)
|
| $31$ |
\( (T + 176)^{2} \)
|
| $37$ |
\( T^{2} + 430T + 92450 \)
|
| $41$ |
\( T^{2} + 40000 \)
|
| $43$ |
\( T^{2} + 200T + 20000 \)
|
| $47$ |
\( (T - 208)^{2} \)
|
| $53$ |
\( T^{2} + 526T + 138338 \)
|
| $59$ |
\( T^{2} - 216T + 23328 \)
|
| $61$ |
\( T^{2} + 678T + 229842 \)
|
| $67$ |
\( T^{2} + 968T + 468512 \)
|
| $71$ |
\( T^{2} + 876096 \)
|
| $73$ |
\( T^{2} + 164836 \)
|
| $79$ |
\( (T - 768)^{2} \)
|
| $83$ |
\( T^{2} + 728T + 264992 \)
|
| $89$ |
\( T^{2} + 1705636 \)
|
| $97$ |
\( (T + 472)^{2} \)
|
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