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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu^{2} + 13\nu + 36 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 8\nu^{2} + 13\nu - 36 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 18 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} - 5\beta_{2} + 26\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(511\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) |
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 | −1.12311 | − | 1.12311i | 0 | 5.24621 | − | 5.24621i | 0 | − | 34.2462i | 0 | − | 24.4773i | 0 | ||||||||||||||||||||||||
| 129.2 | 0 | 7.12311 | + | 7.12311i | 0 | −11.2462 | + | 11.2462i | 0 | − | 17.7538i | 0 | 74.4773i | 0 | ||||||||||||||||||||||||||
| 385.1 | 0 | −1.12311 | + | 1.12311i | 0 | 5.24621 | + | 5.24621i | 0 | 34.2462i | 0 | 24.4773i | 0 | |||||||||||||||||||||||||||
| 385.2 | 0 | 7.12311 | − | 7.12311i | 0 | −11.2462 | − | 11.2462i | 0 | 17.7538i | 0 | − | 74.4773i | 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.o | yes | 4 |
| 4.b | odd | 2 | 1 | 512.4.e.i | ✓ | 4 | |
| 8.b | even | 2 | 1 | 512.4.e.j | yes | 4 | |
| 8.d | odd | 2 | 1 | 512.4.e.p | yes | 4 | |
| 16.e | even | 4 | 1 | 512.4.e.j | yes | 4 | |
| 16.e | even | 4 | 1 | inner | 512.4.e.o | yes | 4 |
| 16.f | odd | 4 | 1 | 512.4.e.i | ✓ | 4 | |
| 16.f | odd | 4 | 1 | 512.4.e.p | yes | 4 | |
| 32.g | even | 8 | 2 | 1024.4.a.i | 4 | ||
| 32.g | even | 8 | 2 | 1024.4.b.e | 4 | ||
| 32.h | odd | 8 | 2 | 1024.4.a.f | 4 | ||
| 32.h | odd | 8 | 2 | 1024.4.b.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 512.4.e.i | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 512.4.e.i | ✓ | 4 | 16.f | odd | 4 | 1 | |
| 512.4.e.j | yes | 4 | 8.b | even | 2 | 1 | |
| 512.4.e.j | yes | 4 | 16.e | even | 4 | 1 | |
| 512.4.e.o | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 512.4.e.o | yes | 4 | 16.e | even | 4 | 1 | inner |
| 512.4.e.p | yes | 4 | 8.d | odd | 2 | 1 | |
| 512.4.e.p | yes | 4 | 16.f | odd | 4 | 1 | |
| 1024.4.a.f | 4 | 32.h | odd | 8 | 2 | ||
| 1024.4.a.i | 4 | 32.g | even | 8 | 2 | ||
| 1024.4.b.e | 4 | 32.g | even | 8 | 2 | ||
| 1024.4.b.h | 4 | 32.h | odd | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(512, [\chi])\):
|
\( T_{3}^{4} - 12T_{3}^{3} + 72T_{3}^{2} + 192T_{3} + 256 \)
|
|
\( T_{5}^{4} + 12T_{5}^{3} + 72T_{5}^{2} - 1416T_{5} + 13924 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 12 T^{3} + \cdots + 256 \)
|
| $5$ |
\( T^{4} + 12 T^{3} + \cdots + 13924 \)
|
| $7$ |
\( T^{4} + 1488 T^{2} + 369664 \)
|
| $11$ |
\( T^{4} - 44 T^{3} + \cdots + 2027776 \)
|
| $13$ |
\( T^{4} + 20 T^{3} + \cdots + 1378276 \)
|
| $17$ |
\( (T^{2} + 104 T + 2432)^{2} \)
|
| $19$ |
\( T^{4} - 116 T^{3} + \cdots + 2715904 \)
|
| $23$ |
\( T^{4} + 39376 T^{2} + 384787456 \)
|
| $29$ |
\( T^{4} + 220 T^{3} + \cdots + 376996 \)
|
| $31$ |
\( (T^{2} - 88 T - 11392)^{2} \)
|
| $37$ |
\( T^{4} - 348 T^{3} + \cdots + 61559716 \)
|
| $41$ |
\( T^{4} + \cdots + 1270779904 \)
|
| $43$ |
\( T^{4} + \cdots + 2649366784 \)
|
| $47$ |
\( (T^{2} + 360 T - 28800)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 19422881956 \)
|
| $59$ |
\( T^{4} + \cdots + 150705451264 \)
|
| $61$ |
\( T^{4} + \cdots + 7627926244 \)
|
| $67$ |
\( T^{4} + \cdots + 383943815424 \)
|
| $71$ |
\( T^{4} + \cdots + 2664211456 \)
|
| $73$ |
\( T^{4} + \cdots + 10880158864 \)
|
| $79$ |
\( (T^{2} - 1600 T + 483328)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 139971760384 \)
|
| $89$ |
\( T^{4} + \cdots + 3852436624 \)
|
| $97$ |
\( (T^{2} - 280 T - 483328)^{2} \)
|
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