Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1024,4,Mod(1,1024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1024.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1024 = 2^{10} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1024.a (trivial) |
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: | yes |
| Analytic conductor: | \(60.4179558459\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{43})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 44x^{2} + 441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 256) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 44x^{2} + 441 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 23\nu ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 65\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 22 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 23\beta_{2} + 65\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.55744 | 0 | −3.61676 | 0 | 2.20255 | 0 | −14.3446 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.55744 | 0 | 3.61676 | 0 | −2.20255 | 0 | −14.3446 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.55744 | 0 | −14.9305 | 0 | 16.3447 | 0 | 64.3446 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.55744 | 0 | 14.9305 | 0 | −16.3447 | 0 | 64.3446 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 1024.4.a.j | 4 | |
| 4.b | odd | 2 | 1 | 1024.4.a.e | 4 | ||
| 8.b | even | 2 | 1 | 1024.4.a.e | 4 | ||
| 8.d | odd | 2 | 1 | inner | 1024.4.a.j | 4 | |
| 16.e | even | 4 | 2 | 1024.4.b.i | 8 | ||
| 16.f | odd | 4 | 2 | 1024.4.b.i | 8 | ||
| 32.g | even | 8 | 2 | 256.4.e.a | ✓ | 8 | |
| 32.g | even | 8 | 2 | 256.4.e.b | yes | 8 | |
| 32.h | odd | 8 | 2 | 256.4.e.a | ✓ | 8 | |
| 32.h | odd | 8 | 2 | 256.4.e.b | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 256.4.e.a | ✓ | 8 | 32.g | even | 8 | 2 | |
| 256.4.e.a | ✓ | 8 | 32.h | odd | 8 | 2 | |
| 256.4.e.b | yes | 8 | 32.g | even | 8 | 2 | |
| 256.4.e.b | yes | 8 | 32.h | odd | 8 | 2 | |
| 1024.4.a.e | 4 | 4.b | odd | 2 | 1 | ||
| 1024.4.a.e | 4 | 8.b | even | 2 | 1 | ||
| 1024.4.a.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 1024.4.a.j | 4 | 8.d | odd | 2 | 1 | inner | |
| 1024.4.b.i | 8 | 16.e | even | 4 | 2 | ||
| 1024.4.b.i | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\):
|
\( T_{3}^{2} - 6T_{3} - 34 \)
|
|
\( T_{5}^{4} - 236T_{5}^{2} + 2916 \)
|
|
\( T_{7}^{4} - 272T_{7}^{2} + 1296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 6 T - 34)^{2} \)
|
| $5$ |
\( T^{4} - 236T^{2} + 2916 \)
|
| $7$ |
\( T^{4} - 272T^{2} + 1296 \)
|
| $11$ |
\( (T^{2} - 2 T - 2106)^{2} \)
|
| $13$ |
\( T^{4} - 2108 T^{2} + 777924 \)
|
| $17$ |
\( (T^{2} + 96 T + 756)^{2} \)
|
| $19$ |
\( (T^{2} - 82 T + 1638)^{2} \)
|
| $23$ |
\( T^{4} - 9776 T^{2} + 22240656 \)
|
| $29$ |
\( T^{4} - 62252 T^{2} + 515199204 \)
|
| $31$ |
\( T^{4} - 105536 T^{2} + 76317696 \)
|
| $37$ |
\( T^{4} + \cdots + 19693631556 \)
|
| $41$ |
\( (T^{2} - 320 T + 24912)^{2} \)
|
| $43$ |
\( (T^{2} + 510 T + 46062)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 4274021376 \)
|
| $53$ |
\( T^{4} - 139388 T^{2} + 453434436 \)
|
| $59$ |
\( (T^{2} - 390 T - 27378)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 104343212484 \)
|
| $67$ |
\( (T^{2} - 646 T + 103942)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 31922254224 \)
|
| $73$ |
\( (T^{2} - 436 T - 592488)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 88223256576 \)
|
| $83$ |
\( (T^{2} - 1490 T + 361998)^{2} \)
|
| $89$ |
\( (T^{2} + 268 T - 44136)^{2} \)
|
| $97$ |
\( (T^{2} + 1120 T + 168948)^{2} \)
|
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