Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,8,Mod(1,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 576.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: | \(179.933774679\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{235}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 235 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 96) |
| 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 \(\beta = 16\sqrt{235}\). We also show the integral \(q\)-expansion of the trace form.
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 | 0 | 0 | −155.275 | 0 | 1742.38 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 335.275 | 0 | −710.377 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 576.8.a.bm | 2 | |
| 3.b | odd | 2 | 1 | 192.8.a.r | 2 | ||
| 4.b | odd | 2 | 1 | 576.8.a.bl | 2 | ||
| 8.b | even | 2 | 1 | 288.8.a.j | 2 | ||
| 8.d | odd | 2 | 1 | 288.8.a.i | 2 | ||
| 12.b | even | 2 | 1 | 192.8.a.u | 2 | ||
| 24.f | even | 2 | 1 | 96.8.a.d | ✓ | 2 | |
| 24.h | odd | 2 | 1 | 96.8.a.g | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 96.8.a.d | ✓ | 2 | 24.f | even | 2 | 1 | |
| 96.8.a.g | yes | 2 | 24.h | odd | 2 | 1 | |
| 192.8.a.r | 2 | 3.b | odd | 2 | 1 | ||
| 192.8.a.u | 2 | 12.b | even | 2 | 1 | ||
| 288.8.a.i | 2 | 8.d | odd | 2 | 1 | ||
| 288.8.a.j | 2 | 8.b | even | 2 | 1 | ||
| 576.8.a.bl | 2 | 4.b | odd | 2 | 1 | ||
| 576.8.a.bm | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(576))\):
|
\( T_{5}^{2} - 180T_{5} - 52060 \)
|
|
\( T_{7}^{2} - 1032T_{7} - 1237744 \)
|
|
\( T_{11}^{2} - 2840T_{11} - 149360 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 180T - 52060 \)
|
| $7$ |
\( T^{2} - 1032 T - 1237744 \)
|
| $11$ |
\( T^{2} - 2840 T - 149360 \)
|
| $13$ |
\( T^{2} - 340 T - 106093340 \)
|
| $17$ |
\( T^{2} + 9780 T - 342101340 \)
|
| $19$ |
\( T^{2} + 32040 T - 38143600 \)
|
| $23$ |
\( T^{2} + \cdots - 6520421376 \)
|
| $29$ |
\( T^{2} + \cdots + 23134731236 \)
|
| $31$ |
\( T^{2} + \cdots - 7317817840 \)
|
| $37$ |
\( T^{2} + \cdots + 257963905060 \)
|
| $41$ |
\( T^{2} + \cdots + 60115458500 \)
|
| $43$ |
\( T^{2} + \cdots - 40457328496 \)
|
| $47$ |
\( T^{2} + \cdots + 159030761536 \)
|
| $53$ |
\( T^{2} + \cdots + 398618486660 \)
|
| $59$ |
\( T^{2} + \cdots - 3051785437040 \)
|
| $61$ |
\( T^{2} + \cdots - 1219589451900 \)
|
| $67$ |
\( T^{2} + \cdots + 6764300717456 \)
|
| $71$ |
\( T^{2} + \cdots + 6518885551360 \)
|
| $73$ |
\( T^{2} + \cdots - 4889157880860 \)
|
| $79$ |
\( T^{2} + \cdots - 16532041465840 \)
|
| $83$ |
\( T^{2} + \cdots - 6199749233136 \)
|
| $89$ |
\( T^{2} + \cdots - 13565931134044 \)
|
| $97$ |
\( T^{2} + \cdots + 2418705617860 \)
|
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