Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,8,Mod(575,576)]
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([1, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.575");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 576.c (of order \(2\), degree \(1\), not 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: | \(179.933774679\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-555})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 283x^{2} - 282x + 18771 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 144) |
| 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
\( x^{4} - 2x^{3} + 283x^{2} - 282x + 18771 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -6\nu^{3} + 9\nu^{2} - 873\nu + 435 ) / 547 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 288\nu^{3} - 432\nu^{2} + 120672\nu - 60264 ) / 547 \)
|
| \(\beta_{3}\) | \(=\) |
\( 216\nu^{2} - 216\nu + 30456 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 48\beta _1 + 72 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 3\beta_{2} + 144\beta _1 - 60696 ) / 432 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 144\beta_{2} - 20040\beta _1 - 30384 ) / 144 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(325\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 575.1 |
|
0 | 0 | 0 | − | 275.772i | 0 | − | 1696.21i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 575.2 | 0 | 0 | 0 | − | 275.772i | 0 | 1696.21i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 575.3 | 0 | 0 | 0 | 275.772i | 0 | − | 1696.21i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 575.4 | 0 | 0 | 0 | 275.772i | 0 | 1696.21i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 576.8.c.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 576.8.c.c | 4 | |
| 4.b | odd | 2 | 1 | inner | 576.8.c.c | 4 | |
| 8.b | even | 2 | 1 | 144.8.c.b | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 144.8.c.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | inner | 576.8.c.c | 4 | |
| 24.f | even | 2 | 1 | 144.8.c.b | ✓ | 4 | |
| 24.h | odd | 2 | 1 | 144.8.c.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 144.8.c.b | ✓ | 4 | 8.b | even | 2 | 1 | |
| 144.8.c.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 144.8.c.b | ✓ | 4 | 24.f | even | 2 | 1 | |
| 144.8.c.b | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 576.8.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 576.8.c.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 576.8.c.c | 4 | 4.b | odd | 2 | 1 | inner | |
| 576.8.c.c | 4 | 12.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 76050 \)
acting on \(S_{8}^{\mathrm{new}}(576, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 76050)^{2} \)
|
| $7$ |
\( (T^{2} + 2877120)^{2} \)
|
| $11$ |
\( (T^{2} - 51788160)^{2} \)
|
| $13$ |
\( (T - 10868)^{4} \)
|
| $17$ |
\( (T^{2} + 215862642)^{2} \)
|
| $19$ |
\( (T^{2} + 184135680)^{2} \)
|
| $23$ |
\( (T^{2} - 8752199040)^{2} \)
|
| $29$ |
\( (T^{2} + 32802312978)^{2} \)
|
| $31$ |
\( (T^{2} + 140978880)^{2} \)
|
| $37$ |
\( (T + 107354)^{4} \)
|
| $41$ |
\( (T^{2} + 96456225762)^{2} \)
|
| $43$ |
\( (T^{2} + 235336907520)^{2} \)
|
| $47$ |
\( (T^{2} - 916080762240)^{2} \)
|
| $53$ |
\( (T^{2} + 1991940136578)^{2} \)
|
| $59$ |
\( (T^{2} - 35008796160)^{2} \)
|
| $61$ |
\( (T - 1107730)^{4} \)
|
| $67$ |
\( (T^{2} + 4877305332480)^{2} \)
|
| $71$ |
\( (T^{2} - 14712446586240)^{2} \)
|
| $73$ |
\( (T + 2589392)^{4} \)
|
| $79$ |
\( (T^{2} + 32617014655680)^{2} \)
|
| $83$ |
\( (T^{2} - 11981760485760)^{2} \)
|
| $89$ |
\( (T^{2} + 5694725265138)^{2} \)
|
| $97$ |
\( (T + 10135736)^{4} \)
|
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