Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(92.3810802123\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{31}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 31 \)
|
| 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 \(\beta = 16\sqrt{31}\). 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 | −71.0842 | 0 | −29.0842 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 107.084 | 0 | 149.084 | 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 | 576.6.a.bn | 2 | |
| 3.b | odd | 2 | 1 | 192.6.a.q | 2 | ||
| 4.b | odd | 2 | 1 | 576.6.a.bm | 2 | ||
| 8.b | even | 2 | 1 | 288.6.a.o | 2 | ||
| 8.d | odd | 2 | 1 | 288.6.a.n | 2 | ||
| 12.b | even | 2 | 1 | 192.6.a.r | 2 | ||
| 24.f | even | 2 | 1 | 96.6.a.g | ✓ | 2 | |
| 24.h | odd | 2 | 1 | 96.6.a.h | yes | 2 | |
| 48.i | odd | 4 | 2 | 768.6.d.s | 4 | ||
| 48.k | even | 4 | 2 | 768.6.d.z | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 96.6.a.g | ✓ | 2 | 24.f | even | 2 | 1 | |
| 96.6.a.h | yes | 2 | 24.h | odd | 2 | 1 | |
| 192.6.a.q | 2 | 3.b | odd | 2 | 1 | ||
| 192.6.a.r | 2 | 12.b | even | 2 | 1 | ||
| 288.6.a.n | 2 | 8.d | odd | 2 | 1 | ||
| 288.6.a.o | 2 | 8.b | even | 2 | 1 | ||
| 576.6.a.bm | 2 | 4.b | odd | 2 | 1 | ||
| 576.6.a.bn | 2 | 1.a | even | 1 | 1 | trivial | |
| 768.6.d.s | 4 | 48.i | odd | 4 | 2 | ||
| 768.6.d.z | 4 | 48.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(576))\):
|
\( T_{5}^{2} - 36T_{5} - 7612 \)
|
|
\( T_{7}^{2} - 120T_{7} - 4336 \)
|
|
\( T_{11}^{2} - 200T_{11} - 275696 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 36T - 7612 \)
|
| $7$ |
\( T^{2} - 120T - 4336 \)
|
| $11$ |
\( T^{2} - 200T - 275696 \)
|
| $13$ |
\( T^{2} + 284T - 265532 \)
|
| $17$ |
\( T^{2} + 2676 T + 1504548 \)
|
| $19$ |
\( T^{2} - 72T - 792304 \)
|
| $23$ |
\( T^{2} - 3840 T - 3456000 \)
|
| $29$ |
\( T^{2} - 10212 T + 23206340 \)
|
| $31$ |
\( T^{2} + 10488 T + 27301136 \)
|
| $37$ |
\( T^{2} + 13148 T - 12778940 \)
|
| $41$ |
\( T^{2} + 4164 T - 15505276 \)
|
| $43$ |
\( T^{2} + 5832 T - 125615344 \)
|
| $47$ |
\( T^{2} - 1520 T - 310545344 \)
|
| $53$ |
\( T^{2} - 9012 T - 19701340 \)
|
| $59$ |
\( T^{2} - 55096 T + 388630288 \)
|
| $61$ |
\( T^{2} - 63444 T + 988000740 \)
|
| $67$ |
\( T^{2} + \cdots - 1551497968 \)
|
| $71$ |
\( T^{2} + \cdots - 1167829760 \)
|
| $73$ |
\( T^{2} + \cdots - 2060240220 \)
|
| $79$ |
\( T^{2} + \cdots + 4557502736 \)
|
| $83$ |
\( T^{2} + \cdots + 2593974672 \)
|
| $89$ |
\( T^{2} + \cdots - 2331558940 \)
|
| $97$ |
\( T^{2} + \cdots + 1164856516 \)
|
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