Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,6,Mod(1,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, 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("216.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.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: | \(34.6429050796\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{185}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 46 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| 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 \(\beta = 6\sqrt{185}\). 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 | −100.609 | 0 | −65.3912 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 62.6088 | 0 | −228.609 | 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 | 216.6.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 216.6.a.h | yes | 2 | |
| 4.b | odd | 2 | 1 | 432.6.a.l | 2 | ||
| 12.b | even | 2 | 1 | 432.6.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.6.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 216.6.a.h | yes | 2 | 3.b | odd | 2 | 1 | |
| 432.6.a.l | 2 | 4.b | odd | 2 | 1 | ||
| 432.6.a.u | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 38T_{5} - 6299 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(216))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 38T - 6299 \)
|
| $7$ |
\( T^{2} + 294T + 14949 \)
|
| $11$ |
\( T^{2} - 130T - 235535 \)
|
| $13$ |
\( T^{2} + 112T - 23504 \)
|
| $17$ |
\( T^{2} + 2584 T + 1642624 \)
|
| $19$ |
\( T^{2} - 1924 T - 779516 \)
|
| $23$ |
\( T^{2} - 4876 T + 4638484 \)
|
| $29$ |
\( T^{2} - 1548 T - 10056924 \)
|
| $31$ |
\( T^{2} - 2806 T - 74281931 \)
|
| $37$ |
\( T^{2} - 4428 T - 75684204 \)
|
| $41$ |
\( T^{2} - 5652 T + 1166436 \)
|
| $43$ |
\( T^{2} - 9476 T - 188566796 \)
|
| $47$ |
\( T^{2} - 14028 T - 34346844 \)
|
| $53$ |
\( T^{2} + 15770 T - 884405915 \)
|
| $59$ |
\( T^{2} + 33752 T - 61414064 \)
|
| $61$ |
\( T^{2} + \cdots + 2225147584 \)
|
| $67$ |
\( T^{2} + \cdots + 1184550100 \)
|
| $71$ |
\( T^{2} + \cdots + 3416143744 \)
|
| $73$ |
\( T^{2} + \cdots - 1223201879 \)
|
| $79$ |
\( T^{2} - 27752 T + 189319936 \)
|
| $83$ |
\( T^{2} + \cdots + 3623071201 \)
|
| $89$ |
\( T^{2} + \cdots + 11166238836 \)
|
| $97$ |
\( T^{2} + \cdots - 6930977279 \)
|
| show more | |
| show less |