Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,6,Mod(1,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, 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("432.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 432.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: | \(69.2858101592\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2292.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 216) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 13x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu^{2} + 12\nu - 58 \)
|
| \(\beta_{2}\) | \(=\) |
\( -18\nu^{2} + 36\nu + 150 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 24 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 3\beta _1 + 324 ) / 36 \)
|
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 | −104.455 | 0 | 93.1555 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 23.4082 | 0 | 106.540 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 45.0468 | 0 | −214.696 | 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 | 432.6.a.w | 3 | |
| 3.b | odd | 2 | 1 | 432.6.a.x | 3 | ||
| 4.b | odd | 2 | 1 | 216.6.a.i | ✓ | 3 | |
| 12.b | even | 2 | 1 | 216.6.a.j | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.6.a.i | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 216.6.a.j | yes | 3 | 12.b | even | 2 | 1 | |
| 432.6.a.w | 3 | 1.a | even | 1 | 1 | trivial | |
| 432.6.a.x | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(432))\):
|
\( T_{5}^{3} + 36T_{5}^{2} - 6096T_{5} + 110144 \)
|
|
\( T_{7}^{3} + 15T_{7}^{2} - 32949T_{7} + 2130813 \)
|
|
\( T_{11}^{3} + 444T_{11}^{2} - 304080T_{11} - 129126464 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 36 T^{2} + \cdots + 110144 \)
|
| $7$ |
\( T^{3} + 15 T^{2} + \cdots + 2130813 \)
|
| $11$ |
\( T^{3} + 444 T^{2} + \cdots - 129126464 \)
|
| $13$ |
\( T^{3} + 669 T^{2} + \cdots - 391677025 \)
|
| $17$ |
\( T^{3} - 36 T^{2} + \cdots + 28100032 \)
|
| $19$ |
\( T^{3} - 747 T^{2} + \cdots + 409906775 \)
|
| $23$ |
\( T^{3} + \cdots - 60450433088 \)
|
| $29$ |
\( T^{3} + \cdots + 5202247680 \)
|
| $31$ |
\( T^{3} + \cdots + 16733167808 \)
|
| $37$ |
\( T^{3} + \cdots - 46578850983 \)
|
| $41$ |
\( T^{3} + \cdots - 789297758208 \)
|
| $43$ |
\( T^{3} + \cdots - 489643309568 \)
|
| $47$ |
\( T^{3} + \cdots - 925726370112 \)
|
| $53$ |
\( T^{3} + \cdots - 561238716928 \)
|
| $59$ |
\( T^{3} + \cdots + 36966632711104 \)
|
| $61$ |
\( T^{3} + \cdots - 793452880915 \)
|
| $67$ |
\( T^{3} + \cdots + 32669722281371 \)
|
| $71$ |
\( T^{3} + \cdots - 17074117554688 \)
|
| $73$ |
\( T^{3} + \cdots + 88144279928063 \)
|
| $79$ |
\( T^{3} + \cdots + 108979442056075 \)
|
| $83$ |
\( T^{3} + \cdots - 132628129059328 \)
|
| $89$ |
\( T^{3} + \cdots + 219760113389760 \)
|
| $97$ |
\( T^{3} + \cdots + 100559005858483 \)
|
| show more | |
| show less |