Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(134.950331009\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1289}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 322 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 108) |
| 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 \(\beta = 9\sqrt{1289}\). 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 | −485.124 | 0 | 1459.37 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 161.124 | 0 | −479.371 | 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 | 432.8.a.i | 2 | |
| 3.b | odd | 2 | 1 | 432.8.a.r | 2 | ||
| 4.b | odd | 2 | 1 | 108.8.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 108.8.a.e | yes | 2 | |
| 36.f | odd | 6 | 2 | 324.8.e.j | 4 | ||
| 36.h | even | 6 | 2 | 324.8.e.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.8.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 108.8.a.e | yes | 2 | 12.b | even | 2 | 1 | |
| 324.8.e.g | 4 | 36.h | even | 6 | 2 | ||
| 324.8.e.j | 4 | 36.f | odd | 6 | 2 | ||
| 432.8.a.i | 2 | 1.a | even | 1 | 1 | trivial | |
| 432.8.a.r | 2 | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\):
|
\( T_{5}^{2} + 324T_{5} - 78165 \)
|
|
\( T_{7}^{2} - 980T_{7} - 699581 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 324T - 78165 \)
|
| $7$ |
\( T^{2} - 980T - 699581 \)
|
| $11$ |
\( T^{2} - 810 T - 1506519 \)
|
| $13$ |
\( T^{2} + 7880 T - 44615984 \)
|
| $17$ |
\( T^{2} - 20736 T - 59558976 \)
|
| $19$ |
\( T^{2} + 10456 T - 66636116 \)
|
| $23$ |
\( T^{2} + 60588 T + 750672036 \)
|
| $29$ |
\( T^{2} + \cdots - 40131731556 \)
|
| $31$ |
\( T^{2} + \cdots - 11797562189 \)
|
| $37$ |
\( T^{2} + \cdots - 3821580476 \)
|
| $41$ |
\( T^{2} + \cdots + 276972137100 \)
|
| $43$ |
\( T^{2} + \cdots - 239610916916 \)
|
| $47$ |
\( T^{2} + \cdots + 485764195716 \)
|
| $53$ |
\( T^{2} + \cdots + 883947989259 \)
|
| $59$ |
\( T^{2} + \cdots + 1945901264400 \)
|
| $61$ |
\( T^{2} + \cdots - 572702600000 \)
|
| $67$ |
\( T^{2} + \cdots + 894770581324 \)
|
| $71$ |
\( T^{2} + \cdots + 473739274944 \)
|
| $73$ |
\( T^{2} + \cdots - 4430654478791 \)
|
| $79$ |
\( T^{2} + \cdots + 211199920000 \)
|
| $83$ |
\( T^{2} + \cdots + 1188922519881 \)
|
| $89$ |
\( T^{2} + \cdots + 3197429581356 \)
|
| $97$ |
\( T^{2} + \cdots - 33701749740911 \)
|
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