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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 257x^{2} - 702x - 63 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 257x^{2} - 702x - 63 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 8\nu^{2} + 194\nu - 219 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{3} + 20\nu^{2} + 1826\nu + 2895 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{3} + 4\nu^{2} - 812\nu - 3768 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - 6\beta _1 + 216 ) / 432 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_{3} + 2\beta_{2} + 90\beta _1 + 55944 ) / 432 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 149\beta_{3} + 202\beta_{2} - 654\beta _1 + 197424 ) / 216 \)
|
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 | −312.039 | 0 | 1405.78 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −92.4765 | 0 | −1121.29 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −13.8479 | 0 | −58.0683 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 522.364 | 0 | 265.581 | 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.y | 4 | |
| 3.b | odd | 2 | 1 | 432.8.a.x | 4 | ||
| 4.b | odd | 2 | 1 | 216.8.a.g | yes | 4 | |
| 12.b | even | 2 | 1 | 216.8.a.f | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.8.a.f | ✓ | 4 | 12.b | even | 2 | 1 | |
| 216.8.a.g | yes | 4 | 4.b | odd | 2 | 1 | |
| 432.8.a.x | 4 | 3.b | odd | 2 | 1 | ||
| 432.8.a.y | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\):
|
\( T_{5}^{4} - 104T_{5}^{3} - 184080T_{5}^{2} - 17600000T_{5} - 208736000 \)
|
|
\( T_{7}^{4} - 492T_{7}^{3} - 1532682T_{7}^{2} + 331488180T_{7} + 24309320913 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 104 T^{3} + \cdots - 208736000 \)
|
| $7$ |
\( T^{4} + \cdots + 24309320913 \)
|
| $11$ |
\( T^{4} + \cdots + 217226999918848 \)
|
| $13$ |
\( T^{4} + \cdots + 47\!\cdots\!73 \)
|
| $17$ |
\( T^{4} + \cdots + 11\!\cdots\!52 \)
|
| $19$ |
\( T^{4} + \cdots + 89\!\cdots\!41 \)
|
| $23$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots - 53\!\cdots\!40 \)
|
| $31$ |
\( T^{4} + \cdots + 29\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!57 \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 84\!\cdots\!52 \)
|
| $47$ |
\( T^{4} + \cdots + 47\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots - 37\!\cdots\!28 \)
|
| $59$ |
\( T^{4} + \cdots + 68\!\cdots\!32 \)
|
| $61$ |
\( T^{4} + \cdots + 18\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!61 \)
|
| $71$ |
\( T^{4} + \cdots - 20\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots - 24\!\cdots\!83 \)
|
| $79$ |
\( T^{4} + \cdots - 71\!\cdots\!15 \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots - 16\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots - 51\!\cdots\!35 \)
|
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