Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,8,Mod(1,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, 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("384.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.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: | \(119.955849786\) |
| 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: |
\( x^{4} - 430x^{2} - 2448x + 12138 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3 \) |
| 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 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
\( x^{4} - 430x^{2} - 2448x + 12138 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -96\nu^{3} + 1360\nu^{2} + 36928\nu - 116144 ) / 119 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -200\nu^{3} + 2516\nu^{2} + 52816\nu - 173740 ) / 119 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{3} + 132\nu^{2} + 1808\nu - 13692 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} - 2\beta_{2} + 7\beta_1 ) / 768 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 38\beta_{3} - 34\beta_{2} + 17\beta _1 + 41280 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 173\beta_{3} - 337\beta_{2} + 338\beta _1 + 176256 ) / 96 \)
|
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 | 27.0000 | 0 | −432.433 | 0 | 113.601 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 27.0000 | 0 | −76.3562 | 0 | 439.494 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 27.0000 | 0 | 170.323 | 0 | −803.427 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 27.0000 | 0 | 530.466 | 0 | 930.332 | 0 | 729.000 | 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 | 384.8.a.s | yes | 4 |
| 4.b | odd | 2 | 1 | 384.8.a.o | yes | 4 | |
| 8.b | even | 2 | 1 | 384.8.a.n | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 384.8.a.r | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.8.a.n | ✓ | 4 | 8.b | even | 2 | 1 | |
| 384.8.a.o | yes | 4 | 4.b | odd | 2 | 1 | |
| 384.8.a.r | yes | 4 | 8.d | odd | 2 | 1 | |
| 384.8.a.s | yes | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(384))\):
|
\( T_{5}^{4} - 192T_{5}^{3} - 233184T_{5}^{2} + 22830080T_{5} + 2983276800 \)
|
|
\( T_{7}^{4} - 680T_{7}^{3} - 627336T_{7}^{2} + 407076832T_{7} - 37318092656 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 27)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 2983276800 \)
|
| $7$ |
\( T^{4} + \cdots - 37318092656 \)
|
| $11$ |
\( T^{4} + \cdots + 552050534693120 \)
|
| $13$ |
\( T^{4} + \cdots - 22\!\cdots\!52 \)
|
| $17$ |
\( T^{4} + \cdots + 50\!\cdots\!12 \)
|
| $19$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots - 10\!\cdots\!48 \)
|
| $29$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 76\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots + 21\!\cdots\!80 \)
|
| $41$ |
\( T^{4} + \cdots + 51\!\cdots\!76 \)
|
| $43$ |
\( T^{4} + \cdots - 38\!\cdots\!68 \)
|
| $47$ |
\( T^{4} + \cdots + 37\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots + 81\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots - 12\!\cdots\!84 \)
|
| $61$ |
\( T^{4} + \cdots + 70\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots + 86\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 52\!\cdots\!72 \)
|
| $73$ |
\( T^{4} + \cdots - 15\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots - 11\!\cdots\!48 \)
|
| $83$ |
\( T^{4} + \cdots - 15\!\cdots\!92 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
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