Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(61.5873868082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.48840.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 56x - 120 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 56x - 120 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu^{2} - 12\nu - 147 \)
|
| \(\beta_{2}\) | \(=\) |
\( -8\nu^{2} + 56\nu + 283 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 11 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 14\beta _1 + 1209 ) / 32 \)
|
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 | 9.00000 | 0 | −87.0960 | 0 | 7.20586 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 31.7259 | 0 | −199.188 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 63.3700 | 0 | 221.982 | 0 | 81.0000 | 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 | 384.6.a.t | yes | 3 |
| 4.b | odd | 2 | 1 | 384.6.a.r | yes | 3 | |
| 8.b | even | 2 | 1 | 384.6.a.q | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 384.6.a.s | yes | 3 | |
| 16.e | even | 4 | 2 | 768.6.d.ba | 6 | ||
| 16.f | odd | 4 | 2 | 768.6.d.bb | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.6.a.q | ✓ | 3 | 8.b | even | 2 | 1 | |
| 384.6.a.r | yes | 3 | 4.b | odd | 2 | 1 | |
| 384.6.a.s | yes | 3 | 8.d | odd | 2 | 1 | |
| 384.6.a.t | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 768.6.d.ba | 6 | 16.e | even | 4 | 2 | ||
| 768.6.d.bb | 6 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(384))\):
|
\( T_{5}^{3} - 8T_{5}^{2} - 6272T_{5} + 175104 \)
|
|
\( T_{7}^{3} - 30T_{7}^{2} - 44052T_{7} + 318616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots + 175104 \)
|
| $7$ |
\( T^{3} - 30 T^{2} + \cdots + 318616 \)
|
| $11$ |
\( T^{3} - 284 T^{2} + \cdots - 14084416 \)
|
| $13$ |
\( T^{3} - 190 T^{2} + \cdots - 125544 \)
|
| $17$ |
\( T^{3} - 610 T^{2} + \cdots + 447326184 \)
|
| $19$ |
\( T^{3} - 1168 T^{2} + \cdots + 546988032 \)
|
| $23$ |
\( T^{3} + \cdots - 5942536000 \)
|
| $29$ |
\( T^{3} + \cdots + 39548118720 \)
|
| $31$ |
\( T^{3} + \cdots - 356750680008 \)
|
| $37$ |
\( T^{3} + \cdots + 180279918344 \)
|
| $41$ |
\( T^{3} + \cdots + 1744312659144 \)
|
| $43$ |
\( T^{3} + \cdots + 336684480000 \)
|
| $47$ |
\( T^{3} + \cdots - 4589384516160 \)
|
| $53$ |
\( T^{3} + \cdots - 20898044043840 \)
|
| $59$ |
\( T^{3} + \cdots - 12313859950144 \)
|
| $61$ |
\( T^{3} + \cdots + 1881061774856 \)
|
| $67$ |
\( T^{3} + \cdots + 25094551872064 \)
|
| $71$ |
\( T^{3} + \cdots - 64363441123008 \)
|
| $73$ |
\( T^{3} + \cdots + 26985987718120 \)
|
| $79$ |
\( T^{3} + \cdots + 38873745575160 \)
|
| $83$ |
\( T^{3} + \cdots + 403547148960192 \)
|
| $89$ |
\( T^{3} + \cdots + 412279413810424 \)
|
| $97$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
|
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