Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(1,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.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: | \(26.4328556826\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 224) |
| 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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{2} + 4\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 8 ) / 2 \)
|
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 | −6.24797 | 0 | 18.9243 | 0 | −7.00000 | 0 | 12.0371 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −1.20293 | 0 | −5.37748 | 0 | −7.00000 | 0 | −25.5530 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 7.45090 | 0 | −7.54680 | 0 | −7.00000 | 0 | 28.5159 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
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 | 448.4.a.v | 3 | |
| 4.b | odd | 2 | 1 | 448.4.a.w | 3 | ||
| 8.b | even | 2 | 1 | 224.4.a.f | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 224.4.a.g | yes | 3 | |
| 24.f | even | 2 | 1 | 2016.4.a.x | 3 | ||
| 24.h | odd | 2 | 1 | 2016.4.a.w | 3 | ||
| 56.e | even | 2 | 1 | 1568.4.a.x | 3 | ||
| 56.h | odd | 2 | 1 | 1568.4.a.w | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.4.a.f | ✓ | 3 | 8.b | even | 2 | 1 | |
| 224.4.a.g | yes | 3 | 8.d | odd | 2 | 1 | |
| 448.4.a.v | 3 | 1.a | even | 1 | 1 | trivial | |
| 448.4.a.w | 3 | 4.b | odd | 2 | 1 | ||
| 1568.4.a.w | 3 | 56.h | odd | 2 | 1 | ||
| 1568.4.a.x | 3 | 56.e | even | 2 | 1 | ||
| 2016.4.a.w | 3 | 24.h | odd | 2 | 1 | ||
| 2016.4.a.x | 3 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3}^{3} - 48T_{3} - 56 \)
|
|
\( T_{5}^{3} - 6T_{5}^{2} - 204T_{5} - 768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 48T - 56 \)
|
| $5$ |
\( T^{3} - 6 T^{2} + \cdots - 768 \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} - 3456T - 48832 \)
|
| $13$ |
\( T^{3} - 6 T^{2} + \cdots + 116816 \)
|
| $17$ |
\( T^{3} + 66 T^{2} + \cdots - 936424 \)
|
| $19$ |
\( T^{3} - 168 T^{2} + \cdots + 1148952 \)
|
| $23$ |
\( T^{3} + 336 T^{2} + \cdots - 215936 \)
|
| $29$ |
\( T^{3} + 90 T^{2} + \cdots + 10616 \)
|
| $31$ |
\( T^{3} + 504 T^{2} + \cdots + 366016 \)
|
| $37$ |
\( T^{3} + 18 T^{2} + \cdots - 2845928 \)
|
| $41$ |
\( T^{3} + 450 T^{2} + \cdots - 7572776 \)
|
| $43$ |
\( T^{3} - 31296 T - 2128448 \)
|
| $47$ |
\( T^{3} + 504 T^{2} + \cdots - 60547648 \)
|
| $53$ |
\( T^{3} - 78 T^{2} + \cdots + 67903192 \)
|
| $59$ |
\( T^{3} + 504 T^{2} + \cdots - 192556616 \)
|
| $61$ |
\( T^{3} + 498 T^{2} + \cdots + 2912608 \)
|
| $67$ |
\( T^{3} + 1008 T^{2} + \cdots + 29986432 \)
|
| $71$ |
\( T^{3} + 504 T^{2} + \cdots - 59523072 \)
|
| $73$ |
\( T^{3} + 234 T^{2} + \cdots - 123555144 \)
|
| $79$ |
\( T^{3} + 168 T^{2} + \cdots + 41853952 \)
|
| $83$ |
\( T^{3} + 3024 T^{2} + \cdots + 883723736 \)
|
| $89$ |
\( T^{3} - 246 T^{2} + \cdots + 3703224 \)
|
| $97$ |
\( T^{3} + \cdots - 1004302696 \)
|
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