Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(71.8519512762\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{61}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 15 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{61}\). 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 | −14.8102 | 0 | 37.6717 | 0 | 49.0000 | 0 | −23.6565 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0.810250 | 0 | −71.6717 | 0 | 49.0000 | 0 | −242.343 | 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 | 448.6.a.s | 2 | |
| 4.b | odd | 2 | 1 | 448.6.a.y | 2 | ||
| 8.b | even | 2 | 1 | 224.6.a.d | yes | 2 | |
| 8.d | odd | 2 | 1 | 224.6.a.c | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.a.c | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 224.6.a.d | yes | 2 | 8.b | even | 2 | 1 | |
| 448.6.a.s | 2 | 1.a | even | 1 | 1 | trivial | |
| 448.6.a.y | 2 | 4.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3}^{2} + 14T_{3} - 12 \)
|
|
\( T_{5}^{2} + 34T_{5} - 2700 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 14T - 12 \)
|
| $5$ |
\( T^{2} + 34T - 2700 \)
|
| $7$ |
\( (T - 49)^{2} \)
|
| $11$ |
\( T^{2} - 420T - 84976 \)
|
| $13$ |
\( T^{2} - 490T - 86436 \)
|
| $17$ |
\( T^{2} + 1056 T + 266828 \)
|
| $19$ |
\( T^{2} - 1246 T - 3333420 \)
|
| $23$ |
\( T^{2} - 504T + 47888 \)
|
| $29$ |
\( T^{2} - 3904 T - 14374772 \)
|
| $31$ |
\( T^{2} + 2044 T - 24097520 \)
|
| $37$ |
\( T^{2} - 7488 T + 14005580 \)
|
| $41$ |
\( T^{2} - 7832 T - 63108260 \)
|
| $43$ |
\( T^{2} - 10332 T - 45786544 \)
|
| $47$ |
\( T^{2} + 41972 T + 396339696 \)
|
| $53$ |
\( T^{2} + 32812 T - 171589148 \)
|
| $59$ |
\( T^{2} - 48398 T + 421726020 \)
|
| $61$ |
\( T^{2} - 718 T - 11734460 \)
|
| $67$ |
\( T^{2} - 12824 T - 792171632 \)
|
| $71$ |
\( T^{2} + \cdots + 2694018240 \)
|
| $73$ |
\( T^{2} + \cdots - 1049693676 \)
|
| $79$ |
\( T^{2} + 64568 T + 967369152 \)
|
| $83$ |
\( T^{2} + 47810 T + 554507556 \)
|
| $89$ |
\( T^{2} + \cdots - 12657841308 \)
|
| $97$ |
\( T^{2} + 97296 T + 62455628 \)
|
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