Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(139.948491417\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{865}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 216 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(\beta = \sqrt{865}\). 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 | 17.5891 | 0 | −17.9456 | 0 | 343.000 | 0 | −1877.62 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 76.4109 | 0 | −312.054 | 0 | 343.000 | 0 | 3651.62 | 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 | 448.8.a.t | 2 | |
| 4.b | odd | 2 | 1 | 448.8.a.k | 2 | ||
| 8.b | even | 2 | 1 | 112.8.a.f | 2 | ||
| 8.d | odd | 2 | 1 | 7.8.a.b | ✓ | 2 | |
| 24.f | even | 2 | 1 | 63.8.a.e | 2 | ||
| 40.e | odd | 2 | 1 | 175.8.a.c | 2 | ||
| 40.k | even | 4 | 2 | 175.8.b.b | 4 | ||
| 56.e | even | 2 | 1 | 49.8.a.c | 2 | ||
| 56.k | odd | 6 | 2 | 49.8.c.e | 4 | ||
| 56.m | even | 6 | 2 | 49.8.c.f | 4 | ||
| 168.e | odd | 2 | 1 | 441.8.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.8.a.b | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 49.8.a.c | 2 | 56.e | even | 2 | 1 | ||
| 49.8.c.e | 4 | 56.k | odd | 6 | 2 | ||
| 49.8.c.f | 4 | 56.m | even | 6 | 2 | ||
| 63.8.a.e | 2 | 24.f | even | 2 | 1 | ||
| 112.8.a.f | 2 | 8.b | even | 2 | 1 | ||
| 175.8.a.c | 2 | 40.e | odd | 2 | 1 | ||
| 175.8.b.b | 4 | 40.k | even | 4 | 2 | ||
| 441.8.a.l | 2 | 168.e | odd | 2 | 1 | ||
| 448.8.a.k | 2 | 4.b | odd | 2 | 1 | ||
| 448.8.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 94T_{3} + 1344 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(448))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 94T + 1344 \)
|
| $5$ |
\( T^{2} + 330T + 5600 \)
|
| $7$ |
\( (T - 343)^{2} \)
|
| $11$ |
\( T^{2} - 2844 T - 887776 \)
|
| $13$ |
\( T^{2} + 2534 T - 166620776 \)
|
| $17$ |
\( T^{2} + 1488 T - 22147524 \)
|
| $19$ |
\( T^{2} - 32810 T + 109928560 \)
|
| $23$ |
\( T^{2} - 6576 T + 10312704 \)
|
| $29$ |
\( T^{2} + \cdots - 18920124100 \)
|
| $31$ |
\( T^{2} + \cdots + 37023636384 \)
|
| $37$ |
\( T^{2} + \cdots - 126010986084 \)
|
| $41$ |
\( T^{2} + \cdots + 13303276364 \)
|
| $43$ |
\( T^{2} + \cdots + 50864711104 \)
|
| $47$ |
\( T^{2} + \cdots + 2090896416 \)
|
| $53$ |
\( T^{2} + \cdots + 2037435782724 \)
|
| $59$ |
\( T^{2} + \cdots - 615374101440 \)
|
| $61$ |
\( T^{2} + \cdots - 529516501136 \)
|
| $67$ |
\( T^{2} + \cdots - 533876854064 \)
|
| $71$ |
\( T^{2} + \cdots + 10359492378624 \)
|
| $73$ |
\( T^{2} + \cdots - 3340687254156 \)
|
| $79$ |
\( T^{2} + \cdots - 6335206025600 \)
|
| $83$ |
\( T^{2} + \cdots + 30181573873584 \)
|
| $89$ |
\( T^{2} + \cdots - 4649674734460 \)
|
| $97$ |
\( T^{2} + \cdots + 27021168617436 \)
|
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