Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,8,Mod(1,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.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: | \(30.6137324974\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \sqrt{2}\). 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 |
|
8.00000 | −21.2132 | 64.0000 | 121.622 | −169.706 | 0 | 512.000 | −1737.00 | 972.979 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | 21.2132 | 64.0000 | −121.622 | 169.706 | 0 | 512.000 | −1737.00 | −972.979 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.8.a.i | ✓ | 2 |
| 7.b | odd | 2 | 1 | inner | 98.8.a.i | ✓ | 2 |
| 7.c | even | 3 | 2 | 98.8.c.j | 4 | ||
| 7.d | odd | 6 | 2 | 98.8.c.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.8.a.i | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 98.8.a.i | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 98.8.c.j | 4 | 7.c | even | 3 | 2 | ||
| 98.8.c.j | 4 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 450 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{2} \)
|
| $3$ |
\( T^{2} - 450 \)
|
| $5$ |
\( T^{2} - 14792 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T + 5434)^{2} \)
|
| $13$ |
\( T^{2} - 8128512 \)
|
| $17$ |
\( T^{2} - 5372642 \)
|
| $19$ |
\( T^{2} - 176306642 \)
|
| $23$ |
\( (T + 16580)^{2} \)
|
| $29$ |
\( (T + 103766)^{2} \)
|
| $31$ |
\( T^{2} - 80806824072 \)
|
| $37$ |
\( (T + 332798)^{2} \)
|
| $41$ |
\( T^{2} - 611107134722 \)
|
| $43$ |
\( (T + 651334)^{2} \)
|
| $47$ |
\( T^{2} - 1507924512200 \)
|
| $53$ |
\( (T - 559046)^{2} \)
|
| $59$ |
\( T^{2} - 305860539938 \)
|
| $61$ |
\( T^{2} - 9193386240200 \)
|
| $67$ |
\( (T - 382884)^{2} \)
|
| $71$ |
\( (T + 1886652)^{2} \)
|
| $73$ |
\( T^{2} - 19835311782722 \)
|
| $79$ |
\( (T - 1301660)^{2} \)
|
| $83$ |
\( T^{2} - 16622559238082 \)
|
| $89$ |
\( T^{2} - 49925687632178 \)
|
| $97$ |
\( T^{2} - 145941402719042 \)
|
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