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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2389}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 597 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(\beta = \sqrt{2389}\). 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 | −20.8774 | 64.0000 | 216.755 | −167.019 | 0 | 512.000 | −1751.13 | 1734.04 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | 76.8774 | 64.0000 | 21.2452 | 615.019 | 0 | 512.000 | 3723.13 | 169.962 | |||||||||||||||||||||||||
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 | 98.8.a.k | 2 | |
| 7.b | odd | 2 | 1 | 98.8.a.h | 2 | ||
| 7.c | even | 3 | 2 | 98.8.c.h | 4 | ||
| 7.d | odd | 6 | 2 | 14.8.c.a | ✓ | 4 | |
| 21.g | even | 6 | 2 | 126.8.g.e | 4 | ||
| 28.f | even | 6 | 2 | 112.8.i.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.8.c.a | ✓ | 4 | 7.d | odd | 6 | 2 | |
| 98.8.a.h | 2 | 7.b | odd | 2 | 1 | ||
| 98.8.a.k | 2 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.h | 4 | 7.c | even | 3 | 2 | ||
| 112.8.i.a | 4 | 28.f | even | 6 | 2 | ||
| 126.8.g.e | 4 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 56T_{3} - 1605 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{2} \)
|
| $3$ |
\( T^{2} - 56T - 1605 \)
|
| $5$ |
\( T^{2} - 238T + 4605 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 5848 T + 8432715 \)
|
| $13$ |
\( T^{2} + 1316 T - 91342860 \)
|
| $17$ |
\( T^{2} - 47642 T + 405943641 \)
|
| $19$ |
\( T^{2} - 41048 T + 280166515 \)
|
| $23$ |
\( T^{2} + \cdots - 1105873137 \)
|
| $29$ |
\( T^{2} + \cdots + 5666758164 \)
|
| $31$ |
\( T^{2} + \cdots - 27316742385 \)
|
| $37$ |
\( T^{2} + \cdots - 285131934827 \)
|
| $41$ |
\( T^{2} + \cdots + 332194626036 \)
|
| $43$ |
\( T^{2} + \cdots - 154524293360 \)
|
| $47$ |
\( T^{2} + \cdots + 249359041791 \)
|
| $53$ |
\( T^{2} + \cdots + 1385251917669 \)
|
| $59$ |
\( T^{2} + \cdots + 271714932627 \)
|
| $61$ |
\( T^{2} + \cdots - 1792085999859 \)
|
| $67$ |
\( T^{2} + \cdots - 4112458315381 \)
|
| $71$ |
\( T^{2} + \cdots + 2760738723264 \)
|
| $73$ |
\( T^{2} + \cdots - 5364808200775 \)
|
| $79$ |
\( T^{2} + \cdots + 20967658995495 \)
|
| $83$ |
\( T^{2} + \cdots - 11352739197744 \)
|
| $89$ |
\( T^{2} + \cdots + 15686015663961 \)
|
| $97$ |
\( T^{2} + \cdots + 6136617007220 \)
|
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