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: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2) |
| 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)
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 | −12.0000 | 64.0000 | 210.000 | 96.0000 | 0 | −512.000 | −2043.00 | −1680.00 | |||||||||||||||||||||
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.a | 1 | |
| 7.b | odd | 2 | 1 | 2.8.a.a | ✓ | 1 | |
| 7.c | even | 3 | 2 | 98.8.c.e | 2 | ||
| 7.d | odd | 6 | 2 | 98.8.c.d | 2 | ||
| 21.c | even | 2 | 1 | 18.8.a.b | 1 | ||
| 28.d | even | 2 | 1 | 16.8.a.b | 1 | ||
| 35.c | odd | 2 | 1 | 50.8.a.g | 1 | ||
| 35.f | even | 4 | 2 | 50.8.b.c | 2 | ||
| 56.e | even | 2 | 1 | 64.8.a.e | 1 | ||
| 56.h | odd | 2 | 1 | 64.8.a.c | 1 | ||
| 63.l | odd | 6 | 2 | 162.8.c.l | 2 | ||
| 63.o | even | 6 | 2 | 162.8.c.a | 2 | ||
| 77.b | even | 2 | 1 | 242.8.a.e | 1 | ||
| 84.h | odd | 2 | 1 | 144.8.a.i | 1 | ||
| 91.b | odd | 2 | 1 | 338.8.a.d | 1 | ||
| 91.i | even | 4 | 2 | 338.8.b.d | 2 | ||
| 105.g | even | 2 | 1 | 450.8.a.c | 1 | ||
| 105.k | odd | 4 | 2 | 450.8.c.g | 2 | ||
| 112.j | even | 4 | 2 | 256.8.b.f | 2 | ||
| 112.l | odd | 4 | 2 | 256.8.b.b | 2 | ||
| 119.d | odd | 2 | 1 | 578.8.a.b | 1 | ||
| 140.c | even | 2 | 1 | 400.8.a.l | 1 | ||
| 140.j | odd | 4 | 2 | 400.8.c.j | 2 | ||
| 168.e | odd | 2 | 1 | 576.8.a.f | 1 | ||
| 168.i | even | 2 | 1 | 576.8.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.8.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
| 16.8.a.b | 1 | 28.d | even | 2 | 1 | ||
| 18.8.a.b | 1 | 21.c | even | 2 | 1 | ||
| 50.8.a.g | 1 | 35.c | odd | 2 | 1 | ||
| 50.8.b.c | 2 | 35.f | even | 4 | 2 | ||
| 64.8.a.c | 1 | 56.h | odd | 2 | 1 | ||
| 64.8.a.e | 1 | 56.e | even | 2 | 1 | ||
| 98.8.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.d | 2 | 7.d | odd | 6 | 2 | ||
| 98.8.c.e | 2 | 7.c | even | 3 | 2 | ||
| 144.8.a.i | 1 | 84.h | odd | 2 | 1 | ||
| 162.8.c.a | 2 | 63.o | even | 6 | 2 | ||
| 162.8.c.l | 2 | 63.l | odd | 6 | 2 | ||
| 242.8.a.e | 1 | 77.b | even | 2 | 1 | ||
| 256.8.b.b | 2 | 112.l | odd | 4 | 2 | ||
| 256.8.b.f | 2 | 112.j | even | 4 | 2 | ||
| 338.8.a.d | 1 | 91.b | odd | 2 | 1 | ||
| 338.8.b.d | 2 | 91.i | even | 4 | 2 | ||
| 400.8.a.l | 1 | 140.c | even | 2 | 1 | ||
| 400.8.c.j | 2 | 140.j | odd | 4 | 2 | ||
| 450.8.a.c | 1 | 105.g | even | 2 | 1 | ||
| 450.8.c.g | 2 | 105.k | odd | 4 | 2 | ||
| 576.8.a.f | 1 | 168.e | odd | 2 | 1 | ||
| 576.8.a.g | 1 | 168.i | even | 2 | 1 | ||
| 578.8.a.b | 1 | 119.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 12 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 8 \)
|
| $3$ |
\( T + 12 \)
|
| $5$ |
\( T - 210 \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T - 1092 \)
|
| $13$ |
\( T + 1382 \)
|
| $17$ |
\( T + 14706 \)
|
| $19$ |
\( T - 39940 \)
|
| $23$ |
\( T - 68712 \)
|
| $29$ |
\( T + 102570 \)
|
| $31$ |
\( T + 227552 \)
|
| $37$ |
\( T - 160526 \)
|
| $41$ |
\( T + 10842 \)
|
| $43$ |
\( T + 630748 \)
|
| $47$ |
\( T + 472656 \)
|
| $53$ |
\( T + 1494018 \)
|
| $59$ |
\( T + 2640660 \)
|
| $61$ |
\( T + 827702 \)
|
| $67$ |
\( T + 126004 \)
|
| $71$ |
\( T + 1414728 \)
|
| $73$ |
\( T + 980282 \)
|
| $79$ |
\( T + 3566800 \)
|
| $83$ |
\( T + 5672892 \)
|
| $89$ |
\( T - 11951190 \)
|
| $97$ |
\( T + 8682146 \)
|
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