Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(15.7176143417\) |
| 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 |
|
−4.00000 | −8.48528 | 16.0000 | 9.89949 | 33.9411 | 0 | −64.0000 | −171.000 | −39.5980 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 8.48528 | 16.0000 | −9.89949 | −33.9411 | 0 | −64.0000 | −171.000 | 39.5980 | |||||||||||||||||||||||||
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 | 98.6.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 882.6.a.bp | 2 | ||
| 4.b | odd | 2 | 1 | 784.6.a.x | 2 | ||
| 7.b | odd | 2 | 1 | inner | 98.6.a.d | ✓ | 2 |
| 7.c | even | 3 | 2 | 98.6.c.h | 4 | ||
| 7.d | odd | 6 | 2 | 98.6.c.h | 4 | ||
| 21.c | even | 2 | 1 | 882.6.a.bp | 2 | ||
| 28.d | even | 2 | 1 | 784.6.a.x | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.6.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 98.6.a.d | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 98.6.c.h | 4 | 7.c | even | 3 | 2 | ||
| 98.6.c.h | 4 | 7.d | odd | 6 | 2 | ||
| 784.6.a.x | 2 | 4.b | odd | 2 | 1 | ||
| 784.6.a.x | 2 | 28.d | even | 2 | 1 | ||
| 882.6.a.bp | 2 | 3.b | odd | 2 | 1 | ||
| 882.6.a.bp | 2 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 72 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{2} \)
|
| $3$ |
\( T^{2} - 72 \)
|
| $5$ |
\( T^{2} - 98 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T - 308)^{2} \)
|
| $13$ |
\( T^{2} - 832050 \)
|
| $17$ |
\( T^{2} - 203522 \)
|
| $19$ |
\( T^{2} - 6845000 \)
|
| $23$ |
\( (T + 2324)^{2} \)
|
| $29$ |
\( (T + 6488)^{2} \)
|
| $31$ |
\( T^{2} - 663552 \)
|
| $37$ |
\( (T + 12200)^{2} \)
|
| $41$ |
\( T^{2} - 278258 \)
|
| $43$ |
\( (T + 15028)^{2} \)
|
| $47$ |
\( T^{2} - 2562848 \)
|
| $53$ |
\( (T - 10778)^{2} \)
|
| $59$ |
\( T^{2} - 2718203912 \)
|
| $61$ |
\( T^{2} - 1351688018 \)
|
| $67$ |
\( (T + 23808)^{2} \)
|
| $71$ |
\( (T - 56448)^{2} \)
|
| $73$ |
\( T^{2} - 2639883122 \)
|
| $79$ |
\( (T + 11224)^{2} \)
|
| $83$ |
\( T^{2} - 1496920328 \)
|
| $89$ |
\( T^{2} - 16429406450 \)
|
| $97$ |
\( T^{2} - 3235779458 \)
|
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