Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(5.78218718056\) |
| Analytic rank: | \(0\) |
| 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 |
|
−2.00000 | −7.07107 | 4.00000 | −19.7990 | 14.1421 | 0 | −8.00000 | 23.0000 | 39.5980 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 7.07107 | 4.00000 | 19.7990 | −14.1421 | 0 | −8.00000 | 23.0000 | −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 | 98.4.a.g | ✓ | 2 |
| 3.b | odd | 2 | 1 | 882.4.a.bg | 2 | ||
| 4.b | odd | 2 | 1 | 784.4.a.y | 2 | ||
| 5.b | even | 2 | 1 | 2450.4.a.bx | 2 | ||
| 7.b | odd | 2 | 1 | inner | 98.4.a.g | ✓ | 2 |
| 7.c | even | 3 | 2 | 98.4.c.h | 4 | ||
| 7.d | odd | 6 | 2 | 98.4.c.h | 4 | ||
| 21.c | even | 2 | 1 | 882.4.a.bg | 2 | ||
| 21.g | even | 6 | 2 | 882.4.g.ba | 4 | ||
| 21.h | odd | 6 | 2 | 882.4.g.ba | 4 | ||
| 28.d | even | 2 | 1 | 784.4.a.y | 2 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.bx | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.4.a.g | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 98.4.a.g | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 98.4.c.h | 4 | 7.c | even | 3 | 2 | ||
| 98.4.c.h | 4 | 7.d | odd | 6 | 2 | ||
| 784.4.a.y | 2 | 4.b | odd | 2 | 1 | ||
| 784.4.a.y | 2 | 28.d | even | 2 | 1 | ||
| 882.4.a.bg | 2 | 3.b | odd | 2 | 1 | ||
| 882.4.a.bg | 2 | 21.c | even | 2 | 1 | ||
| 882.4.g.ba | 4 | 21.g | even | 6 | 2 | ||
| 882.4.g.ba | 4 | 21.h | odd | 6 | 2 | ||
| 2450.4.a.bx | 2 | 5.b | even | 2 | 1 | ||
| 2450.4.a.bx | 2 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 50 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} - 50 \)
|
| $5$ |
\( T^{2} - 392 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T + 14)^{2} \)
|
| $13$ |
\( T^{2} - 2592 \)
|
| $17$ |
\( T^{2} - 2 \)
|
| $19$ |
\( T^{2} - 2 \)
|
| $23$ |
\( (T - 140)^{2} \)
|
| $29$ |
\( (T + 286)^{2} \)
|
| $31$ |
\( T^{2} - 8712 \)
|
| $37$ |
\( (T + 38)^{2} \)
|
| $41$ |
\( T^{2} - 15842 \)
|
| $43$ |
\( (T + 34)^{2} \)
|
| $47$ |
\( T^{2} - 273800 \)
|
| $53$ |
\( (T + 74)^{2} \)
|
| $59$ |
\( T^{2} - 188498 \)
|
| $61$ |
\( T^{2} - 200 \)
|
| $67$ |
\( (T - 684)^{2} \)
|
| $71$ |
\( (T - 588)^{2} \)
|
| $73$ |
\( T^{2} - 72962 \)
|
| $79$ |
\( (T - 1220)^{2} \)
|
| $83$ |
\( T^{2} - 178802 \)
|
| $89$ |
\( T^{2} - 381938 \)
|
| $97$ |
\( T^{2} - 2200802 \)
|
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