Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,5,Mod(97,98)]
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([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.97");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(10.1302563822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
−2.82843 | − | 4.60181i | 8.00000 | 5.79050i | 13.0159i | 0 | −22.6274 | 59.8234 | − | 16.3780i | ||||||||||||||||||||||||||||
| 97.2 | −2.82843 | 4.60181i | 8.00000 | − | 5.79050i | − | 13.0159i | 0 | −22.6274 | 59.8234 | 16.3780i | |||||||||||||||||||||||||||||
| 97.3 | 2.82843 | − | 14.9941i | 8.00000 | − | 25.3864i | − | 42.4098i | 0 | 22.6274 | −143.823 | − | 71.8036i | |||||||||||||||||||||||||||
| 97.4 | 2.82843 | 14.9941i | 8.00000 | 25.3864i | 42.4098i | 0 | 22.6274 | −143.823 | 71.8036i | |||||||||||||||||||||||||||||||
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 | 98.5.b.b | 4 | |
| 3.b | odd | 2 | 1 | 882.5.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 784.5.c.b | 4 | ||
| 7.b | odd | 2 | 1 | inner | 98.5.b.b | 4 | |
| 7.c | even | 3 | 1 | 14.5.d.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 98.5.d.a | 4 | ||
| 7.d | odd | 6 | 1 | 14.5.d.a | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 98.5.d.a | 4 | ||
| 21.c | even | 2 | 1 | 882.5.c.b | 4 | ||
| 21.g | even | 6 | 1 | 126.5.n.a | 4 | ||
| 21.h | odd | 6 | 1 | 126.5.n.a | 4 | ||
| 28.d | even | 2 | 1 | 784.5.c.b | 4 | ||
| 28.f | even | 6 | 1 | 112.5.s.b | 4 | ||
| 28.g | odd | 6 | 1 | 112.5.s.b | 4 | ||
| 35.i | odd | 6 | 1 | 350.5.k.a | 4 | ||
| 35.j | even | 6 | 1 | 350.5.k.a | 4 | ||
| 35.k | even | 12 | 2 | 350.5.i.a | 8 | ||
| 35.l | odd | 12 | 2 | 350.5.i.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.5.d.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 14.5.d.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 98.5.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.5.b.b | 4 | 7.b | odd | 2 | 1 | inner | |
| 98.5.d.a | 4 | 7.c | even | 3 | 1 | ||
| 98.5.d.a | 4 | 7.d | odd | 6 | 1 | ||
| 112.5.s.b | 4 | 28.f | even | 6 | 1 | ||
| 112.5.s.b | 4 | 28.g | odd | 6 | 1 | ||
| 126.5.n.a | 4 | 21.g | even | 6 | 1 | ||
| 126.5.n.a | 4 | 21.h | odd | 6 | 1 | ||
| 350.5.i.a | 8 | 35.k | even | 12 | 2 | ||
| 350.5.i.a | 8 | 35.l | odd | 12 | 2 | ||
| 350.5.k.a | 4 | 35.i | odd | 6 | 1 | ||
| 350.5.k.a | 4 | 35.j | even | 6 | 1 | ||
| 784.5.c.b | 4 | 4.b | odd | 2 | 1 | ||
| 784.5.c.b | 4 | 28.d | even | 2 | 1 | ||
| 882.5.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 882.5.c.b | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 246T_{3}^{2} + 4761 \)
acting on \(S_{5}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8)^{2} \)
|
| $3$ |
\( T^{4} + 246T^{2} + 4761 \)
|
| $5$ |
\( T^{4} + 678 T^{2} + 21609 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 54 T + 441)^{2} \)
|
| $13$ |
\( T^{4} + 62592 T^{2} + 955551744 \)
|
| $17$ |
\( T^{4} + \cdots + 2084013801 \)
|
| $19$ |
\( T^{4} + \cdots + 37762205625 \)
|
| $23$ |
\( (T^{2} - 486 T - 401751)^{2} \)
|
| $29$ |
\( (T^{2} - 1620 T + 651492)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 566643101049 \)
|
| $37$ |
\( (T^{2} - 446 T - 533471)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 1046087110656 \)
|
| $43$ |
\( (T^{2} - 1172 T - 3015836)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 5548271897529 \)
|
| $53$ |
\( (T^{2} + 2754 T - 2896479)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 149611524833241 \)
|
| $61$ |
\( T^{4} + \cdots + 66399241936329 \)
|
| $67$ |
\( (T^{2} - 5062 T - 16012247)^{2} \)
|
| $71$ |
\( (T^{2} - 9396 T + 20407716)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 29440640401929 \)
|
| $79$ |
\( (T^{2} + 794 T - 7400663)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 314640617324544 \)
|
| $89$ |
\( T^{4} + \cdots + 28434692391561 \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
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