Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,8,Mod(67,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not 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: | \(30.6137324974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−4.00000 | − | 6.92820i | −10.6066 | + | 18.3712i | −32.0000 | + | 55.4256i | 60.8112 | + | 105.328i | 169.706 | 0 | 512.000 | 868.500 | + | 1504.29i | 486.489 | − | 842.624i | ||||||||||||||||||
| 67.2 | −4.00000 | − | 6.92820i | 10.6066 | − | 18.3712i | −32.0000 | + | 55.4256i | −60.8112 | − | 105.328i | −169.706 | 0 | 512.000 | 868.500 | + | 1504.29i | −486.489 | + | 842.624i | |||||||||||||||||||
| 79.1 | −4.00000 | + | 6.92820i | −10.6066 | − | 18.3712i | −32.0000 | − | 55.4256i | 60.8112 | − | 105.328i | 169.706 | 0 | 512.000 | 868.500 | − | 1504.29i | 486.489 | + | 842.624i | |||||||||||||||||||
| 79.2 | −4.00000 | + | 6.92820i | 10.6066 | + | 18.3712i | −32.0000 | − | 55.4256i | −60.8112 | + | 105.328i | −169.706 | 0 | 512.000 | 868.500 | − | 1504.29i | −486.489 | − | 842.624i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.8.c.j | 4 | |
| 7.b | odd | 2 | 1 | inner | 98.8.c.j | 4 | |
| 7.c | even | 3 | 1 | 98.8.a.i | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 98.8.c.j | 4 | |
| 7.d | odd | 6 | 1 | 98.8.a.i | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 98.8.c.j | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.8.a.i | ✓ | 2 | 7.c | even | 3 | 1 | |
| 98.8.a.i | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 98.8.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.j | 4 | 7.b | odd | 2 | 1 | inner | |
| 98.8.c.j | 4 | 7.c | even | 3 | 1 | inner | |
| 98.8.c.j | 4 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 450T_{3}^{2} + 202500 \)
acting on \(S_{8}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} + 450 T^{2} + 202500 \)
|
| $5$ |
\( T^{4} + 14792 T^{2} + 218803264 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 5434 T + 29528356)^{2} \)
|
| $13$ |
\( (T^{2} - 8128512)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 28865282060164 \)
|
| $19$ |
\( T^{4} + \cdots + 31\!\cdots\!64 \)
|
| $23$ |
\( (T^{2} - 16580 T + 274896400)^{2} \)
|
| $29$ |
\( (T + 103766)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 65\!\cdots\!84 \)
|
| $37$ |
\( (T^{2} - 332798 T + 110754508804)^{2} \)
|
| $41$ |
\( (T^{2} - 611107134722)^{2} \)
|
| $43$ |
\( (T + 651334)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + 559046 T + 312532430116)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 93\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 84\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + 382884 T + 146600157456)^{2} \)
|
| $71$ |
\( (T + 1886652)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 39\!\cdots\!84 \)
|
| $79$ |
\( (T^{2} + \cdots + 1694318755600)^{2} \)
|
| $83$ |
\( (T^{2} - 16622559238082)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!84 \)
|
| $97$ |
\( (T^{2} - 145941402719042)^{2} \)
|
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