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: | \(\Q(\sqrt{-3}, \sqrt{949})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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
\( x^{4} - x^{3} + 238x^{2} + 237x + 56169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 238\nu^{2} - 238\nu + 56169 ) / 56406 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 238\nu^{2} + 113050\nu - 56169 ) / 56406 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 356 ) / 119 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 475\beta _1 - 475 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 119\beta_{3} - 356 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
4.00000 | + | 6.92820i | −1.40292 | + | 2.42993i | −32.0000 | + | 55.4256i | −219.141 | − | 379.563i | −22.4467 | 0 | −512.000 | 1089.56 | + | 1887.18i | 1753.13 | − | 3036.51i | ||||||||||||||||||
| 67.2 | 4.00000 | + | 6.92820i | 29.4029 | − | 50.9274i | −32.0000 | + | 55.4256i | 212.141 | + | 367.439i | 470.447 | 0 | −512.000 | −635.564 | − | 1100.83i | −1697.13 | + | 2939.51i | |||||||||||||||||||
| 79.1 | 4.00000 | − | 6.92820i | −1.40292 | − | 2.42993i | −32.0000 | − | 55.4256i | −219.141 | + | 379.563i | −22.4467 | 0 | −512.000 | 1089.56 | − | 1887.18i | 1753.13 | + | 3036.51i | |||||||||||||||||||
| 79.2 | 4.00000 | − | 6.92820i | 29.4029 | + | 50.9274i | −32.0000 | − | 55.4256i | 212.141 | − | 367.439i | 470.447 | 0 | −512.000 | −635.564 | + | 1100.83i | −1697.13 | − | 2939.51i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.8.c.m | 4 | |
| 7.b | odd | 2 | 1 | 14.8.c.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | 98.8.a.d | 2 | ||
| 7.c | even | 3 | 1 | inner | 98.8.c.m | 4 | |
| 7.d | odd | 6 | 1 | 14.8.c.b | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 98.8.a.f | 2 | ||
| 21.c | even | 2 | 1 | 126.8.g.d | 4 | ||
| 21.g | even | 6 | 1 | 126.8.g.d | 4 | ||
| 28.d | even | 2 | 1 | 112.8.i.b | 4 | ||
| 28.f | even | 6 | 1 | 112.8.i.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.8.c.b | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 14.8.c.b | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 98.8.a.d | 2 | 7.c | even | 3 | 1 | ||
| 98.8.a.f | 2 | 7.d | odd | 6 | 1 | ||
| 98.8.c.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.m | 4 | 7.c | even | 3 | 1 | inner | |
| 112.8.i.b | 4 | 28.d | even | 2 | 1 | ||
| 112.8.i.b | 4 | 28.f | even | 6 | 1 | ||
| 126.8.g.d | 4 | 21.c | even | 2 | 1 | ||
| 126.8.g.d | 4 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 56T_{3}^{3} + 3301T_{3}^{2} + 9240T_{3} + 27225 \)
acting on \(S_{8}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} - 56 T^{3} + \cdots + 27225 \)
|
| $5$ |
\( T^{4} + \cdots + 34579262025 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 18\!\cdots\!25 \)
|
| $13$ |
\( (T^{2} + 10724 T + 26913780)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 39\!\cdots\!01 \)
|
| $19$ |
\( T^{4} + \cdots + 39\!\cdots\!25 \)
|
| $23$ |
\( T^{4} + \cdots + 79\!\cdots\!69 \)
|
| $29$ |
\( (T^{2} + 95660 T - 302210796)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 31\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 21\!\cdots\!49 \)
|
| $41$ |
\( (T^{2} + 805980 T + 154982022516)^{2} \)
|
| $43$ |
\( (T^{2} - 267976 T - 9474621680)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 60\!\cdots\!81 \)
|
| $53$ |
\( T^{4} + \cdots + 77\!\cdots\!81 \)
|
| $59$ |
\( T^{4} + \cdots + 95\!\cdots\!09 \)
|
| $61$ |
\( T^{4} + \cdots + 13\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} + \cdots - 20052968986176)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 20\!\cdots\!25 \)
|
| $79$ |
\( T^{4} + \cdots + 14\!\cdots\!25 \)
|
| $83$ |
\( (T^{2} + \cdots + 14206197364176)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 38\!\cdots\!81 \)
|
| $97$ |
\( (T^{2} + \cdots - 89006519008780)^{2} \)
|
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