Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [198,2,Mod(67,198)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(198, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("198.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 198.e (of order \(3\), degree \(2\), 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: | \(1.58103796002\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(155\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 |
|
−0.500000 | − | 0.866025i | −0.724745 | − | 1.57313i | −0.500000 | + | 0.866025i | 1.72474 | − | 2.98735i | −1.00000 | + | 1.41421i | −0.224745 | − | 0.389270i | 1.00000 | −1.94949 | + | 2.28024i | −3.44949 | ||||||||||||||||
| 67.2 | −0.500000 | − | 0.866025i | 1.72474 | − | 0.158919i | −0.500000 | + | 0.866025i | −0.724745 | + | 1.25529i | −1.00000 | − | 1.41421i | 2.22474 | + | 3.85337i | 1.00000 | 2.94949 | − | 0.548188i | 1.44949 | |||||||||||||||||
| 133.1 | −0.500000 | + | 0.866025i | −0.724745 | + | 1.57313i | −0.500000 | − | 0.866025i | 1.72474 | + | 2.98735i | −1.00000 | − | 1.41421i | −0.224745 | + | 0.389270i | 1.00000 | −1.94949 | − | 2.28024i | −3.44949 | |||||||||||||||||
| 133.2 | −0.500000 | + | 0.866025i | 1.72474 | + | 0.158919i | −0.500000 | − | 0.866025i | −0.724745 | − | 1.25529i | −1.00000 | + | 1.41421i | 2.22474 | − | 3.85337i | 1.00000 | 2.94949 | + | 0.548188i | 1.44949 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 198.2.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 594.2.e.c | 4 | ||
| 9.c | even | 3 | 1 | inner | 198.2.e.c | ✓ | 4 |
| 9.c | even | 3 | 1 | 1782.2.a.o | 2 | ||
| 9.d | odd | 6 | 1 | 594.2.e.c | 4 | ||
| 9.d | odd | 6 | 1 | 1782.2.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 198.2.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 198.2.e.c | ✓ | 4 | 9.c | even | 3 | 1 | inner |
| 594.2.e.c | 4 | 3.b | odd | 2 | 1 | ||
| 594.2.e.c | 4 | 9.d | odd | 6 | 1 | ||
| 1782.2.a.m | 2 | 9.d | odd | 6 | 1 | ||
| 1782.2.a.o | 2 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 2T_{5}^{3} + 9T_{5}^{2} + 10T_{5} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(198, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 9 \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 25 \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 4 \)
|
| $11$ |
\( (T^{2} + T + 1)^{2} \)
|
| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 4 \)
|
| $17$ |
\( (T^{2} + 4 T - 2)^{2} \)
|
| $19$ |
\( (T + 6)^{4} \)
|
| $23$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots + 2500 \)
|
| $31$ |
\( (T^{2} + T + 1)^{2} \)
|
| $37$ |
\( (T^{2} - 14 T + 43)^{2} \)
|
| $41$ |
\( T^{4} - 8 T^{3} + \cdots + 1444 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 4 \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 529 \)
|
| $53$ |
\( (T^{2} + 10 T - 29)^{2} \)
|
| $59$ |
\( T^{4} - 6 T^{3} + \cdots + 9 \)
|
| $61$ |
\( T^{4} + 8 T^{3} + \cdots + 6400 \)
|
| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 2025 \)
|
| $71$ |
\( (T^{2} + 2 T - 95)^{2} \)
|
| $73$ |
\( (T^{2} + 12 T - 18)^{2} \)
|
| $79$ |
\( T^{4} - 4 T^{3} + \cdots + 2500 \)
|
| $83$ |
\( (T^{2} - 8 T + 64)^{2} \)
|
| $89$ |
\( (T^{2} + 16 T - 32)^{2} \)
|
| $97$ |
\( T^{4} + 2 T^{3} + \cdots + 9025 \)
|
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