Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [366,2,Mod(13,366)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(366, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("366.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 366 = 2 \cdot 3 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 366.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: | \(2.92252471398\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.65370672.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 10x^{4} - 13x^{3} + 92x^{2} - 99x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 10x^{4} - 13x^{3} + 92x^{2} - 99x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 92\nu^{2} - 99\nu + 990 ) / 821 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{5} - 80\nu^{4} - 21\nu^{3} - 736\nu^{2} + 792\nu - 4636 ) / 821 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -90\nu^{5} + 79\nu^{4} - 790\nu^{3} + 70\nu^{2} - 7268\nu - 1210 ) / 9031 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -531\nu^{5} - 437\nu^{4} - 4661\nu^{3} + 413\nu^{2} - 32044\nu - 7139 ) / 9031 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 7\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} - 8\beta_{2} + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -10\beta_{5} + 59\beta_{4} + 12\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} - 54\beta_{4} + 8\beta_{3} + 71\beta_{2} - 71\beta _1 - 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/366\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −1.74506 | + | 3.02254i | −0.500000 | + | 0.866025i | 2.04030 | − | 3.53391i | −1.00000 | 1.00000 | 1.74506 | + | 3.02254i | ||||||||||||||||||||||||
| 13.2 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.947448 | + | 1.64103i | −0.500000 | + | 0.866025i | −0.904893 | + | 1.56732i | −1.00000 | 1.00000 | 0.947448 | + | 1.64103i | |||||||||||||||||||||||||
| 13.3 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 2.19251 | − | 3.79754i | −0.500000 | + | 0.866025i | −0.135409 | + | 0.234536i | −1.00000 | 1.00000 | −2.19251 | − | 3.79754i | |||||||||||||||||||||||||
| 169.1 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −1.74506 | − | 3.02254i | −0.500000 | − | 0.866025i | 2.04030 | + | 3.53391i | −1.00000 | 1.00000 | 1.74506 | − | 3.02254i | |||||||||||||||||||||||||
| 169.2 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.947448 | − | 1.64103i | −0.500000 | − | 0.866025i | −0.904893 | − | 1.56732i | −1.00000 | 1.00000 | 0.947448 | − | 1.64103i | |||||||||||||||||||||||||
| 169.3 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 2.19251 | + | 3.79754i | −0.500000 | − | 0.866025i | −0.135409 | − | 0.234536i | −1.00000 | 1.00000 | −2.19251 | + | 3.79754i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 366.2.e.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1098.2.f.f | 6 | ||
| 61.c | even | 3 | 1 | inner | 366.2.e.d | ✓ | 6 |
| 183.k | odd | 6 | 1 | 1098.2.f.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 366.2.e.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 366.2.e.d | ✓ | 6 | 61.c | even | 3 | 1 | inner |
| 1098.2.f.f | 6 | 3.b | odd | 2 | 1 | ||
| 1098.2.f.f | 6 | 183.k | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + T_{5}^{5} + 18T_{5}^{4} + 41T_{5}^{3} + 318T_{5}^{2} + 493T_{5} + 841 \)
acting on \(S_{2}^{\mathrm{new}}(366, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{3} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} + T^{5} + \cdots + 841 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots + 4 \)
|
| $11$ |
\( (T^{3} - 2 T^{2} - 8 T - 2)^{2} \)
|
| $13$ |
\( T^{6} + 2 T^{5} + \cdots + 7744 \)
|
| $17$ |
\( T^{6} + 12 T^{5} + \cdots + 17424 \)
|
| $19$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $23$ |
\( (T^{3} - 4 T^{2} - 10 T + 22)^{2} \)
|
| $29$ |
\( T^{6} - 7 T^{5} + \cdots + 121 \)
|
| $31$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $37$ |
\( (T^{3} + 7 T^{2} + T - 27)^{2} \)
|
| $41$ |
\( (T^{3} - 11 T^{2} + \cdots - 23)^{2} \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 14884 \)
|
| $47$ |
\( T^{6} + 10 T^{5} + \cdots + 948676 \)
|
| $53$ |
\( (T^{3} + 11 T^{2} + \cdots - 723)^{2} \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots + 9216 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots + 226981 \)
|
| $67$ |
\( T^{6} + 2 T^{5} + \cdots + 36 \)
|
| $71$ |
\( T^{6} - 10 T^{5} + \cdots + 576 \)
|
| $73$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $79$ |
\( T^{6} + 6 T^{5} + \cdots + 627264 \)
|
| $83$ |
\( T^{6} + 14 T^{5} + \cdots + 4356 \)
|
| $89$ |
\( (T^{3} - 13 T^{2} + \cdots + 3309)^{2} \)
|
| $97$ |
\( T^{6} - 27 T^{5} + \cdots + 259081 \)
|
| show more | |
| show less |