Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(568,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("819.568");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 819.o (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: | \(6.53974792554\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.6040683.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 5x^{4} - 2x^{3} + 25x^{2} - 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{6} + 5x^{4} - 2x^{3} + 25x^{2} - 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 1 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 5\nu^{2} - \nu + 15 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{3} - \nu^{2} + 25\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} - 15\nu^{3} + 8\nu^{2} - 75\nu + 15 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} - 15\beta_{4} + 5\beta_{3} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} - 25\beta_{2} - 25\beta _1 - 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(379\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 568.1 |
|
−1.16503 | + | 2.01789i | 0 | −1.71459 | − | 2.96975i | −0.900885 | 0 | −0.500000 | − | 0.866025i | 3.33006 | 0 | 1.04956 | − | 1.81789i | ||||||||||||||||||||||||||||
| 568.2 | 0.100820 | − | 0.174625i | 0 | 0.979671 | + | 1.69684i | −3.75770 | 0 | −0.500000 | − | 0.866025i | 0.798360 | 0 | −0.378851 | + | 0.656189i | |||||||||||||||||||||||||||||
| 568.3 | 1.06421 | − | 1.84326i | 0 | −1.26508 | − | 2.19119i | 2.65859 | 0 | −0.500000 | − | 0.866025i | −1.12842 | 0 | 2.82929 | − | 4.90048i | |||||||||||||||||||||||||||||
| 757.1 | −1.16503 | − | 2.01789i | 0 | −1.71459 | + | 2.96975i | −0.900885 | 0 | −0.500000 | + | 0.866025i | 3.33006 | 0 | 1.04956 | + | 1.81789i | |||||||||||||||||||||||||||||
| 757.2 | 0.100820 | + | 0.174625i | 0 | 0.979671 | − | 1.69684i | −3.75770 | 0 | −0.500000 | + | 0.866025i | 0.798360 | 0 | −0.378851 | − | 0.656189i | |||||||||||||||||||||||||||||
| 757.3 | 1.06421 | + | 1.84326i | 0 | −1.26508 | + | 2.19119i | 2.65859 | 0 | −0.500000 | + | 0.866025i | −1.12842 | 0 | 2.82929 | + | 4.90048i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 819.2.o.f | 6 | |
| 3.b | odd | 2 | 1 | 273.2.k.b | ✓ | 6 | |
| 13.c | even | 3 | 1 | inner | 819.2.o.f | 6 | |
| 39.h | odd | 6 | 1 | 3549.2.a.l | 3 | ||
| 39.i | odd | 6 | 1 | 273.2.k.b | ✓ | 6 | |
| 39.i | odd | 6 | 1 | 3549.2.a.m | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.k.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 273.2.k.b | ✓ | 6 | 39.i | odd | 6 | 1 | |
| 819.2.o.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 819.2.o.f | 6 | 13.c | even | 3 | 1 | inner | |
| 3549.2.a.l | 3 | 39.h | odd | 6 | 1 | ||
| 3549.2.a.m | 3 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):
|
\( T_{2}^{6} + 5T_{2}^{4} - 2T_{2}^{3} + 25T_{2}^{2} - 5T_{2} + 1 \)
|
|
\( T_{11}^{6} - 8T_{11}^{5} + 53T_{11}^{4} - 110T_{11}^{3} + 209T_{11}^{2} + 121T_{11} + 121 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 5 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} + 2 T^{2} - 9 T - 9)^{2} \)
|
| $7$ |
\( (T^{2} + T + 1)^{3} \)
|
| $11$ |
\( T^{6} - 8 T^{5} + \cdots + 121 \)
|
| $13$ |
\( T^{6} - 10 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( T^{6} - 12 T^{5} + \cdots + 1849 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 729 \)
|
| $23$ |
\( T^{6} + 7 T^{5} + \cdots + 225 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 169 \)
|
| $31$ |
\( (T^{3} - 5 T^{2} + \cdots + 169)^{2} \)
|
| $37$ |
\( T^{6} + 89 T^{4} + \cdots + 16129 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 46656 \)
|
| $43$ |
\( T^{6} - 7 T^{5} + \cdots + 225 \)
|
| $47$ |
\( (T^{3} + 9 T^{2} + \cdots - 405)^{2} \)
|
| $53$ |
\( (T^{3} - 13 T^{2} + 23 T - 3)^{2} \)
|
| $59$ |
\( T^{6} - 11 T^{5} + \cdots + 2673225 \)
|
| $61$ |
\( T^{6} + 19 T^{5} + \cdots + 75625 \)
|
| $67$ |
\( T^{6} + 21 T^{5} + \cdots + 1 \)
|
| $71$ |
\( T^{6} - 22 T^{5} + \cdots + 47961 \)
|
| $73$ |
\( (T^{3} + 14 T^{2} + \cdots + 71)^{2} \)
|
| $79$ |
\( (T^{3} + T^{2} - 62 T + 163)^{2} \)
|
| $83$ |
\( (T^{3} + 32 T^{2} + \cdots + 365)^{2} \)
|
| $89$ |
\( T^{6} + 3 T^{5} + \cdots + 20449 \)
|
| $97$ |
\( T^{6} - 21 T^{5} + \cdots + 139129 \)
|
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