Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [618,2,Mod(355,618)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(618, 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("618.355");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 618 = 2 \cdot 3 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 618.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: | \(4.93475484492\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.2091141441.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - 6\nu^{4} + 6\nu^{3} + 9\nu^{2} - 9\nu ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} + \nu^{4} + 3\nu^{3} - 6\nu^{2} + 9\nu + 9 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - 2\nu^{5} - 3\nu^{4} + 3\nu^{3} - 18\nu^{2} + 9\nu + 54 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 2\nu^{6} - 7\nu^{5} + 15\nu^{4} - 12\nu^{3} - 9\nu^{2} + 72\nu - 54 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{7} - 5\nu^{6} - 4\nu^{5} + 12\nu^{4} - 21\nu^{3} + 36\nu^{2} + 72\nu - 81 ) / 27 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 5\nu^{4} - 12\nu^{3} + 6\nu^{2} + 36\nu - 36 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 8\nu^{6} - 8\nu^{5} - 15\nu^{4} + 12\nu^{3} - 72\nu^{2} + 9\nu + 216 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{4} + 2\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} - 2\beta_{6} + 2\beta_{4} + 9\beta_{3} + 2\beta_{2} + 4\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -3\beta_{7} - \beta_{6} - 3\beta_{5} + 4\beta_{4} + \beta_{2} - 4\beta _1 + 18 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9\beta_{7} + 10\beta_{6} - 12\beta_{5} - 7\beta_{4} + 18\beta_{3} + 8\beta_{2} + \beta _1 + 18 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3\beta_{7} + 2\beta_{6} + 9\beta_{5} - 20\beta_{4} - 9\beta_{3} + 16\beta_{2} - 22\beta _1 - 27 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -24\beta_{7} - 8\beta_{6} + 3\beta_{5} + 5\beta_{4} + 108\beta_{3} - 19\beta_{2} - 5\beta _1 - 18 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/618\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(413\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 355.1 |
|
0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −2.19047 | + | 3.79401i | −0.500000 | − | 0.866025i | 0.518550 | − | 0.898154i | −1.00000 | 1.00000 | −4.38095 | ||||||||||||||||||||||||||||||||
| 355.2 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.300268 | + | 0.520080i | −0.500000 | − | 0.866025i | 1.39013 | − | 2.40777i | −1.00000 | 1.00000 | −0.600537 | |||||||||||||||||||||||||||||||||
| 355.3 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.164508 | + | 0.284936i | −0.500000 | − | 0.866025i | −1.63934 | + | 2.83942i | −1.00000 | 1.00000 | −0.329016 | |||||||||||||||||||||||||||||||||
| 355.4 | 0.500000 | + | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 1.15525 | − | 2.00095i | −0.500000 | − | 0.866025i | −1.26934 | + | 2.19856i | −1.00000 | 1.00000 | 2.31050 | |||||||||||||||||||||||||||||||||
| 571.1 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −2.19047 | − | 3.79401i | −0.500000 | + | 0.866025i | 0.518550 | + | 0.898154i | −1.00000 | 1.00000 | −4.38095 | |||||||||||||||||||||||||||||||||
| 571.2 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.300268 | − | 0.520080i | −0.500000 | + | 0.866025i | 1.39013 | + | 2.40777i | −1.00000 | 1.00000 | −0.600537 | |||||||||||||||||||||||||||||||||
| 571.3 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.164508 | − | 0.284936i | −0.500000 | + | 0.866025i | −1.63934 | − | 2.83942i | −1.00000 | 1.00000 | −0.329016 | |||||||||||||||||||||||||||||||||
| 571.4 | 0.500000 | − | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 1.15525 | + | 2.00095i | −0.500000 | + | 0.866025i | −1.26934 | − | 2.19856i | −1.00000 | 1.00000 | 2.31050 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 103.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 618.2.e.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1854.2.f.h | 8 | ||
| 103.c | even | 3 | 1 | inner | 618.2.e.e | ✓ | 8 |
| 309.h | odd | 6 | 1 | 1854.2.f.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 618.2.e.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 618.2.e.e | ✓ | 8 | 103.c | even | 3 | 1 | inner |
| 1854.2.f.h | 8 | 3.b | odd | 2 | 1 | ||
| 1854.2.f.h | 8 | 309.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 3T_{5}^{7} + 17T_{5}^{6} - 6T_{5}^{5} + 93T_{5}^{4} + 84T_{5}^{3} + 65T_{5}^{2} + 18T_{5} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots + 4 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 576 \)
|
| $11$ |
\( T^{8} + 21 T^{6} + \cdots + 2916 \)
|
| $13$ |
\( (T^{4} - 3 T^{3} - 11 T^{2} + \cdots + 10)^{2} \)
|
| $17$ |
\( T^{8} - 2 T^{7} + \cdots + 729 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots + 169744 \)
|
| $23$ |
\( (T^{4} + 12 T^{3} + \cdots + 27)^{2} \)
|
| $29$ |
\( T^{8} + 27 T^{6} + \cdots + 1296 \)
|
| $31$ |
\( (T^{4} + 6 T^{3} + \cdots - 383)^{2} \)
|
| $37$ |
\( (T^{4} - 4 T^{3} + \cdots - 636)^{2} \)
|
| $41$ |
\( T^{8} - 6 T^{7} + \cdots + 961 \)
|
| $43$ |
\( T^{8} + T^{7} + \cdots + 3600 \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots + 483025 \)
|
| $53$ |
\( T^{8} - 9 T^{7} + \cdots + 324 \)
|
| $59$ |
\( T^{8} - 4 T^{7} + \cdots + 324 \)
|
| $61$ |
\( (T^{4} + 13 T^{3} + \cdots - 9036)^{2} \)
|
| $67$ |
\( T^{8} + 14 T^{7} + \cdots + 5363856 \)
|
| $71$ |
\( T^{8} - 12 T^{7} + \cdots + 68121 \)
|
| $73$ |
\( (T^{4} + 9 T^{3} + 19 T^{2} + \cdots - 47)^{2} \)
|
| $79$ |
\( (T^{4} + 3 T^{3} - 27 T^{2} + \cdots - 9)^{2} \)
|
| $83$ |
\( T^{8} - 5 T^{7} + \cdots + 57600 \)
|
| $89$ |
\( (T^{4} - 2 T^{3} + \cdots + 1801)^{2} \)
|
| $97$ |
\( T^{8} - 6 T^{7} + \cdots + 12117361 \)
|
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