Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [418,2,Mod(45,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.45");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 418.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: | \(3.33774680449\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.101617200.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 8x^{4} - 4x^{3} + 64x^{2} - 16x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} - 2\nu + 40 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{3} - 2\nu^{2} + 64\nu ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} - 40\nu^{3} + 26\nu^{2} - 320\nu + 80 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{5} - 40\beta_{4} + 8\beta_{3} + 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 26\beta_{4} - 64\beta_{2} - 64\beta _1 - 26 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).
| \(n\) | \(287\) | \(343\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 45.1 |
|
0.500000 | + | 0.866025i | −1.34700 | − | 2.33307i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | 1.34700 | − | 2.33307i | −0.693995 | −1.00000 | −2.12880 | + | 3.68720i | 1.00000 | − | 1.73205i | ||||||||||||||||||||||
| 45.2 | 0.500000 | + | 0.866025i | −0.126000 | − | 0.218239i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | 0.126000 | − | 0.218239i | 1.74800 | −1.00000 | 1.46825 | − | 2.54308i | 1.00000 | − | 1.73205i | |||||||||||||||||||||||
| 45.3 | 0.500000 | + | 0.866025i | 1.47300 | + | 2.55131i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | −1.47300 | + | 2.55131i | 4.94600 | −1.00000 | −2.83944 | + | 4.91806i | 1.00000 | − | 1.73205i | |||||||||||||||||||||||
| 353.1 | 0.500000 | − | 0.866025i | −1.34700 | + | 2.33307i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | 1.34700 | + | 2.33307i | −0.693995 | −1.00000 | −2.12880 | − | 3.68720i | 1.00000 | + | 1.73205i | |||||||||||||||||||||||
| 353.2 | 0.500000 | − | 0.866025i | −0.126000 | + | 0.218239i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | 0.126000 | + | 0.218239i | 1.74800 | −1.00000 | 1.46825 | + | 2.54308i | 1.00000 | + | 1.73205i | |||||||||||||||||||||||
| 353.3 | 0.500000 | − | 0.866025i | 1.47300 | − | 2.55131i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | −1.47300 | − | 2.55131i | 4.94600 | −1.00000 | −2.83944 | − | 4.91806i | 1.00000 | + | 1.73205i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 418.2.e.i | ✓ | 6 |
| 19.c | even | 3 | 1 | inner | 418.2.e.i | ✓ | 6 |
| 19.c | even | 3 | 1 | 7942.2.a.bd | 3 | ||
| 19.d | odd | 6 | 1 | 7942.2.a.bj | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 418.2.e.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 418.2.e.i | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 7942.2.a.bd | 3 | 19.c | even | 3 | 1 | ||
| 7942.2.a.bj | 3 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\):
|
\( T_{3}^{6} + 8T_{3}^{4} + 4T_{3}^{3} + 64T_{3}^{2} + 16T_{3} + 4 \)
|
|
\( T_{5}^{2} + 2T_{5} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{3} \)
|
| $3$ |
\( T^{6} + 8 T^{4} + \cdots + 4 \)
|
| $5$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $7$ |
\( (T^{3} - 6 T^{2} + 4 T + 6)^{2} \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 7 T^{5} + \cdots + 5041 \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 2500 \)
|
| $19$ |
\( T^{6} + 5 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} + 12 T^{5} + \cdots + 6400 \)
|
| $29$ |
\( (T^{2} - T + 1)^{3} \)
|
| $31$ |
\( (T^{3} - 2 T^{2} + \cdots + 116)^{2} \)
|
| $37$ |
\( (T^{3} - 4 T^{2} + \cdots - 150)^{2} \)
|
| $41$ |
\( T^{6} + 10 T^{5} + \cdots + 2916 \)
|
| $43$ |
\( T^{6} + 5 T^{5} + \cdots + 6561 \)
|
| $47$ |
\( T^{6} - 11 T^{5} + \cdots + 251001 \)
|
| $53$ |
\( T^{6} + 128 T^{4} + \cdots + 16384 \)
|
| $59$ |
\( T^{6} \)
|
| $61$ |
\( T^{6} + 11 T^{5} + \cdots + 140625 \)
|
| $67$ |
\( T^{6} + 72 T^{4} + \cdots + 2916 \)
|
| $71$ |
\( T^{6} - 5 T^{5} + \cdots + 6561 \)
|
| $73$ |
\( T^{6} + 8 T^{5} + \cdots + 10404 \)
|
| $79$ |
\( T^{6} + 6 T^{5} + \cdots + 36100 \)
|
| $83$ |
\( (T^{3} - 7 T^{2} + \cdots - 205)^{2} \)
|
| $89$ |
\( T^{6} + T^{5} + \cdots + 1849 \)
|
| $97$ |
\( T^{6} - 7 T^{5} + \cdots + 17161 \)
|
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