Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,2,Mod(221,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, 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("430.221");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 430.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: | \(3.43356728692\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).
| \(n\) | \(87\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 221.1 |
|
1.00000 | −1.04307 | + | 1.80664i | 1.00000 | 0.500000 | − | 0.866025i | −1.04307 | + | 1.80664i | 0.324030 | + | 0.561237i | 1.00000 | −0.675970 | − | 1.17081i | 0.500000 | − | 0.866025i | ||||||||||||||||||||||||
| 221.2 | 1.00000 | 0.285997 | − | 0.495361i | 1.00000 | 0.500000 | − | 0.866025i | 0.285997 | − | 0.495361i | 2.33641 | + | 4.04678i | 1.00000 | 1.33641 | + | 2.31473i | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 221.3 | 1.00000 | 1.25707 | − | 2.17731i | 1.00000 | 0.500000 | − | 0.866025i | 1.25707 | − | 2.17731i | −0.660442 | − | 1.14392i | 1.00000 | −1.66044 | − | 2.87597i | 0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 251.1 | 1.00000 | −1.04307 | − | 1.80664i | 1.00000 | 0.500000 | + | 0.866025i | −1.04307 | − | 1.80664i | 0.324030 | − | 0.561237i | 1.00000 | −0.675970 | + | 1.17081i | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 251.2 | 1.00000 | 0.285997 | + | 0.495361i | 1.00000 | 0.500000 | + | 0.866025i | 0.285997 | + | 0.495361i | 2.33641 | − | 4.04678i | 1.00000 | 1.33641 | − | 2.31473i | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 251.3 | 1.00000 | 1.25707 | + | 2.17731i | 1.00000 | 0.500000 | + | 0.866025i | 1.25707 | + | 2.17731i | −0.660442 | + | 1.14392i | 1.00000 | −1.66044 | + | 2.87597i | 0.500000 | + | 0.866025i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 430.2.e.e | ✓ | 6 |
| 43.c | even | 3 | 1 | inner | 430.2.e.e | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 430.2.e.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 430.2.e.e | ✓ | 6 | 43.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - T_{3}^{5} + 6T_{3}^{4} - T_{3}^{3} + 28T_{3}^{2} - 15T_{3} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \)
|
| $5$ |
\( (T^{2} - T + 1)^{3} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 16 \)
|
| $11$ |
\( (T^{3} + 4 T^{2} - 6)^{2} \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 21316 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} + 2 T^{5} + \cdots + 4 \)
|
| $23$ |
\( T^{6} + 8 T^{5} + \cdots + 144 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots + 91809 \)
|
| $31$ |
\( T^{6} - 8 T^{5} + \cdots + 36 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 14884 \)
|
| $41$ |
\( (T^{3} + T^{2} - 45 T + 27)^{2} \)
|
| $43$ |
\( T^{6} - 5 T^{5} + \cdots + 79507 \)
|
| $47$ |
\( (T^{3} + 9 T^{2} + \cdots - 549)^{2} \)
|
| $53$ |
\( T^{6} - 8 T^{5} + \cdots + 20736 \)
|
| $59$ |
\( (T^{3} + 12 T^{2} + \cdots - 432)^{2} \)
|
| $61$ |
\( T^{6} - 12 T^{5} + \cdots + 4624 \)
|
| $67$ |
\( T^{6} + 7 T^{5} + \cdots + 826281 \)
|
| $71$ |
\( T^{6} + 12 T^{5} + \cdots + 324 \)
|
| $73$ |
\( T^{6} + 14 T^{5} + \cdots + 158404 \)
|
| $79$ |
\( T^{6} + 4 T^{5} + \cdots + 331776 \)
|
| $83$ |
\( T^{6} - 15 T^{5} + \cdots + 59049 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots + 1225449 \)
|
| $97$ |
\( (T^{3} + 10 T^{2} + \cdots - 328)^{2} \)
|
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