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: | 6.0.27870912.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).
| \(n\) | \(87\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 221.1 |
|
1.00000 | −0.927886 | + | 1.60715i | 1.00000 | −0.500000 | + | 0.866025i | −0.927886 | + | 1.60715i | 1.00000 | + | 1.73205i | 1.00000 | −0.221946 | − | 0.384421i | −0.500000 | + | 0.866025i | ||||||||||||||||||||||||
| 221.2 | 1.00000 | 0.339181 | − | 0.587479i | 1.00000 | −0.500000 | + | 0.866025i | 0.339181 | − | 0.587479i | 1.00000 | + | 1.73205i | 1.00000 | 1.26991 | + | 2.19955i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 221.3 | 1.00000 | 1.58870 | − | 2.75172i | 1.00000 | −0.500000 | + | 0.866025i | 1.58870 | − | 2.75172i | 1.00000 | + | 1.73205i | 1.00000 | −3.54797 | − | 6.14526i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||
| 251.1 | 1.00000 | −0.927886 | − | 1.60715i | 1.00000 | −0.500000 | − | 0.866025i | −0.927886 | − | 1.60715i | 1.00000 | − | 1.73205i | 1.00000 | −0.221946 | + | 0.384421i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 251.2 | 1.00000 | 0.339181 | + | 0.587479i | 1.00000 | −0.500000 | − | 0.866025i | 0.339181 | + | 0.587479i | 1.00000 | − | 1.73205i | 1.00000 | 1.26991 | − | 2.19955i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||
| 251.3 | 1.00000 | 1.58870 | + | 2.75172i | 1.00000 | −0.500000 | − | 0.866025i | 1.58870 | + | 2.75172i | 1.00000 | − | 1.73205i | 1.00000 | −3.54797 | + | 6.14526i | −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.f | ✓ | 6 |
| 43.c | even | 3 | 1 | inner | 430.2.e.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 430.2.e.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 430.2.e.f | ✓ | 6 | 43.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} + 9T_{3}^{4} + 2T_{3}^{3} + 33T_{3}^{2} - 20T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 16 \)
|
| $5$ |
\( (T^{2} + T + 1)^{3} \)
|
| $7$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $11$ |
\( (T^{3} + T^{2} - 6 T - 2)^{2} \)
|
| $13$ |
\( T^{6} + 14 T^{4} + \cdots + 256 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 256 \)
|
| $19$ |
\( T^{6} + 7 T^{5} + \cdots + 66564 \)
|
| $23$ |
\( T^{6} - 8 T^{5} + \cdots + 2304 \)
|
| $29$ |
\( T^{6} - 10 T^{5} + \cdots + 82944 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 16 \)
|
| $37$ |
\( T^{6} + 10 T^{5} + \cdots + 15376 \)
|
| $41$ |
\( (T^{3} + 5 T^{2} + \cdots - 169)^{2} \)
|
| $43$ |
\( T^{6} + 7 T^{5} + \cdots + 79507 \)
|
| $47$ |
\( (T^{3} - 4 T^{2} - 11 T + 26)^{2} \)
|
| $53$ |
\( T^{6} - 8 T^{5} + \cdots + 43264 \)
|
| $59$ |
\( (T^{3} - 3 T^{2} + \cdots + 356)^{2} \)
|
| $61$ |
\( T^{6} + 4 T^{5} + \cdots + 1557504 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots + 613089 \)
|
| $71$ |
\( T^{6} - 4 T^{5} + \cdots + 4096 \)
|
| $73$ |
\( T^{6} - 21 T^{5} + \cdots + 3364 \)
|
| $79$ |
\( T^{6} - 4 T^{5} + \cdots + 16384 \)
|
| $83$ |
\( T^{6} - 10 T^{5} + \cdots + 36864 \)
|
| $89$ |
\( T^{6} - 7 T^{5} + \cdots + 7569 \)
|
| $97$ |
\( (T^{3} - 5 T^{2} + \cdots + 432)^{2} \)
|
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