Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2006,2,Mod(237,2006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2006, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2006.237");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2006 = 2 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2006.b (of order \(2\), degree \(1\), 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: | \(16.0179906455\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 2\nu - 4 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - \nu^{4} + 4\nu^{3} - 5\nu^{2} + 2\nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 3\beta_{4} - \beta_{2} - 2\beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} - 3\beta_{4} + \beta_{3} - 2\beta_{2} - 3\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2006\mathbb{Z}\right)^\times\).
| \(n\) | \(1123\) | \(1771\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 237.1 |
|
−1.00000 | − | 2.77846i | 1.00000 | 0.529317i | 2.77846i | 1.47068i | −1.00000 | −4.71982 | − | 0.529317i | ||||||||||||||||||||||||||||||||||
| 237.2 | −1.00000 | − | 2.48929i | 1.00000 | 1.34292i | 2.48929i | − | 3.34292i | −1.00000 | −3.19656 | − | 1.34292i | ||||||||||||||||||||||||||||||||||
| 237.3 | −1.00000 | − | 0.289169i | 1.00000 | − | 2.81361i | 0.289169i | 0.813607i | −1.00000 | 2.91638 | 2.81361i | |||||||||||||||||||||||||||||||||||
| 237.4 | −1.00000 | 0.289169i | 1.00000 | 2.81361i | − | 0.289169i | − | 0.813607i | −1.00000 | 2.91638 | − | 2.81361i | ||||||||||||||||||||||||||||||||||
| 237.5 | −1.00000 | 2.48929i | 1.00000 | − | 1.34292i | − | 2.48929i | 3.34292i | −1.00000 | −3.19656 | 1.34292i | |||||||||||||||||||||||||||||||||||
| 237.6 | −1.00000 | 2.77846i | 1.00000 | − | 0.529317i | − | 2.77846i | − | 1.47068i | −1.00000 | −4.71982 | 0.529317i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2006.2.b.i | ✓ | 6 |
| 17.b | even | 2 | 1 | inner | 2006.2.b.i | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2006.2.b.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2006.2.b.i | ✓ | 6 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2006, [\chi])\):
|
\( T_{3}^{6} + 14T_{3}^{4} + 49T_{3}^{2} + 4 \)
|
|
\( T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} + 10 T^{4} + \cdots + 4 \)
|
| $7$ |
\( T^{6} + 14 T^{4} + \cdots + 16 \)
|
| $11$ |
\( T^{6} + 48 T^{4} + \cdots + 256 \)
|
| $13$ |
\( (T^{3} - 8 T^{2} + 6 T + 44)^{2} \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} - 6 T^{2} + 5 T + 4)^{2} \)
|
| $23$ |
\( T^{6} + 116 T^{4} + \cdots + 15376 \)
|
| $29$ |
\( T^{6} + 82 T^{4} + \cdots + 7396 \)
|
| $31$ |
\( T^{6} + 132 T^{4} + \cdots + 71824 \)
|
| $37$ |
\( T^{6} + 144 T^{4} + \cdots + 16384 \)
|
| $41$ |
\( T^{6} + 66 T^{4} + \cdots + 16 \)
|
| $43$ |
\( (T^{3} - 26 T^{2} + \cdots - 496)^{2} \)
|
| $47$ |
\( (T^{3} + 4 T^{2} - 24 T - 64)^{2} \)
|
| $53$ |
\( (T^{3} + 2 T^{2} + \cdots - 422)^{2} \)
|
| $59$ |
\( (T + 1)^{6} \)
|
| $61$ |
\( T^{6} + 80 T^{4} + \cdots + 18496 \)
|
| $67$ |
\( (T^{3} + 18 T^{2} + \cdots - 256)^{2} \)
|
| $71$ |
\( T^{6} + 168 T^{4} + \cdots + 30976 \)
|
| $73$ |
\( T^{6} + 100 T^{4} + \cdots + 13456 \)
|
| $79$ |
\( T^{6} + 198 T^{4} + \cdots + 53824 \)
|
| $83$ |
\( (T^{3} + 10 T^{2} + \cdots - 1000)^{2} \)
|
| $89$ |
\( (T^{3} - 2 T^{2} + \cdots + 296)^{2} \)
|
| $97$ |
\( T^{6} + 140 T^{4} + \cdots + 64 \)
|
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