Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2006,2,Mod(1,2006)]
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([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2006 = 2 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2006.a (trivial) |
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: | yes |
| Analytic conductor: | \(16.0179906455\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13676.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 6x^{2} + 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} + 5\beta _1 - 4 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −3.22628 | 1.00000 | 3.22628 | −3.22628 | −3.00000 | 1.00000 | 7.40890 | 3.22628 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.91023 | 1.00000 | 1.91023 | −1.91023 | −3.00000 | 1.00000 | 0.648991 | 1.91023 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.493910 | 1.00000 | −0.493910 | 0.493910 | −3.00000 | 1.00000 | −2.75605 | −0.493910 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.64261 | 1.00000 | −1.64261 | 1.64261 | −3.00000 | 1.00000 | −0.301842 | −1.64261 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(-1\) |
| \(59\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2006.2.a.s | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2006.2.a.s | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\):
|
\( T_{3}^{4} + 3T_{3}^{3} - 4T_{3}^{2} - 9T_{3} + 5 \)
|
|
\( T_{5}^{4} - 3T_{5}^{3} - 4T_{5}^{2} + 9T_{5} + 5 \)
|
|
\( T_{11}^{4} + 5T_{11}^{3} - 17T_{11}^{2} - 84T_{11} - 8 \)
|
|
\( T_{31}^{4} + 8T_{31}^{3} - 93T_{31}^{2} - 804T_{31} - 1252 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 5 \)
|
| $5$ |
\( T^{4} - 3 T^{3} + \cdots + 5 \)
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| $7$ |
\( (T + 3)^{4} \)
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| $11$ |
\( T^{4} + 5 T^{3} + \cdots - 8 \)
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| $13$ |
\( T^{4} + T^{3} + \cdots + 172 \)
|
| $17$ |
\( (T - 1)^{4} \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots - 61 \)
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| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 128 \)
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| $29$ |
\( T^{4} + 18 T^{3} + \cdots + 208 \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots - 1252 \)
|
| $37$ |
\( T^{4} - 7 T^{3} + \cdots - 512 \)
|
| $41$ |
\( T^{4} + 8 T^{3} + \cdots - 2068 \)
|
| $43$ |
\( T^{4} + 18 T^{3} + \cdots - 3308 \)
|
| $47$ |
\( T^{4} + 16 T^{3} + \cdots - 8 \)
|
| $53$ |
\( T^{4} - 11 T^{3} + \cdots + 11 \)
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| $59$ |
\( (T + 1)^{4} \)
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| $61$ |
\( T^{4} + 20 T^{3} + \cdots - 592 \)
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| $67$ |
\( T^{4} - 4 T^{3} + \cdots + 2264 \)
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| $71$ |
\( T^{4} + 22 T^{3} + \cdots - 4544 \)
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| $73$ |
\( T^{4} + 4 T^{3} + \cdots + 1868 \)
|
| $79$ |
\( T^{4} + 31 T^{3} + \cdots - 2365 \)
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| $83$ |
\( T^{4} + 27 T^{3} + \cdots - 13912 \)
|
| $89$ |
\( T^{4} - 18 T^{3} + \cdots - 1948 \)
|
| $97$ |
\( T^{4} - 9 T^{3} + \cdots + 4804 \)
|
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