Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4004,2,Mod(1,4004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("4004.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4004.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: | \(31.9721009693\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.246302029.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 9\nu^{2} + 5\nu + 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 12\nu^{2} - 6\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 13\nu^{2} + 6\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 6\nu^{4} - 13\nu^{3} + 42\nu^{2} - 3\nu - 29 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 9\beta_{4} + 11\beta_{3} + 3\beta_{2} + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} + 6\beta_{4} + 25\beta_{3} + 22\beta_{2} + 34\beta _1 + 18 \)
|
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 |
|
0 | −3.12467 | 0 | −1.82266 | 0 | 1.00000 | 0 | 6.76356 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.73486 | 0 | 1.63787 | 0 | 1.00000 | 0 | 0.00974910 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.23083 | 0 | −3.39686 | 0 | 1.00000 | 0 | −1.48506 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.714665 | 0 | 2.43635 | 0 | 1.00000 | 0 | −2.48925 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.820857 | 0 | 0.0215726 | 0 | 1.00000 | 0 | −2.32619 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.55484 | 0 | −1.87627 | 0 | 1.00000 | 0 | 3.52720 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
| \(13\) | \(1\) |
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 | 4004.2.a.g | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4004.2.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 2T_{3}^{5} - 9T_{3}^{4} - 14T_{3}^{3} + 15T_{3}^{2} + 13T_{3} - 10 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 10 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots - 256 \)
|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots + 4 \)
|
| $23$ |
\( T^{6} - 5 T^{5} + \cdots + 1 \)
|
| $29$ |
\( T^{6} + 14 T^{5} + \cdots + 544 \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots + 275 \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots + 2538 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 13024 \)
|
| $43$ |
\( T^{6} + 14 T^{5} + \cdots - 54 \)
|
| $47$ |
\( T^{6} - 2 T^{5} + \cdots - 26352 \)
|
| $53$ |
\( T^{6} + 3 T^{5} + \cdots - 482 \)
|
| $59$ |
\( T^{6} - 2 T^{5} + \cdots + 1040 \)
|
| $61$ |
\( T^{6} + 26 T^{5} + \cdots - 60056 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots - 50630 \)
|
| $71$ |
\( T^{6} - 3 T^{5} + \cdots - 47200 \)
|
| $73$ |
\( T^{6} + 7 T^{5} + \cdots + 29266 \)
|
| $79$ |
\( T^{6} - 140 T^{4} + \cdots + 18292 \)
|
| $83$ |
\( T^{6} + 15 T^{5} + \cdots - 148838 \)
|
| $89$ |
\( T^{6} + T^{5} + \cdots - 5333 \)
|
| $97$ |
\( T^{6} + 16 T^{5} + \cdots - 701 \)
|
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