Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1640,2,Mod(1,1640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1640, 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("1640.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1640 = 2^{3} \cdot 5 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1640.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: | \(13.0954659315\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.135076.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 4\nu^{2} - 2\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 6\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{2} + 7 \)
|
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 | −3.03288 | 0 | −1.00000 | 0 | −3.35123 | 0 | 6.19838 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.60810 | 0 | −1.00000 | 0 | 0.0904211 | 0 | −0.414002 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.423072 | 0 | −1.00000 | 0 | 2.64382 | 0 | −2.82101 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.35357 | 0 | −1.00000 | 0 | −2.74439 | 0 | −1.16786 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.86435 | 0 | −1.00000 | 0 | −3.63863 | 0 | 5.20449 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(41\) | \( -1 \) |
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 | 1640.2.a.m | ✓ | 5 |
| 4.b | odd | 2 | 1 | 3280.2.a.bi | 5 | ||
| 5.b | even | 2 | 1 | 8200.2.a.bd | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1640.2.a.m | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 3280.2.a.bi | 5 | 4.b | odd | 2 | 1 | ||
| 8200.2.a.bd | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1640))\):
|
\( T_{3}^{5} - 11T_{3}^{3} + 2T_{3}^{2} + 20T_{3} - 8 \)
|
|
\( T_{7}^{5} + 7T_{7}^{4} + 5T_{7}^{3} - 50T_{7}^{2} - 84T_{7} + 8 \)
|
|
\( T_{11}^{5} - 2T_{11}^{4} - 29T_{11}^{3} + 78T_{11}^{2} + 100T_{11} - 296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 11 T^{3} + \cdots - 8 \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( T^{5} + 7 T^{4} + \cdots + 8 \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots - 296 \)
|
| $13$ |
\( T^{5} + 7 T^{4} + \cdots - 8 \)
|
| $17$ |
\( T^{5} - 35 T^{3} + \cdots + 8 \)
|
| $19$ |
\( T^{5} + 3 T^{4} + \cdots - 664 \)
|
| $23$ |
\( T^{5} + 12 T^{4} + \cdots - 64 \)
|
| $29$ |
\( T^{5} - 5 T^{4} + \cdots - 8 \)
|
| $31$ |
\( T^{5} + 11 T^{4} + \cdots + 512 \)
|
| $37$ |
\( T^{5} + 9 T^{4} + \cdots + 17288 \)
|
| $41$ |
\( (T - 1)^{5} \)
|
| $43$ |
\( T^{5} - 3 T^{4} + \cdots - 6848 \)
|
| $47$ |
\( T^{5} + 35 T^{4} + \cdots + 11864 \)
|
| $53$ |
\( T^{5} + 4 T^{4} + \cdots - 3232 \)
|
| $59$ |
\( T^{5} + 15 T^{4} + \cdots + 12608 \)
|
| $61$ |
\( T^{5} - 10 T^{4} + \cdots - 27688 \)
|
| $67$ |
\( T^{5} + 7 T^{4} + \cdots - 10664 \)
|
| $71$ |
\( T^{5} + 4 T^{4} + \cdots - 4856 \)
|
| $73$ |
\( T^{5} + 27 T^{4} + \cdots - 4232 \)
|
| $79$ |
\( T^{5} + 23 T^{4} + \cdots + 4856 \)
|
| $83$ |
\( T^{5} + 17 T^{4} + \cdots + 5504 \)
|
| $89$ |
\( T^{5} + 8 T^{4} + \cdots - 4136 \)
|
| $97$ |
\( T^{5} + 12 T^{4} + \cdots - 144992 \)
|
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