Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8040,2,Mod(1,8040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8040.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8040.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: | \(64.1997232251\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.630757.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 3x^{2} + 9x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 7x^{3} + 3x^{2} + 9x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + \nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + 11\beta _1 + 15 \)
|
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 | −1.00000 | 0 | 1.00000 | 0 | −1.33987 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | 1.00000 | 0 | −0.884038 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 1.00000 | 0 | 2.10529 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.00000 | 0 | 2.69035 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 1.00000 | 0 | 3.42827 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 8040.2.a.o | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8040.2.a.o | ✓ | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):
|
\( T_{7}^{5} - 6T_{7}^{4} + 5T_{7}^{3} + 20T_{7}^{2} - 17T_{7} - 23 \)
|
|
\( T_{11}^{5} + 2T_{11}^{4} - 21T_{11}^{3} + 8T_{11}^{2} + 61T_{11} - 49 \)
|
|
\( T_{13}^{5} + 5T_{13}^{4} - 7T_{13}^{3} - 15T_{13}^{2} + 9T_{13} + 2 \)
|
|
\( T_{17}^{5} + 8T_{17}^{4} + 6T_{17}^{3} - 45T_{17}^{2} - 27T_{17} + 46 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 6 T^{4} + \cdots - 23 \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots - 49 \)
|
| $13$ |
\( T^{5} + 5 T^{4} + \cdots + 2 \)
|
| $17$ |
\( T^{5} + 8 T^{4} + \cdots + 46 \)
|
| $19$ |
\( T^{5} - T^{4} + \cdots - 98 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots + 100 \)
|
| $29$ |
\( T^{5} + 13 T^{4} + \cdots - 44 \)
|
| $31$ |
\( T^{5} - 5 T^{4} + \cdots - 1316 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 10609 \)
|
| $41$ |
\( T^{5} + 15 T^{4} + \cdots + 878 \)
|
| $43$ |
\( T^{5} - 7 T^{4} + \cdots + 2206 \)
|
| $47$ |
\( T^{5} + T^{4} + \cdots - 814 \)
|
| $53$ |
\( T^{5} + 11 T^{4} + \cdots - 16 \)
|
| $59$ |
\( T^{5} + 21 T^{4} + \cdots + 1238 \)
|
| $61$ |
\( T^{5} + 8 T^{4} + \cdots - 1619 \)
|
| $67$ |
\( (T - 1)^{5} \)
|
| $71$ |
\( T^{5} + 13 T^{4} + \cdots + 215 \)
|
| $73$ |
\( T^{5} + 24 T^{4} + \cdots + 1784 \)
|
| $79$ |
\( T^{5} + 5 T^{4} + \cdots - 1138 \)
|
| $83$ |
\( T^{5} + 5 T^{4} + \cdots + 7379 \)
|
| $89$ |
\( T^{5} + 23 T^{4} + \cdots - 28489 \)
|
| $97$ |
\( T^{5} + 8 T^{4} + \cdots - 103 \)
|
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