Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1062,4,Mod(1,1062)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1062, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1062.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1062 = 2 \cdot 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1062.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: | \(62.6600284261\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.47277.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 48x - 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 354) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 48x - 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 2\nu - 32 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 2\beta _1 + 32 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | −2.88648 | 0 | −22.9482 | −8.00000 | 0 | 5.77297 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 4.13667 | 0 | 4.92477 | −8.00000 | 0 | −8.27334 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 16.7498 | 0 | 24.0234 | −8.00000 | 0 | −33.4996 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(59\) | \( -1 \) |
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 | 1062.4.a.i | 3 | |
| 3.b | odd | 2 | 1 | 354.4.a.g | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.4.a.g | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 1062.4.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 18T_{5}^{2} + 9T_{5} + 200 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1062))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 18 T^{2} + \cdots + 200 \)
|
| $7$ |
\( T^{3} - 6 T^{2} + \cdots + 2715 \)
|
| $11$ |
\( T^{3} - 9 T^{2} + \cdots - 31415 \)
|
| $13$ |
\( T^{3} + 45 T^{2} + \cdots - 90561 \)
|
| $17$ |
\( T^{3} - 96 T^{2} + \cdots + 359573 \)
|
| $19$ |
\( T^{3} + 171 T^{2} + \cdots - 261448 \)
|
| $23$ |
\( T^{3} - 177 T^{2} + \cdots + 447350 \)
|
| $29$ |
\( T^{3} - 351 T^{2} + \cdots - 33082 \)
|
| $31$ |
\( T^{3} + 267 T^{2} + \cdots - 471056 \)
|
| $37$ |
\( T^{3} + 222 T^{2} + \cdots - 3442019 \)
|
| $41$ |
\( T^{3} - 780 T^{2} + \cdots - 13036477 \)
|
| $43$ |
\( T^{3} - 21 T^{2} + \cdots - 1216743 \)
|
| $47$ |
\( T^{3} - 717 T^{2} + \cdots - 2660896 \)
|
| $53$ |
\( T^{3} - 234 T^{2} + \cdots + 280692 \)
|
| $59$ |
\( (T - 59)^{3} \)
|
| $61$ |
\( T^{3} - 615 T^{2} + \cdots - 1849294 \)
|
| $67$ |
\( T^{3} + 144 T^{2} + \cdots + 5182508 \)
|
| $71$ |
\( T^{3} + 981 T^{2} + \cdots + 15591315 \)
|
| $73$ |
\( T^{3} + 63 T^{2} + \cdots - 31365664 \)
|
| $79$ |
\( T^{3} - 555 T^{2} + \cdots - 146656073 \)
|
| $83$ |
\( T^{3} + 66 T^{2} + \cdots - 138813551 \)
|
| $89$ |
\( T^{3} - 1389 T^{2} + \cdots + 169366104 \)
|
| $97$ |
\( T^{3} - 960 T^{2} + \cdots + 172378396 \)
|
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