Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1386,4,Mod(1,1386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1386.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1386.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: | \(81.7766472680\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.768425.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 218x + 792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 462) |
| 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
\( x^{3} - x^{2} - 218x + 792 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 7\nu - 150 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 5\nu + 144 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{2} + 5\beta _1 + 293 ) / 2 \)
|
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 |
|
2.00000 | 0 | 4.00000 | −16.6244 | 0 | 7.00000 | 8.00000 | 0 | −33.2487 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 3.51833 | 0 | 7.00000 | 8.00000 | 0 | 7.03666 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 20.1060 | 0 | 7.00000 | 8.00000 | 0 | 40.2121 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1386.4.a.bi | 3 | |
| 3.b | odd | 2 | 1 | 462.4.a.q | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.4.a.q | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 1386.4.a.bi | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):
|
\( T_{5}^{3} - 7T_{5}^{2} - 322T_{5} + 1176 \)
|
|
\( T_{13}^{3} - 51T_{13}^{2} + 52T_{13} + 3932 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 7 T^{2} + \cdots + 1176 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} - 51 T^{2} + \cdots + 3932 \)
|
| $17$ |
\( T^{3} + 74 T^{2} + \cdots - 359328 \)
|
| $19$ |
\( T^{3} - 25 T^{2} + \cdots + 760520 \)
|
| $23$ |
\( T^{3} - 82 T^{2} + \cdots - 159744 \)
|
| $29$ |
\( T^{3} + 81 T^{2} + \cdots + 659868 \)
|
| $31$ |
\( T^{3} - 210 T^{2} + \cdots + 11130880 \)
|
| $37$ |
\( T^{3} - 253 T^{2} + \cdots + 13829684 \)
|
| $41$ |
\( T^{3} + 370 T^{2} + \cdots - 44916000 \)
|
| $43$ |
\( T^{3} - 626 T^{2} + \cdots + 42380512 \)
|
| $47$ |
\( T^{3} - 685 T^{2} + \cdots + 43208640 \)
|
| $53$ |
\( T^{3} + 170 T^{2} + \cdots + 28952280 \)
|
| $59$ |
\( T^{3} + 385 T^{2} + \cdots + 16299360 \)
|
| $61$ |
\( T^{3} - 266 T^{2} + \cdots + 17512 \)
|
| $67$ |
\( T^{3} - 841 T^{2} + \cdots + 120967792 \)
|
| $71$ |
\( T^{3} - 224 T^{2} + \cdots + 221119488 \)
|
| $73$ |
\( T^{3} - 1147 T^{2} + \cdots + 25092416 \)
|
| $79$ |
\( T^{3} - 1340 T^{2} + \cdots - 16773120 \)
|
| $83$ |
\( T^{3} + 1006 T^{2} + \cdots - 30441312 \)
|
| $89$ |
\( T^{3} + 890 T^{2} + \cdots - 215502120 \)
|
| $97$ |
\( T^{3} + \cdots + 1152318440 \)
|
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