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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{137}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 34 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{137})\). We also show the integral \(q\)-expansion of the trace form.
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 | −2.35235 | 0 | 7.00000 | 8.00000 | 0 | −4.70470 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 9.35235 | 0 | 7.00000 | 8.00000 | 0 | 18.7047 | |||||||||||||||||||||||||
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.ba | 2 | |
| 3.b | odd | 2 | 1 | 154.4.a.f | ✓ | 2 | |
| 12.b | even | 2 | 1 | 1232.4.a.p | 2 | ||
| 21.c | even | 2 | 1 | 1078.4.a.j | 2 | ||
| 33.d | even | 2 | 1 | 1694.4.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.a.f | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 1078.4.a.j | 2 | 21.c | even | 2 | 1 | ||
| 1232.4.a.p | 2 | 12.b | even | 2 | 1 | ||
| 1386.4.a.ba | 2 | 1.a | even | 1 | 1 | trivial | |
| 1694.4.a.l | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1386))\):
|
\( T_{5}^{2} - 7T_{5} - 22 \)
|
|
\( T_{13}^{2} - 46T_{13} - 2896 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 7T - 22 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( (T - 11)^{2} \)
|
| $13$ |
\( T^{2} - 46T - 2896 \)
|
| $17$ |
\( T^{2} - 88T + 1388 \)
|
| $19$ |
\( T^{2} + 46T - 6184 \)
|
| $23$ |
\( T^{2} - 115T - 4400 \)
|
| $29$ |
\( T^{2} - 372T + 25828 \)
|
| $31$ |
\( T^{2} + 137T + 4384 \)
|
| $37$ |
\( T^{2} + 19T - 766 \)
|
| $41$ |
\( T^{2} - 440T + 34700 \)
|
| $43$ |
\( T^{2} - 192T + 7024 \)
|
| $47$ |
\( T^{2} + 728T + 112768 \)
|
| $53$ |
\( T^{2} - 808T + 149516 \)
|
| $59$ |
\( T^{2} - 35T - 172348 \)
|
| $61$ |
\( T^{2} - 312T - 318164 \)
|
| $67$ |
\( T^{2} + 301T - 779108 \)
|
| $71$ |
\( T^{2} - 137T - 530464 \)
|
| $73$ |
\( T^{2} - 788T - 63964 \)
|
| $79$ |
\( T^{2} + 366T + 32256 \)
|
| $83$ |
\( T^{2} - 2324 T + 1336544 \)
|
| $89$ |
\( T^{2} + 1235 T + 366202 \)
|
| $97$ |
\( T^{2} - 709T - 996394 \)
|
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