Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,4,Mod(1,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, 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("342.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.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: | \(20.1786532220\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{273}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 68 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{273})\). 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 | −13.7614 | 0 | −3.76136 | −8.00000 | 0 | 27.5227 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 2.76136 | 0 | 12.7614 | −8.00000 | 0 | −5.52271 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
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 | 342.4.a.g | 2 | |
| 3.b | odd | 2 | 1 | 114.4.a.f | ✓ | 2 | |
| 12.b | even | 2 | 1 | 912.4.a.j | 2 | ||
| 57.d | even | 2 | 1 | 2166.4.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.a.f | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 342.4.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 912.4.a.j | 2 | 12.b | even | 2 | 1 | ||
| 2166.4.a.l | 2 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 11T_{5} - 38 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 11T - 38 \)
|
| $7$ |
\( T^{2} - 9T - 48 \)
|
| $11$ |
\( T^{2} - T - 3344 \)
|
| $13$ |
\( T^{2} - 16T - 1028 \)
|
| $17$ |
\( T^{2} + 9T - 594 \)
|
| $19$ |
\( (T + 19)^{2} \)
|
| $23$ |
\( T^{2} - 4T - 1088 \)
|
| $29$ |
\( T^{2} - 20T - 4268 \)
|
| $31$ |
\( T^{2} + 238T - 31976 \)
|
| $37$ |
\( T^{2} + 262T + 10336 \)
|
| $41$ |
\( T^{2} - 122T - 116672 \)
|
| $43$ |
\( T^{2} + 427T - 4172 \)
|
| $47$ |
\( T^{2} + 125T - 122288 \)
|
| $53$ |
\( T^{2} + 140T - 34412 \)
|
| $59$ |
\( T^{2} + 96T - 211728 \)
|
| $61$ |
\( T^{2} + 551T + 74194 \)
|
| $67$ |
\( T^{2} + 912T - 6096 \)
|
| $71$ |
\( T^{2} + 704T - 155648 \)
|
| $73$ |
\( T^{2} + 321T + 14226 \)
|
| $79$ |
\( T^{2} + 512T - 672656 \)
|
| $83$ |
\( T^{2} + 1188 T - 128736 \)
|
| $89$ |
\( T^{2} + 1776 T + 787452 \)
|
| $97$ |
\( T^{2} - 1772 T + 780628 \)
|
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