Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [343,4,Mod(1,343)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("343.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 343.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: | \(20.2376551320\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-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 \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 |
|
−5.18598 | 0 | 18.8944 | 0 | 0 | 0 | −56.4981 | −27.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.46681 | 0 | −5.84846 | 0 | 0 | 0 | 20.3131 | −27.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 5.65279 | 0 | 23.9541 | 0 | 0 | 0 | 90.1850 | −27.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 343.4.a.a | ✓ | 3 |
| 7.b | odd | 2 | 1 | CM | 343.4.a.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 343.4.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 343.4.a.a | ✓ | 3 | 7.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\):
|
\( T_{2}^{3} + T_{2}^{2} - 30T_{2} - 43 \)
|
|
\( T_{3} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} + \cdots - 43 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 69 T^{2} + \cdots + 44239 \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} - 307 T^{2} + \cdots + 1852549 \)
|
| $29$ |
\( T^{3} + 267 T^{2} + \cdots - 18131863 \)
|
| $31$ |
\( T^{3} \)
|
| $37$ |
\( T^{3} + 239 T^{2} + \cdots - 12055667 \)
|
| $41$ |
\( T^{3} \)
|
| $43$ |
\( T^{3} - 797 T^{2} + \cdots + 11954039 \)
|
| $47$ |
\( T^{3} \)
|
| $53$ |
\( T^{3} - 363 T^{2} + \cdots + 200013071 \)
|
| $59$ |
\( T^{3} \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} - 1441 T^{2} + \cdots - 51634997 \)
|
| $71$ |
\( T^{3} - 951 T^{2} + \cdots + 999683761 \)
|
| $73$ |
\( T^{3} \)
|
| $79$ |
\( T^{3} - 1007 T^{2} + \cdots + 418017473 \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} \)
|
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