Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2842,2,Mod(1,2842)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2842, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2842.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2842 = 2 \cdot 7^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2842.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: | \(22.6934842544\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.401917952.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 14x^{4} + 49x^{2} - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 14x^{4} + 49x^{2} - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 7\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{2} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{2} + 15 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} + 32\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{4} + 10\beta_{3} + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{5} + 36\beta_{2} + 52\beta_1 \)
|
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 |
|
1.00000 | −2.90839 | 1.00000 | −1.49418 | −2.90839 | 0 | 1.00000 | 5.45876 | −1.49418 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.26409 | 1.00000 | −3.67831 | −2.26409 | 0 | 1.00000 | 2.12612 | −3.67831 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.644301 | 1.00000 | −2.05851 | −0.644301 | 0 | 1.00000 | −2.58488 | −2.05851 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.644301 | 1.00000 | 2.05851 | 0.644301 | 0 | 1.00000 | −2.58488 | 2.05851 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.26409 | 1.00000 | 3.67831 | 2.26409 | 0 | 1.00000 | 2.12612 | 3.67831 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.90839 | 1.00000 | 1.49418 | 2.90839 | 0 | 1.00000 | 5.45876 | 1.49418 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(29\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2842.2.a.bb | ✓ | 6 |
| 7.b | odd | 2 | 1 | inner | 2842.2.a.bb | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2842.2.a.bb | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2842.2.a.bb | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2842))\):
|
\( T_{3}^{6} - 14T_{3}^{4} + 49T_{3}^{2} - 18 \)
|
|
\( T_{5}^{6} - 20T_{5}^{4} + 97T_{5}^{2} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 14 T^{4} + \cdots - 18 \)
|
| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 128 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} - 4 T^{2} - 7 T + 18)^{2} \)
|
| $13$ |
\( T^{6} - 60 T^{4} + \cdots - 5832 \)
|
| $17$ |
\( T^{6} - 50 T^{4} + \cdots - 512 \)
|
| $19$ |
\( T^{6} - 62 T^{4} + \cdots - 5000 \)
|
| $23$ |
\( (T^{3} + 4 T^{2} - 44 T - 32)^{2} \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( T^{6} - 68 T^{4} + \cdots - 2592 \)
|
| $37$ |
\( (T + 2)^{6} \)
|
| $41$ |
\( T^{6} - 130 T^{4} + \cdots - 512 \)
|
| $43$ |
\( (T^{3} - 6 T^{2} + \cdots + 958)^{2} \)
|
| $47$ |
\( T^{6} - 68 T^{4} + \cdots - 2592 \)
|
| $53$ |
\( (T^{3} - 8 T^{2} + 9 T + 10)^{2} \)
|
| $59$ |
\( T^{6} - 226 T^{4} + \cdots - 359552 \)
|
| $61$ |
\( T^{6} - 376 T^{4} + \cdots - 1936512 \)
|
| $67$ |
\( (T^{3} + 4 T^{2} + \cdots - 144)^{2} \)
|
| $71$ |
\( (T - 8)^{6} \)
|
| $73$ |
\( T^{6} - 226 T^{4} + \cdots - 359552 \)
|
| $79$ |
\( (T^{3} - 32 T^{2} + \cdots - 1032)^{2} \)
|
| $83$ |
\( T^{6} - 130 T^{4} + \cdots - 512 \)
|
| $89$ |
\( T^{6} - 146 T^{4} + \cdots - 2592 \)
|
| $97$ |
\( T^{6} - 386 T^{4} + \cdots - 3200 \)
|
| show more | |
| show less |