Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6422,2,Mod(1,6422)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, 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("6422.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6422 = 2 \cdot 13^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6422.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: | \(51.2799281781\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.316645497.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 494) |
| 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
\( x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 12\nu^{3} + 36\nu^{2} + 10\nu - 31 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} - 24\nu^{3} + 33\nu^{2} + 29\nu - 14 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 27\nu^{3} - 33\nu^{2} - 56\nu + 17 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 13\nu^{3} + 12\nu^{2} + 25\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + 9\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{5} + 13\beta_{4} - 11\beta_{3} + 9\beta_{2} - 3\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{5} + 27\beta_{4} + 27\beta_{3} - 3\beta_{2} + 89\beta _1 - 17 \)
|
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 | −3.28714 | 1.00000 | 2.88198 | −3.28714 | 3.02408 | 1.00000 | 7.80527 | 2.88198 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.13360 | 1.00000 | 2.43014 | −1.13360 | −3.63269 | 1.00000 | −1.71494 | 2.43014 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.260281 | 1.00000 | −4.36375 | 0.260281 | 0.775135 | 1.00000 | −2.93225 | −4.36375 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.724358 | 1.00000 | −1.67578 | 0.724358 | −2.46211 | 1.00000 | −2.47531 | −1.67578 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.19765 | 1.00000 | 3.12703 | 2.19765 | 2.17944 | 1.00000 | 1.82965 | 3.12703 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.23845 | 1.00000 | −0.399620 | 3.23845 | 1.11615 | 1.00000 | 7.48758 | −0.399620 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(1\) |
| \(19\) | \(1\) |
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 | 6422.2.a.bd | 6 | |
| 13.b | even | 2 | 1 | 6422.2.a.bc | 6 | ||
| 13.c | even | 3 | 2 | 494.2.g.f | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 494.2.g.f | ✓ | 12 | 13.c | even | 3 | 2 | |
| 6422.2.a.bc | 6 | 13.b | even | 2 | 1 | ||
| 6422.2.a.bd | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 24T_{3}^{3} + 13T_{3}^{2} - 24T_{3} + 5 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 21T_{5}^{4} + 51T_{5}^{3} + 64T_{5}^{2} - 144T_{5} - 64 \)
|
|
\( T_{7}^{6} - T_{7}^{5} - 17T_{7}^{4} + 25T_{7}^{3} + 57T_{7}^{2} - 117T_{7} + 51 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 5 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 64 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots + 51 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots - 64 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 120 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 8 T^{5} + \cdots - 137 \)
|
| $29$ |
\( T^{6} - 20 T^{5} + \cdots + 37 \)
|
| $31$ |
\( T^{6} - 3 T^{5} + \cdots - 16376 \)
|
| $37$ |
\( T^{6} + 9 T^{5} + \cdots - 43096 \)
|
| $41$ |
\( T^{6} - 3 T^{5} + \cdots - 296 \)
|
| $43$ |
\( T^{6} - 13 T^{5} + \cdots + 200 \)
|
| $47$ |
\( T^{6} - 20 T^{5} + \cdots - 919 \)
|
| $53$ |
\( T^{6} - 25 T^{5} + \cdots - 99821 \)
|
| $59$ |
\( T^{6} - 220 T^{4} + \cdots + 42513 \)
|
| $61$ |
\( T^{6} - 6 T^{5} + \cdots + 184136 \)
|
| $67$ |
\( T^{6} + 32 T^{5} + \cdots + 172488 \)
|
| $71$ |
\( T^{6} - 39 T^{5} + \cdots - 116072 \)
|
| $73$ |
\( T^{6} - 7 T^{5} + \cdots + 3033 \)
|
| $79$ |
\( T^{6} - 18 T^{5} + \cdots - 87032 \)
|
| $83$ |
\( T^{6} + 7 T^{5} + \cdots - 92760 \)
|
| $89$ |
\( T^{6} - 9 T^{5} + \cdots - 648 \)
|
| $97$ |
\( T^{6} + 4 T^{5} + \cdots + 1150912 \)
|
| show more | |
| show less |