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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 16x^{5} + 29x^{4} + 60x^{3} - 79x^{2} - 47x - 2 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 16x^{5} + 29x^{4} + 60x^{3} - 79x^{2} - 47x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 15\nu^{4} + 14\nu^{3} + 50\nu^{2} - 37\nu - 30 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 15\nu^{4} - 14\nu^{3} - 40\nu^{2} + 27\nu - 20 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 17\nu^{4} - 18\nu^{3} - 66\nu^{2} + 71\nu + 16 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} + 16\nu^{4} - 28\nu^{3} - 57\nu^{2} + 71\nu + 24 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 15\nu^{4} - 43\nu^{3} - 48\nu^{2} + 116\nu + 27 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} - 2\beta_{4} + 9\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} + 4\beta_{5} + \beta_{4} + 8\beta_{3} + 13\beta_{2} + 9\beta _1 + 45 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{6} + 29\beta_{5} - 29\beta_{4} - \beta_{3} + 4\beta_{2} + 90\beta _1 - 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( -28\beta_{6} + 61\beta_{5} + 14\beta_{4} + 69\beta_{3} + 159\beta_{2} + 86\beta _1 + 451 \)
|
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 | −3.29705 | 1.00000 | −3.53833 | −3.29705 | −0.150781 | 1.00000 | 7.87054 | −3.53833 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.95679 | 1.00000 | 0.613661 | −1.95679 | 0.465830 | 1.00000 | 0.829012 | 0.613661 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.431028 | 1.00000 | 0.638027 | −0.431028 | 2.88689 | 1.00000 | −2.81422 | 0.638027 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.0462763 | 1.00000 | 2.81822 | −0.0462763 | −4.11903 | 1.00000 | −2.99786 | 2.81822 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.77366 | 1.00000 | −0.491653 | 1.77366 | 4.65991 | 1.00000 | 0.145874 | −0.491653 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.64643 | 1.00000 | 4.02894 | 2.64643 | 1.07639 | 1.00000 | 4.00358 | 4.02894 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.31105 | 1.00000 | −2.06886 | 3.31105 | −3.81920 | 1.00000 | 7.96307 | −2.06886 | ||||||||||||||||||||||||||||||||||||||||
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.bf | 7 | |
| 13.b | even | 2 | 1 | 6422.2.a.be | 7 | ||
| 13.d | odd | 4 | 2 | 494.2.d.c | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 494.2.d.c | ✓ | 14 | 13.d | odd | 4 | 2 | |
| 6422.2.a.be | 7 | 13.b | even | 2 | 1 | ||
| 6422.2.a.bf | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):
|
\( T_{3}^{7} - 2T_{3}^{6} - 16T_{3}^{5} + 29T_{3}^{4} + 60T_{3}^{3} - 79T_{3}^{2} - 47T_{3} - 2 \)
|
|
\( T_{5}^{7} - 2T_{5}^{6} - 19T_{5}^{5} + 29T_{5}^{4} + 77T_{5}^{3} - 70T_{5}^{2} - 16T_{5} + 16 \)
|
|
\( T_{7}^{7} - T_{7}^{6} - 31T_{7}^{5} + 31T_{7}^{4} + 220T_{7}^{3} - 300T_{7}^{2} + 56T_{7} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots - 2 \)
|
| $5$ |
\( T^{7} - 2 T^{6} + \cdots + 16 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots + 16 \)
|
| $11$ |
\( T^{7} - 5 T^{6} + \cdots + 3328 \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} - 16 T^{6} + \cdots + 22622 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} - 3 T^{6} + \cdots + 14944 \)
|
| $29$ |
\( T^{7} + 7 T^{6} + \cdots - 1352 \)
|
| $31$ |
\( T^{7} - 11 T^{6} + \cdots + 173056 \)
|
| $37$ |
\( T^{7} + 3 T^{6} + \cdots + 36352 \)
|
| $41$ |
\( T^{7} - 15 T^{6} + \cdots + 4160 \)
|
| $43$ |
\( T^{7} - 23 T^{6} + \cdots + 3328 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots - 107968 \)
|
| $53$ |
\( T^{7} - 21 T^{6} + \cdots + 100256 \)
|
| $59$ |
\( T^{7} - 8 T^{6} + \cdots - 2276864 \)
|
| $61$ |
\( T^{7} + 6 T^{6} + \cdots - 58240 \)
|
| $67$ |
\( T^{7} - 28 T^{6} + \cdots - 848704 \)
|
| $71$ |
\( T^{7} - 4 T^{6} + \cdots - 78784 \)
|
| $73$ |
\( T^{7} - 12 T^{6} + \cdots - 690304 \)
|
| $79$ |
\( T^{7} + 4 T^{6} + \cdots - 2882560 \)
|
| $83$ |
\( T^{7} - 13 T^{6} + \cdots - 6258400 \)
|
| $89$ |
\( T^{7} + 15 T^{6} + \cdots - 13514752 \)
|
| $97$ |
\( T^{7} + 37 T^{6} + \cdots - 84736 \)
|
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