Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8470,2,Mod(1,8470)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8470.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8470.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: | \(67.6332905120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.13298000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 3x^{3} + 26x^{2} + 13x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 3x^{3} + 26x^{2} + 13x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 8\nu^{3} - 22\nu^{2} - 20\nu - 3 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 16\nu^{3} - 14\nu^{2} - 30\nu - 6 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 21\nu^{3} - 14\nu^{2} - 60\nu - 16 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 7\nu^{4} - 24\nu^{3} + 41\nu^{2} + 45\nu - 11 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + 5\beta_{3} + 8\beta_{2} + 8\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 8\beta_{4} - 10\beta_{3} + 5\beta_{2} + 38\beta _1 + 21 \)
|
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 | −2.80837 | 1.00000 | 1.00000 | −2.80837 | 1.00000 | 1.00000 | 4.88695 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.64424 | 1.00000 | 1.00000 | −2.64424 | 1.00000 | 1.00000 | 3.99201 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.418877 | 1.00000 | 1.00000 | −0.418877 | 1.00000 | 1.00000 | −2.82454 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.17142 | 1.00000 | 1.00000 | 1.17142 | 1.00000 | 1.00000 | −1.62777 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.44508 | 1.00000 | 1.00000 | 2.44508 | 1.00000 | 1.00000 | 2.97843 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.25498 | 1.00000 | 1.00000 | 3.25498 | 1.00000 | 1.00000 | 7.59492 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 8470.2.a.de | 6 | |
| 11.b | odd | 2 | 1 | 8470.2.a.cy | 6 | ||
| 11.c | even | 5 | 2 | 770.2.n.i | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.n.i | ✓ | 12 | 11.c | even | 5 | 2 | |
| 8470.2.a.cy | 6 | 11.b | odd | 2 | 1 | ||
| 8470.2.a.de | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):
|
\( T_{3}^{6} - T_{3}^{5} - 16T_{3}^{4} + 13T_{3}^{3} + 66T_{3}^{2} - 45T_{3} - 29 \)
|
|
\( T_{13}^{6} - 2T_{13}^{5} - 32T_{13}^{4} + 94T_{13}^{3} + 160T_{13}^{2} - 704T_{13} + 484 \)
|
|
\( T_{17}^{6} - 7T_{17}^{5} - 20T_{17}^{4} + 183T_{17}^{3} - 308T_{17}^{2} + 139T_{17} + 11 \)
|
|
\( T_{19}^{6} - 11T_{19}^{5} + 20T_{19}^{4} + 177T_{19}^{3} - 896T_{19}^{2} + 1445T_{19} - 725 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots - 29 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 484 \)
|
| $17$ |
\( T^{6} - 7 T^{5} + \cdots + 11 \)
|
| $19$ |
\( T^{6} - 11 T^{5} + \cdots - 725 \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots - 1856 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 3020 \)
|
| $31$ |
\( T^{6} - 42 T^{4} + \cdots - 284 \)
|
| $37$ |
\( T^{6} + 14 T^{5} + \cdots - 4 \)
|
| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 3649 \)
|
| $43$ |
\( T^{6} - 19 T^{5} + \cdots - 20719 \)
|
| $47$ |
\( T^{6} - 22 T^{5} + \cdots + 49856 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots - 244 \)
|
| $59$ |
\( T^{6} + 7 T^{5} + \cdots + 95 \)
|
| $61$ |
\( T^{6} - 22 T^{5} + \cdots - 112156 \)
|
| $67$ |
\( T^{6} - 5 T^{5} + \cdots - 5821 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 13036 \)
|
| $73$ |
\( T^{6} - 13 T^{5} + \cdots + 71779 \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots - 848020 \)
|
| $83$ |
\( T^{6} + 5 T^{5} + \cdots - 53629 \)
|
| $89$ |
\( T^{6} - T^{5} + \cdots - 362975 \)
|
| $97$ |
\( T^{6} + 3 T^{5} + \cdots - 4649 \)
|
| show more | |
| show less |