Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9405,2,Mod(1,9405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9405.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: | \(75.0993031010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.31252925.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 25x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3135) |
| 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
\( x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 25x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 9\nu^{3} - 2\nu^{2} + 16\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{5} + \nu^{4} + 17\nu^{3} - 4\nu^{2} - 29\nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + \nu^{4} + 18\nu^{3} - 4\nu^{2} - 34\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 8\beta_{2} + 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} - 9\beta_{4} + \beta_{3} + 2\beta_{2} + 29\beta _1 + 3 \)
|
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 |
|
−2.31060 | 0 | 3.33889 | 1.00000 | 0 | −1.50337 | −3.09365 | 0 | −2.31060 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.58358 | 0 | 0.507741 | 1.00000 | 0 | 2.90367 | 2.36312 | 0 | −1.58358 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.0840094 | 0 | −1.99294 | 1.00000 | 0 | 2.44733 | −0.335445 | 0 | 0.0840094 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.47133 | 0 | 0.164816 | 1.00000 | 0 | 4.11945 | −2.70016 | 0 | 1.47133 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.63619 | 0 | 0.677103 | 1.00000 | 0 | −2.69753 | −2.16451 | 0 | 1.63619 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70266 | 0 | 5.30439 | 1.00000 | 0 | −0.269551 | 8.93065 | 0 | 2.70266 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
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 | 9405.2.a.ba | 6 | |
| 3.b | odd | 2 | 1 | 3135.2.a.n | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3135.2.a.n | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9405.2.a.ba | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 15T_{2}^{3} + 13T_{2}^{2} - 25T_{2} + 2 \)
|
|
\( T_{7}^{6} - 5T_{7}^{5} - 8T_{7}^{4} + 53T_{7}^{3} + 10T_{7}^{2} - 120T_{7} - 32 \)
|
|
\( T_{13}^{6} - 7T_{13}^{5} + 3T_{13}^{4} + 64T_{13}^{3} - 115T_{13}^{2} + 44 \)
|
|
\( T_{17}^{6} - 11T_{17}^{5} + 19T_{17}^{4} + 96T_{17}^{3} - 115T_{17}^{2} - 432T_{17} - 256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 2 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots - 32 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 44 \)
|
| $17$ |
\( T^{6} - 11 T^{5} + \cdots - 256 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 12 T^{5} + \cdots + 776 \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots - 7232 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 1240 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots + 160 \)
|
| $41$ |
\( T^{6} - 7 T^{5} + \cdots - 1600 \)
|
| $43$ |
\( T^{6} - 14 T^{5} + \cdots + 1600 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 1024 \)
|
| $53$ |
\( T^{6} - 20 T^{5} + \cdots - 2852 \)
|
| $59$ |
\( T^{6} + 9 T^{5} + \cdots + 1916 \)
|
| $61$ |
\( T^{6} - 15 T^{5} + \cdots + 157376 \)
|
| $67$ |
\( T^{6} + 3 T^{5} + \cdots - 256 \)
|
| $71$ |
\( T^{6} - 20 T^{5} + \cdots + 10060 \)
|
| $73$ |
\( T^{6} + 14 T^{5} + \cdots + 2816 \)
|
| $79$ |
\( T^{6} - 3 T^{5} + \cdots - 704 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots - 55408 \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 12100 \)
|
| $97$ |
\( T^{6} + 14 T^{5} + \cdots + 54304 \)
|
| show more | |
| show less |