Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8700,2,Mod(349,8700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8700, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8700.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8700.g (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(69.4698497585\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.4227136.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 9x^{4} + 22x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 5\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{4} - 6\beta_{3} + 17\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8700\mathbb{Z}\right)^\times\).
| \(n\) | \(901\) | \(4177\) | \(4351\) | \(5801\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
0 | − | 1.00000i | 0 | 0 | 0 | − | 2.91223i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 349.2 | 0 | − | 1.00000i | 0 | 0 | 0 | − | 0.286462i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 349.3 | 0 | − | 1.00000i | 0 | 0 | 0 | 1.19869i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 349.4 | 0 | 1.00000i | 0 | 0 | 0 | − | 1.19869i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 349.5 | 0 | 1.00000i | 0 | 0 | 0 | 0.286462i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 349.6 | 0 | 1.00000i | 0 | 0 | 0 | 2.91223i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8700.2.g.u | 6 | |
| 5.b | even | 2 | 1 | inner | 8700.2.g.u | 6 | |
| 5.c | odd | 4 | 1 | 8700.2.a.bb | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 8700.2.a.bc | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8700.2.a.bb | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 8700.2.a.bc | yes | 3 | 5.c | odd | 4 | 1 | |
| 8700.2.g.u | 6 | 1.a | even | 1 | 1 | trivial | |
| 8700.2.g.u | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8700, [\chi])\):
|
\( T_{7}^{6} + 10T_{7}^{4} + 13T_{7}^{2} + 1 \)
|
|
\( T_{11}^{3} - 5T_{11} - 3 \)
|
|
\( T_{13}^{6} + 22T_{13}^{4} + 81T_{13}^{2} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 10 T^{4} + \cdots + 1 \)
|
| $11$ |
\( (T^{3} - 5 T - 3)^{2} \)
|
| $13$ |
\( T^{6} + 22 T^{4} + \cdots + 81 \)
|
| $17$ |
\( T^{6} + 50 T^{4} + \cdots + 441 \)
|
| $19$ |
\( (T - 4)^{6} \)
|
| $23$ |
\( T^{6} + 68 T^{4} + \cdots + 576 \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 8 T + 56)^{2} \)
|
| $37$ |
\( T^{6} + 184 T^{4} + \cdots + 153664 \)
|
| $41$ |
\( (T^{3} + 11 T^{2} + \cdots - 75)^{2} \)
|
| $43$ |
\( T^{6} + 56 T^{4} + \cdots + 1600 \)
|
| $47$ |
\( T^{6} + 282 T^{4} + \cdots + 335241 \)
|
| $53$ |
\( T^{6} + 272 T^{4} + \cdots + 36864 \)
|
| $59$ |
\( (T^{3} - 14 T^{2} + \cdots + 120)^{2} \)
|
| $61$ |
\( (T^{3} - 6 T^{2} + \cdots + 344)^{2} \)
|
| $67$ |
\( T^{6} + 254 T^{4} + \cdots + 390625 \)
|
| $71$ |
\( (T^{3} + 2 T^{2} - 40 T + 72)^{2} \)
|
| $73$ |
\( T^{6} + 404 T^{4} + \cdots + 2383936 \)
|
| $79$ |
\( (T^{3} - 28 T^{2} + \cdots - 328)^{2} \)
|
| $83$ |
\( T^{6} + 396 T^{4} + \cdots + 2143296 \)
|
| $89$ |
\( (T^{3} - 8 T^{2} + \cdots + 225)^{2} \)
|
| $97$ |
\( T^{6} + 196 T^{4} + \cdots + 129600 \)
|
| show more | |
| show less |