Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,5,Mod(197,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.197");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.b (of order \(2\), degree \(1\), 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: | \(91.1723074400\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-29})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 28x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 28x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 13\nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 172\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 56 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 56 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 13\beta_{2} + 172\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 21.5407i | 0 | 0 | 22.6274i | 0 | −60.9262 | ||||||||||||||||||||||||||||
| 197.2 | − | 2.82843i | 0 | −8.00000 | 21.5407i | 0 | 0 | 22.6274i | 0 | 60.9262 | ||||||||||||||||||||||||||||||
| 197.3 | 2.82843i | 0 | −8.00000 | − | 21.5407i | 0 | 0 | − | 22.6274i | 0 | 60.9262 | |||||||||||||||||||||||||||||
| 197.4 | 2.82843i | 0 | −8.00000 | 21.5407i | 0 | 0 | − | 22.6274i | 0 | −60.9262 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 882.5.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 882.5.b.b | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 882.5.b.b | ✓ | 4 |
| 21.c | even | 2 | 1 | inner | 882.5.b.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 882.5.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 882.5.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 882.5.b.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 882.5.b.b | ✓ | 4 | 21.c | 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):
|
\( T_{5}^{2} + 464 \)
|
|
\( T_{13}^{2} - 92800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 464)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 3362)^{2} \)
|
| $13$ |
\( (T^{2} - 92800)^{2} \)
|
| $17$ |
\( (T^{2} + 104400)^{2} \)
|
| $19$ |
\( (T^{2} - 23200)^{2} \)
|
| $23$ |
\( (T^{2} + 290322)^{2} \)
|
| $29$ |
\( (T^{2} + 6962)^{2} \)
|
| $31$ |
\( (T^{2} - 580000)^{2} \)
|
| $37$ |
\( (T + 600)^{4} \)
|
| $41$ |
\( (T^{2} + 2610000)^{2} \)
|
| $43$ |
\( (T + 1160)^{4} \)
|
| $47$ |
\( (T^{2} + 1160000)^{2} \)
|
| $53$ |
\( (T^{2} + 1219922)^{2} \)
|
| $59$ |
\( (T^{2} + 13409600)^{2} \)
|
| $61$ |
\( (T^{2} - 6704800)^{2} \)
|
| $67$ |
\( (T - 8278)^{4} \)
|
| $71$ |
\( (T^{2} + 3385202)^{2} \)
|
| $73$ |
\( (T^{2} - 86327200)^{2} \)
|
| $79$ |
\( (T - 3082)^{4} \)
|
| $83$ |
\( (T^{2} + 184161600)^{2} \)
|
| $89$ |
\( (T^{2} + 4187600)^{2} \)
|
| $97$ |
\( (T^{2} - 580000)^{2} \)
|
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