Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [163,3,Mod(162,163)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(163, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("163.162");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 163 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 163.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: | yes |
| Analytic conductor: | \(4.44142830907\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/163\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 162.1 |
|
0 | 0 | 4.00000 | 0 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 163.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-163}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 163.3.b.a | ✓ | 1 |
| 163.b | odd | 2 | 1 | CM | 163.3.b.a | ✓ | 1 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 163.3.b.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 163.3.b.a | ✓ | 1 | 163.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(163, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T \)
|
| $41$ |
\( T + 81 \)
|
| $43$ |
\( T + 77 \)
|
| $47$ |
\( T + 69 \)
|
| $53$ |
\( T + 57 \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T + 41 \)
|
| $67$ |
\( T \)
|
| $71$ |
\( T + 21 \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T \)
|
| $83$ |
\( T - 3 \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T - 31 \)
|
| show more | |
| show less |
Additional information
The Fourier coefficient $a_{n^2}$ for this newform is $n^2$ for all $n < 41$, and $a_p = 0$ for $p < 41$, yielding a striking beginning to the $q$-expansion.