Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,1,Mod(6,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 18]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.6");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 161.l (of order \(22\), degree \(10\), 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: | \(0.0803494670339\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\Q(\zeta_{22})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{11}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).
| \(n\) | \(24\) | \(120\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{22}^{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−0.544078 | + | 1.19136i | 0 | −0.468468 | − | 0.540641i | 0 | 0 | −0.142315 | + | 0.989821i | −0.357685 | + | 0.105026i | 0.841254 | + | 0.540641i | 0 | ||||||||||||||||||||||||||||||||||||||
| 13.1 | 0.186393 | + | 0.215109i | 0 | 0.130785 | − | 0.909632i | 0 | 0 | −0.959493 | − | 0.281733i | 0.459493 | − | 0.295298i | 0.415415 | + | 0.909632i | 0 | |||||||||||||||||||||||||||||||||||||||
| 27.1 | −0.544078 | − | 1.19136i | 0 | −0.468468 | + | 0.540641i | 0 | 0 | −0.142315 | − | 0.989821i | −0.357685 | − | 0.105026i | 0.841254 | − | 0.540641i | 0 | |||||||||||||||||||||||||||||||||||||||
| 41.1 | −1.61435 | − | 0.474017i | 0 | 1.54019 | + | 0.989821i | 0 | 0 | 0.415415 | + | 0.909632i | −0.915415 | − | 1.05645i | −0.142315 | − | 0.989821i | 0 | |||||||||||||||||||||||||||||||||||||||
| 48.1 | 0.698939 | + | 0.449181i | 0 | −0.128663 | − | 0.281733i | 0 | 0 | −0.654861 | + | 0.755750i | 0.154861 | − | 1.07708i | −0.959493 | + | 0.281733i | 0 | |||||||||||||||||||||||||||||||||||||||
| 55.1 | −1.61435 | + | 0.474017i | 0 | 1.54019 | − | 0.989821i | 0 | 0 | 0.415415 | − | 0.909632i | −0.915415 | + | 1.05645i | −0.142315 | + | 0.989821i | 0 | |||||||||||||||||||||||||||||||||||||||
| 62.1 | 0.186393 | − | 0.215109i | 0 | 0.130785 | + | 0.909632i | 0 | 0 | −0.959493 | + | 0.281733i | 0.459493 | + | 0.295298i | 0.415415 | − | 0.909632i | 0 | |||||||||||||||||||||||||||||||||||||||
| 104.1 | 0.698939 | − | 0.449181i | 0 | −0.128663 | + | 0.281733i | 0 | 0 | −0.654861 | − | 0.755750i | 0.154861 | + | 1.07708i | −0.959493 | − | 0.281733i | 0 | |||||||||||||||||||||||||||||||||||||||
| 118.1 | 0.273100 | + | 1.89945i | 0 | −2.57385 | + | 0.755750i | 0 | 0 | 0.841254 | − | 0.540641i | −1.34125 | − | 2.93694i | −0.654861 | − | 0.755750i | 0 | |||||||||||||||||||||||||||||||||||||||
| 146.1 | 0.273100 | − | 1.89945i | 0 | −2.57385 | − | 0.755750i | 0 | 0 | 0.841254 | + | 0.540641i | −1.34125 | + | 2.93694i | −0.654861 | + | 0.755750i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 23.c | even | 11 | 1 | inner |
| 161.l | odd | 22 | 1 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(161, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots + 1 \)
|
| $11$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} \)
|
| $19$ |
\( T^{10} \)
|
| $23$ |
\( T^{10} + T^{9} + \cdots + 1 \)
|
| $29$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $31$ |
\( T^{10} \)
|
| $37$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $41$ |
\( T^{10} \)
|
| $43$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $47$ |
\( T^{10} \)
|
| $53$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $59$ |
\( T^{10} \)
|
| $61$ |
\( T^{10} \)
|
| $67$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $71$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $73$ |
\( T^{10} \)
|
| $79$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $83$ |
\( T^{10} \)
|
| $89$ |
\( T^{10} \)
|
| $97$ |
\( T^{10} \)
|
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