Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3703,1,Mod(118,3703)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3703, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 16]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3703.118");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3703 = 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3703.l (of order \(22\), degree \(10\), 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: | \(1.84803774178\) |
| 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: | no (minimal twist has level 161) |
| 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 \(\left(\mathbb{Z}/3703\mathbb{Z}\right)^\times\).
| \(n\) | \(2117\) | \(3179\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{22}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 118.1 |
|
−0.118239 | − | 0.822373i | 0 | 0.297176 | − | 0.0872586i | 0 | 0 | 0.841254 | − | 0.540641i | −0.452036 | − | 0.989821i | −0.654861 | − | 0.755750i | 0 | ||||||||||||||||||||||||||||||||||||||
| 699.1 | 1.25667 | − | 0.368991i | 0 | 0.601808 | − | 0.386758i | 0 | 0 | 0.415415 | − | 0.909632i | −0.244123 | + | 0.281733i | −0.142315 | + | 0.989821i | 0 | |||||||||||||||||||||||||||||||||||||||
| 706.1 | −1.10181 | + | 1.27155i | 0 | −0.260554 | − | 1.81219i | 0 | 0 | −0.959493 | + | 0.281733i | 1.17597 | + | 0.755750i | 0.415415 | − | 0.909632i | 0 | |||||||||||||||||||||||||||||||||||||||
| 1392.1 | −0.239446 | + | 0.153882i | 0 | −0.381761 | + | 0.835939i | 0 | 0 | −0.654861 | − | 0.755750i | −0.0777324 | − | 0.540641i | −0.959493 | − | 0.281733i | 0 | |||||||||||||||||||||||||||||||||||||||
| 2582.1 | −0.797176 | + | 1.74557i | 0 | −1.75667 | − | 2.02730i | 0 | 0 | −0.142315 | + | 0.989821i | 3.09792 | − | 0.909632i | 0.841254 | + | 0.540641i | 0 | |||||||||||||||||||||||||||||||||||||||
| 2603.1 | −0.797176 | − | 1.74557i | 0 | −1.75667 | + | 2.02730i | 0 | 0 | −0.142315 | − | 0.989821i | 3.09792 | + | 0.909632i | 0.841254 | − | 0.540641i | 0 | |||||||||||||||||||||||||||||||||||||||
| 2617.1 | 1.25667 | + | 0.368991i | 0 | 0.601808 | + | 0.386758i | 0 | 0 | 0.415415 | + | 0.909632i | −0.244123 | − | 0.281733i | −0.142315 | − | 0.989821i | 0 | |||||||||||||||||||||||||||||||||||||||
| 2911.1 | −1.10181 | − | 1.27155i | 0 | −0.260554 | + | 1.81219i | 0 | 0 | −0.959493 | − | 0.281733i | 1.17597 | − | 0.755750i | 0.415415 | + | 0.909632i | 0 | |||||||||||||||||||||||||||||||||||||||
| 3044.1 | −0.118239 | + | 0.822373i | 0 | 0.297176 | + | 0.0872586i | 0 | 0 | 0.841254 | + | 0.540641i | −0.452036 | + | 0.989821i | −0.654861 | + | 0.755750i | 0 | |||||||||||||||||||||||||||||||||||||||
| 3429.1 | −0.239446 | − | 0.153882i | 0 | −0.381761 | − | 0.835939i | 0 | 0 | −0.654861 | + | 0.755750i | −0.0777324 | + | 0.540641i | −0.959493 | + | 0.281733i | 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 can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3703, [\chi])\):
|
\( T_{2}^{10} + 2T_{2}^{9} + 4T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} - 12T_{2}^{5} + 9T_{2}^{4} + 7T_{2}^{3} + 14T_{2}^{2} + 6T_{2} + 1 \)
|
|
\( T_{11}^{10} - 9 T_{11}^{9} + 37 T_{11}^{8} - 91 T_{11}^{7} + 148 T_{11}^{6} - 166 T_{11}^{5} + 130 T_{11}^{4} + \cdots + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots + 1 \)
|
| $11$ |
\( T^{10} - 9 T^{9} + \cdots + 1 \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} \)
|
| $19$ |
\( T^{10} \)
|
| $23$ |
\( T^{10} \)
|
| $29$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $31$ |
\( T^{10} \)
|
| $37$ |
\( T^{10} - 9 T^{9} + \cdots + 1 \)
|
| $41$ |
\( T^{10} \)
|
| $43$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $47$ |
\( T^{10} \)
|
| $53$ |
\( T^{10} - 9 T^{9} + \cdots + 1 \)
|
| $59$ |
\( T^{10} \)
|
| $61$ |
\( T^{10} \)
|
| $67$ |
\( T^{10} + 2 T^{9} + \cdots + 1 \)
|
| $71$ |
\( T^{10} - 9 T^{9} + \cdots + 1 \)
|
| $73$ |
\( T^{10} \)
|
| $79$ |
\( T^{10} - 9 T^{9} + \cdots + 1 \)
|
| $83$ |
\( T^{10} \)
|
| $89$ |
\( T^{10} \)
|
| $97$ |
\( T^{10} \)
|
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