Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2672,1,Mod(333,2672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2672, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2672.333");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2672 = 2^{4} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2672.i (of order \(4\), degree \(2\), 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: | \(1.33350171376\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{44})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{44}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/2672\mathbb{Z}\right)^\times\).
| \(n\) | \(335\) | \(673\) | \(2005\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\zeta_{44}^{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 333.1 |
|
−0.989821 | + | 0.142315i | −0.300613 | − | 0.300613i | 0.959493 | − | 0.281733i | 0 | 0.340335 | + | 0.254771i | 1.51150i | −0.909632 | + | 0.415415i | − | 0.819264i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.2 | −0.909632 | − | 0.415415i | −0.847507 | − | 0.847507i | 0.654861 | + | 0.755750i | 0 | 0.418852 | + | 1.12299i | − | 1.08128i | −0.281733 | − | 0.959493i | 0.436535i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.3 | −0.755750 | + | 0.654861i | 1.24123 | + | 1.24123i | 0.142315 | − | 0.989821i | 0 | −1.75089 | − | 0.125226i | − | 1.81926i | 0.540641 | + | 0.841254i | 2.08128i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.4 | −0.540641 | − | 0.841254i | 1.41061 | + | 1.41061i | −0.415415 | + | 0.909632i | 0 | 0.424047 | − | 1.94931i | 0.563465i | 0.989821 | − | 0.142315i | 2.97964i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.5 | −0.281733 | + | 0.959493i | −1.32505 | − | 1.32505i | −0.841254 | − | 0.540641i | 0 | 1.64468 | − | 0.898064i | 1.97964i | 0.755750 | − | 0.654861i | 2.51150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.6 | 0.281733 | + | 0.959493i | 0.494217 | + | 0.494217i | −0.841254 | + | 0.540641i | 0 | −0.334961 | + | 0.613435i | − | 1.97964i | −0.755750 | − | 0.654861i | − | 0.511499i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.7 | 0.540641 | − | 0.841254i | −0.100889 | − | 0.100889i | −0.415415 | − | 0.909632i | 0 | −0.139418 | + | 0.0303285i | − | 0.563465i | −0.989821 | − | 0.142315i | − | 0.979643i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.8 | 0.755750 | + | 0.654861i | 0.677760 | + | 0.677760i | 0.142315 | + | 0.989821i | 0 | 0.0683785 | + | 0.956056i | 1.81926i | −0.540641 | + | 0.841254i | − | 0.0812816i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.9 | 0.909632 | − | 0.415415i | 1.13214 | + | 1.13214i | 0.654861 | − | 0.755750i | 0 | 1.50013 | + | 0.559521i | 1.08128i | 0.281733 | − | 0.959493i | 1.56347i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 333.10 | 0.989821 | + | 0.142315i | −1.38189 | − | 1.38189i | 0.959493 | + | 0.281733i | 0 | −1.17116 | − | 1.56449i | − | 1.51150i | 0.909632 | + | 0.415415i | 2.81926i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.1 | −0.989821 | − | 0.142315i | −0.300613 | + | 0.300613i | 0.959493 | + | 0.281733i | 0 | 0.340335 | − | 0.254771i | − | 1.51150i | −0.909632 | − | 0.415415i | 0.819264i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.2 | −0.909632 | + | 0.415415i | −0.847507 | + | 0.847507i | 0.654861 | − | 0.755750i | 0 | 0.418852 | − | 1.12299i | 1.08128i | −0.281733 | + | 0.959493i | − | 0.436535i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.3 | −0.755750 | − | 0.654861i | 1.24123 | − | 1.24123i | 0.142315 | + | 0.989821i | 0 | −1.75089 | + | 0.125226i | 1.81926i | 0.540641 | − | 0.841254i | − | 2.08128i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.4 | −0.540641 | + | 0.841254i | 1.41061 | − | 1.41061i | −0.415415 | − | 0.909632i | 0 | 0.424047 | + | 1.94931i | − | 0.563465i | 0.989821 | + | 0.142315i | − | 2.97964i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.5 | −0.281733 | − | 0.959493i | −1.32505 | + | 1.32505i | −0.841254 | + | 0.540641i | 0 | 1.64468 | + | 0.898064i | − | 1.97964i | 0.755750 | + | 0.654861i | − | 2.51150i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.6 | 0.281733 | − | 0.959493i | 0.494217 | − | 0.494217i | −0.841254 | − | 0.540641i | 0 | −0.334961 | − | 0.613435i | 1.97964i | −0.755750 | + | 0.654861i | 0.511499i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.7 | 0.540641 | + | 0.841254i | −0.100889 | + | 0.100889i | −0.415415 | + | 0.909632i | 0 | −0.139418 | − | 0.0303285i | 0.563465i | −0.989821 | + | 0.142315i | 0.979643i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.8 | 0.755750 | − | 0.654861i | 0.677760 | − | 0.677760i | 0.142315 | − | 0.989821i | 0 | 0.0683785 | − | 0.956056i | − | 1.81926i | −0.540641 | − | 0.841254i | 0.0812816i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.9 | 0.909632 | + | 0.415415i | 1.13214 | − | 1.13214i | 0.654861 | + | 0.755750i | 0 | 1.50013 | − | 0.559521i | − | 1.08128i | 0.281733 | + | 0.959493i | − | 1.56347i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1669.10 | 0.989821 | − | 0.142315i | −1.38189 | + | 1.38189i | 0.959493 | − | 0.281733i | 0 | −1.17116 | + | 1.56449i | 1.51150i | 0.909632 | − | 0.415415i | − | 2.81926i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 167.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-167}) \) |
| 16.e | even | 4 | 1 | inner |
| 2672.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2672.1.i.b | ✓ | 20 |
| 16.e | even | 4 | 1 | inner | 2672.1.i.b | ✓ | 20 |
| 167.b | odd | 2 | 1 | CM | 2672.1.i.b | ✓ | 20 |
| 2672.i | odd | 4 | 1 | inner | 2672.1.i.b | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2672.1.i.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 2672.1.i.b | ✓ | 20 | 16.e | even | 4 | 1 | inner |
| 2672.1.i.b | ✓ | 20 | 167.b | odd | 2 | 1 | CM |
| 2672.1.i.b | ✓ | 20 | 2672.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 2 T_{3}^{19} + 2 T_{3}^{18} + 29 T_{3}^{16} - 58 T_{3}^{15} + 58 T_{3}^{14} + 236 T_{3}^{12} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2672, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1 \)
|
| $3$ |
\( T^{20} - 2 T^{19} + \cdots + 1 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( (T^{10} + 11 T^{8} + \cdots + 11)^{2} \)
|
| $11$ |
\( T^{20} - 2 T^{19} + \cdots + 1 \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} \)
|
| $19$ |
\( T^{20} + 2 T^{19} + \cdots + 1 \)
|
| $23$ |
\( T^{20} \)
|
| $29$ |
\( T^{20} + 2 T^{19} + \cdots + 1 \)
|
| $31$ |
\( (T^{10} - 11 T^{8} + \cdots - 11)^{2} \)
|
| $37$ |
\( T^{20} \)
|
| $41$ |
\( T^{20} \)
|
| $43$ |
\( T^{20} \)
|
| $47$ |
\( (T^{10} - 11 T^{8} + \cdots - 11)^{2} \)
|
| $53$ |
\( T^{20} \)
|
| $59$ |
\( T^{20} \)
|
| $61$ |
\( T^{20} + 2 T^{19} + \cdots + 1 \)
|
| $67$ |
\( T^{20} \)
|
| $71$ |
\( T^{20} \)
|
| $73$ |
\( T^{20} \)
|
| $79$ |
\( T^{20} \)
|
| $83$ |
\( T^{20} \)
|
| $89$ |
\( (T^{10} + 11 T^{8} + \cdots + 11)^{2} \)
|
| $97$ |
\( (T^{10} - 11 T^{8} + \cdots - 11)^{2} \)
|
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