Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1776,1,Mod(221,1776)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1776, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1776.221");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1776 = 2^{4} \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1776.z (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: | \(0.886339462436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(S_{4}\) |
| Projective field: | Galois closure of 4.2.227328.2 |
$q$-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 \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).
| \(n\) | \(223\) | \(593\) | \(1297\) | \(1333\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 221.1 |
|
− | 1.00000i | −0.707107 | + | 0.707107i | −1.00000 | 1.00000 | − | 1.00000i | 0.707107 | + | 0.707107i | − | 1.00000i | 1.00000i | − | 1.00000i | −1.00000 | − | 1.00000i | |||||||||||||||||||
| 221.2 | − | 1.00000i | 0.707107 | − | 0.707107i | −1.00000 | 1.00000 | − | 1.00000i | −0.707107 | − | 0.707107i | − | 1.00000i | 1.00000i | − | 1.00000i | −1.00000 | − | 1.00000i | ||||||||||||||||||||
| 1109.1 | 1.00000i | −0.707107 | − | 0.707107i | −1.00000 | 1.00000 | + | 1.00000i | 0.707107 | − | 0.707107i | 1.00000i | − | 1.00000i | 1.00000i | −1.00000 | + | 1.00000i | ||||||||||||||||||||||
| 1109.2 | 1.00000i | 0.707107 | + | 0.707107i | −1.00000 | 1.00000 | + | 1.00000i | −0.707107 | + | 0.707107i | 1.00000i | − | 1.00000i | 1.00000i | −1.00000 | + | 1.00000i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
| 111.d | odd | 2 | 1 | inner |
| 1776.z | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1776.1.z.b | yes | 4 |
| 3.b | odd | 2 | 1 | 1776.1.z.a | ✓ | 4 | |
| 16.e | even | 4 | 1 | inner | 1776.1.z.b | yes | 4 |
| 37.b | even | 2 | 1 | 1776.1.z.a | ✓ | 4 | |
| 48.i | odd | 4 | 1 | 1776.1.z.a | ✓ | 4 | |
| 111.d | odd | 2 | 1 | inner | 1776.1.z.b | yes | 4 |
| 592.n | even | 4 | 1 | 1776.1.z.a | ✓ | 4 | |
| 1776.z | odd | 4 | 1 | inner | 1776.1.z.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1776.1.z.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1776.1.z.a | ✓ | 4 | 37.b | even | 2 | 1 | |
| 1776.1.z.a | ✓ | 4 | 48.i | odd | 4 | 1 | |
| 1776.1.z.a | ✓ | 4 | 592.n | even | 4 | 1 | |
| 1776.1.z.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1776.1.z.b | yes | 4 | 16.e | even | 4 | 1 | inner |
| 1776.1.z.b | yes | 4 | 111.d | odd | 2 | 1 | inner |
| 1776.1.z.b | yes | 4 | 1776.z | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(1776, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} + 1 \)
|
| $5$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( T^{4} + 1 \)
|
| $13$ |
\( T^{4} + 1 \)
|
| $17$ |
\( (T + 1)^{4} \)
|
| $19$ |
\( T^{4} + 1 \)
|
| $23$ |
\( (T^{2} + 1)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 2)^{2} \)
|
| $37$ |
\( T^{4} + 1 \)
|
| $41$ |
\( (T^{2} - 2)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 2)^{2} \)
|
| $53$ |
\( T^{4} + 1 \)
|
| $59$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( (T^{2} - 2)^{2} \)
|
| $73$ |
\( (T^{2} + 1)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + 1 \)
|
| $89$ |
\( (T^{2} + 1)^{2} \)
|
| $97$ |
\( T^{4} \)
|
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