Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1224,1,Mod(53,1224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 4, 4, 7]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1224.53");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1224.bu (of order \(8\), degree \(4\), 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: | \(0.610855575463\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{8}\) |
| Projective field: | Galois closure of 8.0.153158089019904.1 |
$q$-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 \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).
| \(n\) | \(137\) | \(613\) | \(649\) | \(919\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{8}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
0.707107 | − | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 0.292893 | − | 0.707107i | −0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||
| 485.1 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 0 | 0 | 0.292893 | + | 0.707107i | −0.707107 | + | 0.707107i | 0 | 0 | |||||||||||||||||||||||||
| 773.1 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 0 | 0 | 1.70711 | − | 0.707107i | 0.707107 | − | 0.707107i | 0 | 0 | |||||||||||||||||||||||||
| 1205.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0 | 0 | 1.70711 | + | 0.707107i | 0.707107 | + | 0.707107i | 0 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | RM by \(\Q(\sqrt{2}) \) |
| 51.g | odd | 8 | 1 | inner |
| 408.be | odd | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1224.1.bu.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1224.1.bu.b | yes | 4 | |
| 8.b | even | 2 | 1 | RM | 1224.1.bu.a | ✓ | 4 |
| 17.d | even | 8 | 1 | 1224.1.bu.b | yes | 4 | |
| 24.h | odd | 2 | 1 | 1224.1.bu.b | yes | 4 | |
| 51.g | odd | 8 | 1 | inner | 1224.1.bu.a | ✓ | 4 |
| 136.o | even | 8 | 1 | 1224.1.bu.b | yes | 4 | |
| 408.be | odd | 8 | 1 | inner | 1224.1.bu.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1224.1.bu.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1224.1.bu.a | ✓ | 4 | 8.b | even | 2 | 1 | RM |
| 1224.1.bu.a | ✓ | 4 | 51.g | odd | 8 | 1 | inner |
| 1224.1.bu.a | ✓ | 4 | 408.be | odd | 8 | 1 | inner |
| 1224.1.bu.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 1224.1.bu.b | yes | 4 | 17.d | even | 8 | 1 | |
| 1224.1.bu.b | yes | 4 | 24.h | odd | 2 | 1 | |
| 1224.1.bu.b | yes | 4 | 136.o | even | 8 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{4} + 2T_{23}^{2} - 4T_{23} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(1224, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 1)^{2} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} - 2)^{2} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $73$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $79$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( (T^{2} - 2)^{2} \)
|
| $97$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
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