Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3312,1,Mod(275,3312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 10, 6]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3312.275");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3312.bw (of order \(12\), 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: | \(1.65290332184\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\Q(\zeta_{36})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{6} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{36}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times\).
| \(n\) | \(415\) | \(2305\) | \(2485\) | \(2945\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\zeta_{36}^{9}\) | \(-\zeta_{36}^{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 275.1 |
|
−0.984808 | − | 0.173648i | 0.642788 | + | 0.766044i | 0.939693 | + | 0.342020i | 0 | −0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | −0.173648 | + | 0.984808i | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 275.2 | 0.342020 | + | 0.939693i | −0.984808 | + | 0.173648i | −0.766044 | + | 0.642788i | 0 | −0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | 0.939693 | − | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 275.3 | 0.642788 | − | 0.766044i | 0.342020 | − | 0.939693i | −0.173648 | − | 0.984808i | 0 | −0.500000 | − | 0.866025i | 0 | −0.866025 | − | 0.500000i | −0.766044 | − | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1379.1 | −0.642788 | − | 0.766044i | −0.342020 | − | 0.939693i | −0.173648 | + | 0.984808i | 0 | −0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | −0.766044 | + | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1379.2 | −0.342020 | + | 0.939693i | 0.984808 | + | 0.173648i | −0.766044 | − | 0.642788i | 0 | −0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | 0.939693 | + | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1379.3 | 0.984808 | − | 0.173648i | −0.642788 | + | 0.766044i | 0.939693 | − | 0.342020i | 0 | −0.500000 | + | 0.866025i | 0 | 0.866025 | − | 0.500000i | −0.173648 | − | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1931.1 | −0.642788 | + | 0.766044i | −0.342020 | + | 0.939693i | −0.173648 | − | 0.984808i | 0 | −0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | −0.766044 | − | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1931.2 | −0.342020 | − | 0.939693i | 0.984808 | − | 0.173648i | −0.766044 | + | 0.642788i | 0 | −0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | 0.939693 | − | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1931.3 | 0.984808 | + | 0.173648i | −0.642788 | − | 0.766044i | 0.939693 | + | 0.342020i | 0 | −0.500000 | − | 0.866025i | 0 | 0.866025 | + | 0.500000i | −0.173648 | + | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 3035.1 | −0.984808 | + | 0.173648i | 0.642788 | − | 0.766044i | 0.939693 | − | 0.342020i | 0 | −0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | −0.173648 | − | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 3035.2 | 0.342020 | − | 0.939693i | −0.984808 | − | 0.173648i | −0.766044 | − | 0.642788i | 0 | −0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | 0.939693 | + | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 3035.3 | 0.642788 | + | 0.766044i | 0.342020 | + | 0.939693i | −0.173648 | + | 0.984808i | 0 | −0.500000 | + | 0.866025i | 0 | −0.866025 | + | 0.500000i | −0.766044 | + | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 144.u | even | 12 | 1 | inner |
| 3312.bw | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3312.1.bw.a | ✓ | 12 |
| 9.d | odd | 6 | 1 | 3312.1.bw.b | yes | 12 | |
| 16.f | odd | 4 | 1 | 3312.1.bw.b | yes | 12 | |
| 23.b | odd | 2 | 1 | CM | 3312.1.bw.a | ✓ | 12 |
| 144.u | even | 12 | 1 | inner | 3312.1.bw.a | ✓ | 12 |
| 207.g | even | 6 | 1 | 3312.1.bw.b | yes | 12 | |
| 368.i | even | 4 | 1 | 3312.1.bw.b | yes | 12 | |
| 3312.bw | odd | 12 | 1 | inner | 3312.1.bw.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3312.1.bw.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 3312.1.bw.a | ✓ | 12 | 23.b | odd | 2 | 1 | CM |
| 3312.1.bw.a | ✓ | 12 | 144.u | even | 12 | 1 | inner |
| 3312.1.bw.a | ✓ | 12 | 3312.bw | odd | 12 | 1 | inner |
| 3312.1.bw.b | yes | 12 | 9.d | odd | 6 | 1 | |
| 3312.1.bw.b | yes | 12 | 16.f | odd | 4 | 1 | |
| 3312.1.bw.b | yes | 12 | 207.g | even | 6 | 1 | |
| 3312.1.bw.b | yes | 12 | 368.i | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{12} + 2 T_{13}^{9} - 9 T_{13}^{8} - 12 T_{13}^{7} + 2 T_{13}^{6} + 18 T_{13}^{5} + 69 T_{13}^{4} + \cdots + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{6} + 1 \)
|
| $3$ |
\( T^{12} - T^{6} + 1 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + 2 T^{9} + \cdots + 1 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( (T^{4} - T^{2} + 1)^{3} \)
|
| $29$ |
\( T^{12} + 2 T^{9} + \cdots + 1 \)
|
| $31$ |
\( T^{12} - 6 T^{10} + \cdots + 1 \)
|
| $37$ |
\( T^{12} \)
|
| $41$ |
\( T^{12} + 6 T^{10} + \cdots + 9 \)
|
| $43$ |
\( T^{12} \)
|
| $47$ |
\( (T^{6} + 3 T^{4} - 2 T^{3} + \cdots + 1)^{2} \)
|
| $53$ |
\( T^{12} \)
|
| $59$ |
\( (T^{4} - 4 T^{3} + 5 T^{2} + \cdots + 1)^{3} \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{2} \)
|
| $73$ |
\( (T^{6} + 6 T^{4} + 9 T^{2} + 1)^{2} \)
|
| $79$ |
\( T^{12} \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} \)
|
| $97$ |
\( T^{12} \)
|
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