Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3248,1,Mod(27,3248)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3248, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([14, 7, 14, 15]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3248.27");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3248 = 2^{4} \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3248.dt (of order \(28\), degree \(12\), 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: | \(1.62096316103\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{28})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{28}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{28} + \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/3248\mathbb{Z}\right)^\times\).
| \(n\) | \(465\) | \(785\) | \(2031\) | \(2437\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{28}^{9}\) | \(-1\) | \(\zeta_{28}^{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 27.1 |
|
−0.623490 | + | 0.781831i | 0 | −0.222521 | − | 0.974928i | 0 | 0 | −0.222521 | − | 0.974928i | 0.900969 | + | 0.433884i | 0.900969 | − | 0.433884i | 0 | ||||||||||||||||||||||||||||||||||||||||||||
| 195.1 | 0.900969 | + | 0.433884i | 0 | 0.623490 | + | 0.781831i | 0 | 0 | 0.623490 | + | 0.781831i | 0.222521 | + | 0.974928i | 0.222521 | − | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 363.1 | −0.623490 | − | 0.781831i | 0 | −0.222521 | + | 0.974928i | 0 | 0 | −0.222521 | + | 0.974928i | 0.900969 | − | 0.433884i | 0.900969 | + | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1203.1 | −0.623490 | − | 0.781831i | 0 | −0.222521 | + | 0.974928i | 0 | 0 | −0.222521 | + | 0.974928i | 0.900969 | − | 0.433884i | 0.900969 | + | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1371.1 | 0.900969 | + | 0.433884i | 0 | 0.623490 | + | 0.781831i | 0 | 0 | 0.623490 | + | 0.781831i | 0.222521 | + | 0.974928i | 0.222521 | − | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1539.1 | −0.623490 | + | 0.781831i | 0 | −0.222521 | − | 0.974928i | 0 | 0 | −0.222521 | − | 0.974928i | 0.900969 | + | 0.433884i | 0.900969 | − | 0.433884i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1875.1 | 0.222521 | + | 0.974928i | 0 | −0.900969 | + | 0.433884i | 0 | 0 | −0.900969 | + | 0.433884i | −0.623490 | − | 0.781831i | −0.623490 | + | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 2099.1 | 0.900969 | − | 0.433884i | 0 | 0.623490 | − | 0.781831i | 0 | 0 | 0.623490 | − | 0.781831i | 0.222521 | − | 0.974928i | 0.222521 | + | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 2323.1 | 0.222521 | − | 0.974928i | 0 | −0.900969 | − | 0.433884i | 0 | 0 | −0.900969 | − | 0.433884i | −0.623490 | + | 0.781831i | −0.623490 | − | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 2491.1 | 0.222521 | − | 0.974928i | 0 | −0.900969 | − | 0.433884i | 0 | 0 | −0.900969 | − | 0.433884i | −0.623490 | + | 0.781831i | −0.623490 | − | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 2715.1 | 0.900969 | − | 0.433884i | 0 | 0.623490 | − | 0.781831i | 0 | 0 | 0.623490 | − | 0.781831i | 0.222521 | − | 0.974928i | 0.222521 | + | 0.974928i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 2939.1 | 0.222521 | + | 0.974928i | 0 | −0.900969 | + | 0.433884i | 0 | 0 | −0.900969 | + | 0.433884i | −0.623490 | − | 0.781831i | −0.623490 | + | 0.781831i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 464.bm | even | 28 | 1 | inner |
| 3248.dt | odd | 28 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3248.1.dt.b | ✓ | 12 |
| 7.b | odd | 2 | 1 | CM | 3248.1.dt.b | ✓ | 12 |
| 16.f | odd | 4 | 1 | 3248.1.ep.b | yes | 12 | |
| 29.f | odd | 28 | 1 | 3248.1.ep.b | yes | 12 | |
| 112.j | even | 4 | 1 | 3248.1.ep.b | yes | 12 | |
| 203.r | even | 28 | 1 | 3248.1.ep.b | yes | 12 | |
| 464.bm | even | 28 | 1 | inner | 3248.1.dt.b | ✓ | 12 |
| 3248.dt | odd | 28 | 1 | inner | 3248.1.dt.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3248.1.dt.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 3248.1.dt.b | ✓ | 12 | 7.b | odd | 2 | 1 | CM |
| 3248.1.dt.b | ✓ | 12 | 464.bm | even | 28 | 1 | inner |
| 3248.1.dt.b | ✓ | 12 | 3248.dt | odd | 28 | 1 | inner |
| 3248.1.ep.b | yes | 12 | 16.f | odd | 4 | 1 | |
| 3248.1.ep.b | yes | 12 | 29.f | odd | 28 | 1 | |
| 3248.1.ep.b | yes | 12 | 112.j | even | 4 | 1 | |
| 3248.1.ep.b | yes | 12 | 203.r | even | 28 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{12} + 7T_{11}^{10} + 21T_{11}^{8} + 35T_{11}^{6} + 49T_{11}^{4} - 49T_{11}^{2} + 49 \)
acting on \(S_{1}^{\mathrm{new}}(3248, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{12} + 7 T^{10} + \cdots + 49 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} + 14 T^{8} + \cdots + 49 \)
|
| $29$ |
\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
|
| $31$ |
\( T^{12} \)
|
| $37$ |
\( T^{12} + 14 T^{8} + \cdots + 49 \)
|
| $41$ |
\( T^{12} \)
|
| $43$ |
\( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 64)^{2} \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $59$ |
\( T^{12} \)
|
| $61$ |
\( T^{12} \)
|
| $67$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $71$ |
\( T^{12} - 4 T^{10} + \cdots + 1 \)
|
| $73$ |
\( T^{12} \)
|
| $79$ |
\( T^{12} + 12 T^{11} + \cdots + 1 \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} \)
|
| $97$ |
\( T^{12} \)
|
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