Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3968,1,Mod(433,3968)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3968, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5, 4]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3968.433");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3968 = 2^{7} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3968.y (of order \(8\), degree \(4\), not 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: | \(1.98028997013\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 992) |
| Projective image: | \(D_{24}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/3968\mathbb{Z}\right)^\times\).
| \(n\) | \(2047\) | \(2049\) | \(3845\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\zeta_{24}^{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 433.1 |
|
0 | 0 | 0 | −0.758819 | + | 1.83195i | 0 | −1.22474 | − | 1.22474i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||||||||||||||||||||||||||||
| 433.2 | 0 | 0 | 0 | 0.465926 | − | 1.12484i | 0 | 1.22474 | + | 1.22474i | 0 | 0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 1425.1 | 0 | 0 | 0 | −1.46593 | + | 0.607206i | 0 | −1.22474 | + | 1.22474i | 0 | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 1425.2 | 0 | 0 | 0 | −0.241181 | + | 0.0999004i | 0 | 1.22474 | − | 1.22474i | 0 | −0.707107 | − | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 2417.1 | 0 | 0 | 0 | −1.46593 | − | 0.607206i | 0 | −1.22474 | − | 1.22474i | 0 | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 2417.2 | 0 | 0 | 0 | −0.241181 | − | 0.0999004i | 0 | 1.22474 | + | 1.22474i | 0 | −0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 3409.1 | 0 | 0 | 0 | −0.758819 | − | 1.83195i | 0 | −1.22474 | + | 1.22474i | 0 | 0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
| 3409.2 | 0 | 0 | 0 | 0.465926 | + | 1.12484i | 0 | 1.22474 | − | 1.22474i | 0 | 0.707107 | + | 0.707107i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
| 32.g | even | 8 | 1 | inner |
| 992.y | odd | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3968.1.y.b | 8 | |
| 4.b | odd | 2 | 1 | 992.1.y.b | ✓ | 8 | |
| 31.b | odd | 2 | 1 | CM | 3968.1.y.b | 8 | |
| 32.g | even | 8 | 1 | inner | 3968.1.y.b | 8 | |
| 32.h | odd | 8 | 1 | 992.1.y.b | ✓ | 8 | |
| 124.d | even | 2 | 1 | 992.1.y.b | ✓ | 8 | |
| 992.w | even | 8 | 1 | 992.1.y.b | ✓ | 8 | |
| 992.y | odd | 8 | 1 | inner | 3968.1.y.b | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 992.1.y.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 992.1.y.b | ✓ | 8 | 32.h | odd | 8 | 1 | |
| 992.1.y.b | ✓ | 8 | 124.d | even | 2 | 1 | |
| 992.1.y.b | ✓ | 8 | 992.w | even | 8 | 1 | |
| 3968.1.y.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 3968.1.y.b | 8 | 31.b | odd | 2 | 1 | CM | |
| 3968.1.y.b | 8 | 32.g | even | 8 | 1 | inner | |
| 3968.1.y.b | 8 | 992.y | odd | 8 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 4T_{5}^{7} + 10T_{5}^{6} + 16T_{5}^{5} + 18T_{5}^{4} + 20T_{5}^{3} + 22T_{5}^{2} + 8T_{5} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(3968, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} + 9)^{2} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} - 2 T^{6} + \cdots + 1 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( (T^{4} + 9)^{2} \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( (T^{2} + 2)^{4} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 1 \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \)
|
| $71$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( (T^{4} - 4 T^{2} + 1)^{2} \)
|
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