Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,4,Mod(3,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.i (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: | \(9.67631324094\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{8}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(129\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−2.00000 | + | 2.00000i | 0 | − | 8.00000i | 2.82843 | + | 2.82843i | 0 | 0 | 16.0000 | + | 16.0000i | −19.0919 | − | 19.0919i | −11.3137 | |||||||||||||||||||||
| 27.1 | −2.00000 | − | 2.00000i | 0 | 8.00000i | −2.82843 | + | 2.82843i | 0 | 0 | 16.0000 | − | 16.0000i | 19.0919 | − | 19.0919i | 11.3137 | |||||||||||||||||||||||
| 55.1 | −2.00000 | − | 2.00000i | 0 | 8.00000i | 2.82843 | − | 2.82843i | 0 | 0 | 16.0000 | − | 16.0000i | −19.0919 | + | 19.0919i | −11.3137 | |||||||||||||||||||||||
| 79.1 | −2.00000 | + | 2.00000i | 0 | − | 8.00000i | −2.82843 | − | 2.82843i | 0 | 0 | 16.0000 | + | 16.0000i | 19.0919 | + | 19.0919i | 11.3137 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 41.e | odd | 8 | 1 | inner |
| 164.i | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 164.4.i.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | CM | 164.4.i.a | ✓ | 4 |
| 41.e | odd | 8 | 1 | inner | 164.4.i.a | ✓ | 4 |
| 164.i | even | 8 | 1 | inner | 164.4.i.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.4.i.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 164.4.i.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
| 164.4.i.a | ✓ | 4 | 41.e | odd | 8 | 1 | inner |
| 164.4.i.a | ✓ | 4 | 164.i | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{4}^{\mathrm{new}}(164, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 256 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 184 T^{3} + \cdots + 2913698 \)
|
| $17$ |
\( T^{4} + 188 T^{3} + \cdots + 38596898 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + \cdots + 2366582402 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 186050)^{2} \)
|
| $41$ |
\( (T^{2} + 230 T + 68921)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 53048713538 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} + 1298 T + 842402)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 122825015296 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} + \cdots + 1399267092962 \)
|
| $97$ |
\( T^{4} + \cdots + 77541582818 \)
|
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