Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1304,1,Mod(325,1304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1304, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1304.325");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1304 = 2^{3} \cdot 163 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1304.h (of order \(2\), degree \(1\), 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: | yes |
| Analytic conductor: | \(0.650780776473\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\Q(\zeta_{22})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{11}\) |
| Projective field: | Galois closure of 11.1.3770404603265024.1 |
$q$-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 basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1304\mathbb{Z}\right)^\times\).
| \(n\) | \(327\) | \(653\) | \(817\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 325.1 |
|
−1.00000 | −1.68251 | 1.00000 | 1.30972 | 1.68251 | 0 | −1.00000 | 1.83083 | −1.30972 | |||||||||||||||||||||||||||||||||
| 325.2 | −1.00000 | −0.830830 | 1.00000 | 0.284630 | 0.830830 | 0 | −1.00000 | −0.309721 | −0.284630 | ||||||||||||||||||||||||||||||||||
| 325.3 | −1.00000 | 0.284630 | 1.00000 | −1.68251 | −0.284630 | 0 | −1.00000 | −0.918986 | 1.68251 | ||||||||||||||||||||||||||||||||||
| 325.4 | −1.00000 | 1.30972 | 1.00000 | 1.91899 | −1.30972 | 0 | −1.00000 | 0.715370 | −1.91899 | ||||||||||||||||||||||||||||||||||
| 325.5 | −1.00000 | 1.91899 | 1.00000 | −0.830830 | −1.91899 | 0 | −1.00000 | 2.68251 | 0.830830 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 1304.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-326}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1304.1.h.a | ✓ | 5 |
| 8.b | even | 2 | 1 | 1304.1.h.b | yes | 5 | |
| 163.b | odd | 2 | 1 | 1304.1.h.b | yes | 5 | |
| 1304.h | odd | 2 | 1 | CM | 1304.1.h.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1304.1.h.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1304.1.h.a | ✓ | 5 | 1304.h | odd | 2 | 1 | CM |
| 1304.1.h.b | yes | 5 | 8.b | even | 2 | 1 | |
| 1304.1.h.b | yes | 5 | 163.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - T_{3}^{4} - 4T_{3}^{3} + 3T_{3}^{2} + 3T_{3} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(1304, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $13$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $31$ |
\( T^{5} \)
|
| $37$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $41$ |
\( T^{5} + T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $43$ |
\( T^{5} \)
|
| $47$ |
\( T^{5} + T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $53$ |
\( T^{5} \)
|
| $59$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} - T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $71$ |
\( T^{5} + T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $73$ |
\( T^{5} \)
|
| $79$ |
\( T^{5} \)
|
| $83$ |
\( T^{5} \)
|
| $89$ |
\( T^{5} \)
|
| $97$ |
\( T^{5} + T^{4} - 4 T^{3} + \cdots + 1 \)
|
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