Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1280,2,Mod(321,1280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1280.321");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1280 = 2^{8} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1280.l (of order \(4\), degree \(2\), 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: | \(10.2208514587\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} + \zeta_{24}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{7} + \zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{4} + \zeta_{24}^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( -\beta_{6} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( \beta_{7} - \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).
| \(n\) | \(257\) | \(261\) | \(511\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 321.1 |
|
0 | −2.07313 | − | 2.07313i | 0 | −0.707107 | + | 0.707107i | 0 | 4.34607i | 0 | 5.59575i | 0 | ||||||||||||||||||||||||||||||||||||||
| 321.2 | 0 | −0.658919 | − | 0.658919i | 0 | 0.707107 | − | 0.707107i | 0 | − | 2.34607i | 0 | − | 2.13165i | 0 | |||||||||||||||||||||||||||||||||||||
| 321.3 | 0 | −0.341081 | − | 0.341081i | 0 | −0.707107 | + | 0.707107i | 0 | 1.89658i | 0 | − | 2.76733i | 0 | ||||||||||||||||||||||||||||||||||||||
| 321.4 | 0 | 1.07313 | + | 1.07313i | 0 | 0.707107 | − | 0.707107i | 0 | 0.103425i | 0 | − | 0.696775i | 0 | ||||||||||||||||||||||||||||||||||||||
| 961.1 | 0 | −2.07313 | + | 2.07313i | 0 | −0.707107 | − | 0.707107i | 0 | − | 4.34607i | 0 | − | 5.59575i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.2 | 0 | −0.658919 | + | 0.658919i | 0 | 0.707107 | + | 0.707107i | 0 | 2.34607i | 0 | 2.13165i | 0 | |||||||||||||||||||||||||||||||||||||||
| 961.3 | 0 | −0.341081 | + | 0.341081i | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.89658i | 0 | 2.76733i | 0 | ||||||||||||||||||||||||||||||||||||||
| 961.4 | 0 | 1.07313 | − | 1.07313i | 0 | 0.707107 | + | 0.707107i | 0 | − | 0.103425i | 0 | 0.696775i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1280.2.l.d | yes | 8 |
| 4.b | odd | 2 | 1 | 1280.2.l.g | yes | 8 | |
| 8.b | even | 2 | 1 | 1280.2.l.f | yes | 8 | |
| 8.d | odd | 2 | 1 | 1280.2.l.a | ✓ | 8 | |
| 16.e | even | 4 | 1 | inner | 1280.2.l.d | yes | 8 |
| 16.e | even | 4 | 1 | 1280.2.l.f | yes | 8 | |
| 16.f | odd | 4 | 1 | 1280.2.l.a | ✓ | 8 | |
| 16.f | odd | 4 | 1 | 1280.2.l.g | yes | 8 | |
| 32.g | even | 8 | 1 | 5120.2.a.b | 4 | ||
| 32.g | even | 8 | 1 | 5120.2.a.q | 4 | ||
| 32.h | odd | 8 | 1 | 5120.2.a.a | 4 | ||
| 32.h | odd | 8 | 1 | 5120.2.a.r | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1280.2.l.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 1280.2.l.a | ✓ | 8 | 16.f | odd | 4 | 1 | |
| 1280.2.l.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 1280.2.l.d | yes | 8 | 16.e | even | 4 | 1 | inner |
| 1280.2.l.f | yes | 8 | 8.b | even | 2 | 1 | |
| 1280.2.l.f | yes | 8 | 16.e | even | 4 | 1 | |
| 1280.2.l.g | yes | 8 | 4.b | odd | 2 | 1 | |
| 1280.2.l.g | yes | 8 | 16.f | odd | 4 | 1 | |
| 5120.2.a.a | 4 | 32.h | odd | 8 | 1 | ||
| 5120.2.a.b | 4 | 32.g | even | 8 | 1 | ||
| 5120.2.a.q | 4 | 32.g | even | 8 | 1 | ||
| 5120.2.a.r | 4 | 32.h | odd | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):
|
\( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 8T_{3}^{4} + 24T_{3}^{3} + 32T_{3}^{2} + 16T_{3} + 4 \)
|
|
\( T_{13}^{8} - 8T_{13}^{7} + 32T_{13}^{6} - 48T_{13}^{5} + 32T_{13}^{4} + 128T_{13}^{2} - 128T_{13} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 4 T^{7} + \cdots + 4 \)
|
| $5$ |
\( (T^{4} + 1)^{2} \)
|
| $7$ |
\( T^{8} + 28 T^{6} + \cdots + 4 \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots + 8464 \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 64 \)
|
| $17$ |
\( (T^{4} + 4 T^{3} - 28 T^{2} + \cdots - 8)^{2} \)
|
| $19$ |
\( T^{8} - 96 T^{5} + \cdots + 150544 \)
|
| $23$ |
\( T^{8} + 100 T^{6} + \cdots + 37636 \)
|
| $29$ |
\( T^{8} - 8 T^{7} + \cdots + 446224 \)
|
| $31$ |
\( (T^{4} - 40 T^{2} + 16)^{2} \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 135424 \)
|
| $41$ |
\( T^{8} + 144 T^{6} + \cdots + 150544 \)
|
| $43$ |
\( T^{8} - 44 T^{7} + \cdots + 9721924 \)
|
| $47$ |
\( (T^{4} - 4 T^{3} + \cdots + 2062)^{2} \)
|
| $53$ |
\( T^{8} + 16 T^{7} + \cdots + 602176 \)
|
| $59$ |
\( T^{8} + 16 T^{7} + \cdots + 13278736 \)
|
| $61$ |
\( T^{8} + 8 T^{7} + \cdots + 541696 \)
|
| $67$ |
\( T^{8} + 12 T^{7} + \cdots + 324 \)
|
| $71$ |
\( T^{8} + 80 T^{6} + \cdots + 1024 \)
|
| $73$ |
\( T^{8} + 440 T^{6} + \cdots + 74787904 \)
|
| $79$ |
\( (T^{4} - 48 T^{3} + \cdots + 18448)^{2} \)
|
| $83$ |
\( T^{8} - 28 T^{7} + \cdots + 2116 \)
|
| $89$ |
\( T^{8} + 368 T^{6} + \cdots + 7139584 \)
|
| $97$ |
\( (T^{4} + 12 T^{3} + \cdots + 1912)^{2} \)
|
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