Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [560,3,Mod(321,560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(560, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("560.321");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 560.f (of order \(2\), degree \(1\), 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: | \(15.2588948042\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-5}, \sqrt{-13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 140) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 9x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 9\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(351\) | \(421\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 321.1 |
|
0 | − | 2.92081i | 0 | 2.23607i | 0 | 5.53113 | − | 4.29029i | 0 | 0.468871 | 0 | |||||||||||||||||||||||||||
| 321.2 | 0 | − | 0.684742i | 0 | − | 2.23607i | 0 | −2.53113 | − | 6.52636i | 0 | 8.53113 | 0 | |||||||||||||||||||||||||||
| 321.3 | 0 | 0.684742i | 0 | 2.23607i | 0 | −2.53113 | + | 6.52636i | 0 | 8.53113 | 0 | |||||||||||||||||||||||||||||
| 321.4 | 0 | 2.92081i | 0 | − | 2.23607i | 0 | 5.53113 | + | 4.29029i | 0 | 0.468871 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 560.3.f.c | 4 | |
| 4.b | odd | 2 | 1 | 140.3.d.a | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 560.3.f.c | 4 | |
| 12.b | even | 2 | 1 | 1260.3.j.a | 4 | ||
| 20.d | odd | 2 | 1 | 700.3.d.d | 4 | ||
| 20.e | even | 4 | 2 | 700.3.h.b | 8 | ||
| 28.d | even | 2 | 1 | 140.3.d.a | ✓ | 4 | |
| 28.f | even | 6 | 2 | 980.3.r.a | 8 | ||
| 28.g | odd | 6 | 2 | 980.3.r.a | 8 | ||
| 84.h | odd | 2 | 1 | 1260.3.j.a | 4 | ||
| 140.c | even | 2 | 1 | 700.3.d.d | 4 | ||
| 140.j | odd | 4 | 2 | 700.3.h.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.3.d.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 140.3.d.a | ✓ | 4 | 28.d | even | 2 | 1 | |
| 560.3.f.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 560.3.f.c | 4 | 7.b | odd | 2 | 1 | inner | |
| 700.3.d.d | 4 | 20.d | odd | 2 | 1 | ||
| 700.3.d.d | 4 | 140.c | even | 2 | 1 | ||
| 700.3.h.b | 8 | 20.e | even | 4 | 2 | ||
| 700.3.h.b | 8 | 140.j | odd | 4 | 2 | ||
| 980.3.r.a | 8 | 28.f | even | 6 | 2 | ||
| 980.3.r.a | 8 | 28.g | odd | 6 | 2 | ||
| 1260.3.j.a | 4 | 12.b | even | 2 | 1 | ||
| 1260.3.j.a | 4 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 9T_{3}^{2} + 4 \)
acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 9T^{2} + 4 \)
|
| $5$ |
\( (T^{2} + 5)^{2} \)
|
| $7$ |
\( T^{4} - 6 T^{3} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} + T - 146)^{2} \)
|
| $13$ |
\( T^{4} + 9T^{2} + 4 \)
|
| $17$ |
\( T^{4} + 365T^{2} + 400 \)
|
| $19$ |
\( T^{4} + 264T^{2} + 784 \)
|
| $23$ |
\( (T^{2} + 18 T + 16)^{2} \)
|
| $29$ |
\( (T^{2} - 27 T - 614)^{2} \)
|
| $31$ |
\( T^{4} + 2964 T^{2} + 43264 \)
|
| $37$ |
\( (T^{2} + 6 T - 3176)^{2} \)
|
| $41$ |
\( T^{4} + 3396 T^{2} + 2096704 \)
|
| $43$ |
\( (T^{2} + 20 T - 940)^{2} \)
|
| $47$ |
\( T^{4} + 6501 T^{2} + 1882384 \)
|
| $53$ |
\( (T^{2} + 80 T + 1340)^{2} \)
|
| $59$ |
\( T^{4} + 10664 T^{2} + 13454224 \)
|
| $61$ |
\( T^{4} + 16644 T^{2} + 29506624 \)
|
| $67$ |
\( (T^{2} + 120 T + 1260)^{2} \)
|
| $71$ |
\( (T^{2} - 20 T - 4060)^{2} \)
|
| $73$ |
\( T^{4} + 17044 T^{2} + 68690944 \)
|
| $79$ |
\( (T^{2} - 21 T + 94)^{2} \)
|
| $83$ |
\( T^{4} + 13416 T^{2} + 2704 \)
|
| $89$ |
\( T^{4} + 37584 T^{2} + 351637504 \)
|
| $97$ |
\( T^{4} + 4189 T^{2} + 595984 \)
|
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