Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,6,Mod(1,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.a (trivial) |
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: | \(51.3228223402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.39180.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 36x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 160) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 36x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 8\nu^{2} - 16\nu - 189 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 4\beta _1 + 193 ) / 8 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −29.2272 | 0 | 25.0000 | 0 | 44.5253 | 0 | 611.232 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.755735 | 0 | 25.0000 | 0 | −172.424 | 0 | −242.429 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 18.4715 | 0 | 25.0000 | 0 | 121.899 | 0 | 98.1968 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.6.a.x | 3 | |
| 4.b | odd | 2 | 1 | 320.6.a.y | 3 | ||
| 8.b | even | 2 | 1 | 160.6.a.g | yes | 3 | |
| 8.d | odd | 2 | 1 | 160.6.a.f | ✓ | 3 | |
| 40.e | odd | 2 | 1 | 800.6.a.o | 3 | ||
| 40.f | even | 2 | 1 | 800.6.a.n | 3 | ||
| 40.i | odd | 4 | 2 | 800.6.c.j | 6 | ||
| 40.k | even | 4 | 2 | 800.6.c.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.6.a.f | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 160.6.a.g | yes | 3 | 8.b | even | 2 | 1 | |
| 320.6.a.x | 3 | 1.a | even | 1 | 1 | trivial | |
| 320.6.a.y | 3 | 4.b | odd | 2 | 1 | ||
| 800.6.a.n | 3 | 40.f | even | 2 | 1 | ||
| 800.6.a.o | 3 | 40.e | odd | 2 | 1 | ||
| 800.6.c.j | 6 | 40.i | odd | 4 | 2 | ||
| 800.6.c.k | 6 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 10T_{3}^{2} - 548T_{3} + 408 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 10 T^{2} + \cdots + 408 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} + 6 T^{2} + \cdots + 935848 \)
|
| $11$ |
\( T^{3} - 396 T^{2} + \cdots + 59934400 \)
|
| $13$ |
\( T^{3} - 354 T^{2} + \cdots - 28863000 \)
|
| $17$ |
\( T^{3} + \cdots + 2743838200 \)
|
| $19$ |
\( T^{3} + 3192 T^{2} + \cdots + 126323200 \)
|
| $23$ |
\( T^{3} + \cdots - 5283148104 \)
|
| $29$ |
\( T^{3} + \cdots - 159249002312 \)
|
| $31$ |
\( T^{3} + \cdots + 229217617600 \)
|
| $37$ |
\( T^{3} + \cdots + 1078137168200 \)
|
| $41$ |
\( T^{3} + \cdots - 1203781400 \)
|
| $43$ |
\( T^{3} + \cdots + 352818354968 \)
|
| $47$ |
\( T^{3} + \cdots - 1628197032152 \)
|
| $53$ |
\( T^{3} + \cdots + 18581204112200 \)
|
| $59$ |
\( T^{3} + \cdots + 887454720000 \)
|
| $61$ |
\( T^{3} + \cdots + 47677792189640 \)
|
| $67$ |
\( T^{3} + \cdots - 57239443872504 \)
|
| $71$ |
\( T^{3} + \cdots - 11664981864000 \)
|
| $73$ |
\( T^{3} + \cdots + 165394983407000 \)
|
| $79$ |
\( T^{3} + \cdots + 1808931596800 \)
|
| $83$ |
\( T^{3} + \cdots + 187637358980920 \)
|
| $89$ |
\( T^{3} + \cdots - 175016035497384 \)
|
| $97$ |
\( T^{3} + \cdots + 83009979455000 \)
|
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