Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1560,4,Mod(1,1560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1560.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1560.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: | \(92.0429796090\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 24x^{3} + 25x^{2} + 76x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 5 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2x^{4} - 24x^{3} + 25x^{2} + 76x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 3\nu - 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 21\nu^{2} + 22\nu + 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 4\nu^{3} + 22\nu^{2} - 67\nu - 54 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 24\nu^{2} + 33\nu + 58 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{2} + 3\beta _1 + 9 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{4} + 3\beta_{2} + 11\beta _1 + 213 ) / 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 12\beta_{4} + 5\beta_{3} - 7\beta_{2} + 31\beta _1 + 163 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -37\beta_{4} + 20\beta_{3} + 77\beta_{2} + 289\beta _1 + 4307 ) / 20 \)
|
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 | −3.00000 | 0 | −5.00000 | 0 | −12.1634 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −5.00000 | 0 | −10.8135 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −5.00000 | 0 | −5.02831 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | −5.00000 | 0 | 4.22487 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −3.00000 | 0 | −5.00000 | 0 | 31.7803 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1560.4.a.p | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1560.4.a.p | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} - 8T_{7}^{4} - 627T_{7}^{3} - 4474T_{7}^{2} + 9360T_{7} + 88800 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} - 8 T^{4} + \cdots + 88800 \)
|
| $11$ |
\( T^{5} - 6 T^{4} + \cdots + 40444032 \)
|
| $13$ |
\( (T - 13)^{5} \)
|
| $17$ |
\( T^{5} - 74 T^{4} + \cdots - 664361472 \)
|
| $19$ |
\( T^{5} + 42 T^{4} + \cdots + 66986240 \)
|
| $23$ |
\( T^{5} + \cdots + 2132416512 \)
|
| $29$ |
\( T^{5} + \cdots - 92942450176 \)
|
| $31$ |
\( T^{5} + \cdots + 317559491328 \)
|
| $37$ |
\( T^{5} + \cdots + 7084252856 \)
|
| $41$ |
\( T^{5} + \cdots + 481449107320 \)
|
| $43$ |
\( T^{5} + \cdots + 316128601088 \)
|
| $47$ |
\( T^{5} + \cdots - 12532546466816 \)
|
| $53$ |
\( T^{5} + \cdots - 216365152712 \)
|
| $59$ |
\( T^{5} + \cdots + 31515413962752 \)
|
| $61$ |
\( T^{5} + \cdots + 1702479130376 \)
|
| $67$ |
\( T^{5} + \cdots - 34955207585792 \)
|
| $71$ |
\( T^{5} + \cdots - 139068788736 \)
|
| $73$ |
\( T^{5} + \cdots - 980650800000 \)
|
| $79$ |
\( T^{5} + \cdots + 34865242483200 \)
|
| $83$ |
\( T^{5} + \cdots - 3333708132352 \)
|
| $89$ |
\( T^{5} + \cdots + 125038227080 \)
|
| $97$ |
\( T^{5} + \cdots + 143156695165856 \)
|
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