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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 113x^{4} - 444x^{3} + 156x^{2} + 2448x + 2304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 113x^{4} - 444x^{3} + 156x^{2} + 2448x + 2304 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} + 97\nu^{3} + 40\nu^{2} - 660\nu - 312 ) / 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} + 11\nu^{4} + 307\nu^{3} + 524\nu^{2} - 1668\nu - 2616 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 125\nu^{3} - 372\nu^{2} + 1212\nu + 2484 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 11\nu^{4} + 307\nu^{3} + 556\nu^{2} - 1860\nu - 3800 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - 13\nu^{4} - 545\nu^{3} - 1340\nu^{2} + 2940\nu + 7164 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} - 4\beta_{2} + 2\beta _1 + 6 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + \beta_{3} - 7\beta_{2} + 2\beta _1 + 154 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{5} + 3\beta_{4} + 7\beta_{3} - 49\beta_{2} + 22\beta _1 + 580 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -186\beta_{5} + 125\beta_{4} + 170\beta_{3} - 1175\beta_{2} + 440\beta _1 + 18592 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2606\beta_{5} + 1247\beta_{4} + 2138\beta_{3} - 15221\beta_{2} + 6280\beta _1 + 209732 ) / 4 \)
|
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 | −27.0152 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 5.00000 | 0 | −20.4676 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 5.00000 | 0 | −0.672909 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 5.00000 | 0 | 16.0286 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.00000 | 0 | 5.00000 | 0 | 17.3093 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.00000 | 0 | 5.00000 | 0 | 34.8178 | 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.u | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1560.4.a.u | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 20T_{7}^{5} - 1259T_{7}^{4} + 20106T_{7}^{3} + 350640T_{7}^{2} - 5114880T_{7} - 3594240 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T - 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 20 T^{5} + \cdots - 3594240 \)
|
| $11$ |
\( T^{6} + \cdots - 2603863040 \)
|
| $13$ |
\( (T - 13)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 30329645184 \)
|
| $19$ |
\( T^{6} + \cdots - 86908540416 \)
|
| $23$ |
\( T^{6} + \cdots - 1072719165440 \)
|
| $29$ |
\( T^{6} + \cdots - 13591908864 \)
|
| $31$ |
\( T^{6} + \cdots - 25212903342080 \)
|
| $37$ |
\( T^{6} + \cdots + 270050062066416 \)
|
| $41$ |
\( T^{6} + \cdots + 8627154066352 \)
|
| $43$ |
\( T^{6} + \cdots + 1423217405952 \)
|
| $47$ |
\( T^{6} + \cdots - 6537497575424 \)
|
| $53$ |
\( T^{6} + \cdots + 37\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 131379289814768 \)
|
| $67$ |
\( T^{6} + \cdots - 15\!\cdots\!08 \)
|
| $71$ |
\( T^{6} + \cdots - 757221651513344 \)
|
| $73$ |
\( T^{6} + \cdots + 76\!\cdots\!24 \)
|
| $79$ |
\( T^{6} + \cdots + 858339331276800 \)
|
| $83$ |
\( T^{6} + \cdots + 70\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots + 51\!\cdots\!92 \)
|
| $97$ |
\( T^{6} + \cdots - 232298166562208 \)
|
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