Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1287,2,Mod(1,1287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1287, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1287.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1287 = 3^{2} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1287.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: | \(10.2767467401\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.368464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 13\beta _1 + 19 \)
|
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 |
|
−2.78948 | 0 | 5.78120 | 2.33540 | 0 | 0.430991 | −10.5476 | 0 | −6.51455 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.09027 | 0 | 2.36924 | −3.20929 | 0 | −0.388134 | −0.771813 | 0 | 6.70830 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.447457 | 0 | −1.79978 | 1.88693 | 0 | −3.78739 | 1.70024 | 0 | −0.844323 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.178368 | 0 | −1.96818 | −2.75203 | 0 | 3.42801 | −0.707798 | 0 | −0.490874 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.14884 | 0 | 2.61752 | −2.26101 | 0 | −3.68348 | 1.32696 | 0 | −4.85856 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1287.2.a.o | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1287.2.a.p | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1287.2.a.o | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1287.2.a.p | yes | 5 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1287))\):
|
\( T_{2}^{5} + 3T_{2}^{4} - 4T_{2}^{3} - 14T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{5}^{5} + 4T_{5}^{4} - 8T_{5}^{3} - 38T_{5}^{2} + 14T_{5} + 88 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 88 \)
|
| $7$ |
\( T^{5} + 4 T^{4} + \cdots + 8 \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( (T - 1)^{5} \)
|
| $17$ |
\( T^{5} + 18 T^{4} + \cdots + 244 \)
|
| $19$ |
\( T^{5} - 36 T^{3} + \cdots + 232 \)
|
| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 3376 \)
|
| $29$ |
\( T^{5} + 20 T^{4} + \cdots - 796 \)
|
| $31$ |
\( T^{5} + 8 T^{4} + \cdots + 232 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots - 14896 \)
|
| $41$ |
\( T^{5} + 8 T^{4} + \cdots - 1096 \)
|
| $43$ |
\( T^{5} + 22 T^{4} + \cdots - 436 \)
|
| $47$ |
\( T^{5} + 6 T^{4} + \cdots - 64 \)
|
| $53$ |
\( T^{5} + 20 T^{4} + \cdots + 11264 \)
|
| $59$ |
\( T^{5} - 16 T^{4} + \cdots - 6016 \)
|
| $61$ |
\( T^{5} + 4 T^{4} + \cdots + 16 \)
|
| $67$ |
\( T^{5} + 20 T^{4} + \cdots + 32 \)
|
| $71$ |
\( T^{5} + 6 T^{4} + \cdots - 256 \)
|
| $73$ |
\( T^{5} + 12 T^{4} + \cdots + 344 \)
|
| $79$ |
\( T^{5} - 6 T^{4} + \cdots - 15076 \)
|
| $83$ |
\( T^{5} + 4 T^{4} + \cdots - 44864 \)
|
| $89$ |
\( T^{5} + 10 T^{4} + \cdots + 9584 \)
|
| $97$ |
\( T^{5} - 4 T^{4} + \cdots + 1744 \)
|
| show more | |
| show less |