Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1587,2,Mod(1,1587)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1587, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1587.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1587 = 3 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1587.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: | \(12.6722588008\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2803712.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 6x^{4} + 8x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 6\nu^{2} - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 6\nu^{3} - 6\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{3} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 9\beta_{3} - 3\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.90321 | 1.00000 | 1.62222 | −3.13152 | −1.90321 | 4.54573 | 0.719004 | 1.00000 | 5.95995 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.90321 | 1.00000 | 1.62222 | 3.13152 | −1.90321 | −4.54573 | 0.719004 | 1.00000 | −5.95995 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.193937 | 1.00000 | −1.96239 | −2.36899 | 0.193937 | 0.954779 | −0.768452 | 1.00000 | −0.459434 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.193937 | 1.00000 | −1.96239 | 2.36899 | 0.193937 | −0.954779 | −0.768452 | 1.00000 | 0.459434 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.70928 | 1.00000 | 5.34017 | −0.762528 | 2.70928 | −0.651685 | 9.04945 | 1.00000 | −2.06590 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70928 | 1.00000 | 5.34017 | 0.762528 | 2.70928 | 0.651685 | 9.04945 | 1.00000 | 2.06590 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(23\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1587.2.a.s | ✓ | 6 |
| 3.b | odd | 2 | 1 | 4761.2.a.bs | 6 | ||
| 23.b | odd | 2 | 1 | inner | 1587.2.a.s | ✓ | 6 |
| 69.c | even | 2 | 1 | 4761.2.a.bs | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1587.2.a.s | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1587.2.a.s | ✓ | 6 | 23.b | odd | 2 | 1 | inner |
| 4761.2.a.bs | 6 | 3.b | odd | 2 | 1 | ||
| 4761.2.a.bs | 6 | 69.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\):
|
\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \)
|
|
\( T_{5}^{6} - 16T_{5}^{4} + 64T_{5}^{2} - 32 \)
|
|
\( T_{7}^{6} - 22T_{7}^{4} + 28T_{7}^{2} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 5 T + 1)^{2} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} - 16 T^{4} + \cdots - 32 \)
|
| $7$ |
\( T^{6} - 22 T^{4} + \cdots - 8 \)
|
| $11$ |
\( T^{6} - 56 T^{4} + \cdots - 512 \)
|
| $13$ |
\( (T^{3} - 4 T^{2} - 16 T + 32)^{2} \)
|
| $17$ |
\( T^{6} - 40 T^{4} + \cdots - 32 \)
|
| $19$ |
\( T^{6} - 86 T^{4} + \cdots - 6728 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 4 T + 8)^{2} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 4 T + 40)^{2} \)
|
| $37$ |
\( T^{6} - 86 T^{4} + \cdots - 2888 \)
|
| $41$ |
\( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \)
|
| $43$ |
\( T^{6} - 70 T^{4} + \cdots - 2312 \)
|
| $47$ |
\( (T^{3} - 4 T^{2} - 88 T - 16)^{2} \)
|
| $53$ |
\( T^{6} - 200 T^{4} + \cdots - 143648 \)
|
| $59$ |
\( (T^{3} - 28 T^{2} + \cdots - 688)^{2} \)
|
| $61$ |
\( T^{6} - 70 T^{4} + \cdots - 7688 \)
|
| $67$ |
\( T^{6} - 150 T^{4} + \cdots - 10952 \)
|
| $71$ |
\( (T + 4)^{6} \)
|
| $73$ |
\( (T^{3} - 20 T^{2} + \cdots + 64)^{2} \)
|
| $79$ |
\( T^{6} - 134 T^{4} + \cdots - 2888 \)
|
| $83$ |
\( T^{6} - 344 T^{4} + \cdots - 86528 \)
|
| $89$ |
\( T^{6} - 240 T^{4} + \cdots - 5408 \)
|
| $97$ |
\( T^{6} - 214 T^{4} + \cdots - 29768 \)
|
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