Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1287,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1287.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1287 = 3^{2} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(75.9354581774\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 45x^{6} + 11x^{5} + 572x^{4} + 122x^{3} - 1556x^{2} + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 429) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 45x^{6} + 11x^{5} + 572x^{4} + 122x^{3} - 1556x^{2} + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 39\nu^{5} + 77\nu^{4} + 454\nu^{3} - 450\nu^{2} - 1280\nu + 520 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 84\nu^{5} - 43\nu^{4} - 992\nu^{3} - 210\nu^{2} + 2248\nu + 280 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{7} - 3\nu^{6} - 84\nu^{5} + 43\nu^{4} + 1016\nu^{3} + 162\nu^{2} - 2704\nu + 8 ) / 24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 51\nu^{5} + 16\nu^{4} - 718\nu^{3} - 324\nu^{2} + 2192\nu + 152 ) / 12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} - 3\nu^{6} - 186\nu^{5} + 35\nu^{4} + 2380\nu^{3} + 306\nu^{2} - 6152\nu + 544 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta_{2} + 21\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{5} + 3\beta_{4} + 27\beta_{2} + 45\beta _1 + 226 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{7} + 5\beta_{6} + 36\beta_{5} + 35\beta_{4} - 4\beta_{3} + 85\beta_{2} + 505\beta _1 + 470 \)
|
| \(\nu^{6}\) | \(=\) |
\( 43\beta_{7} + 47\beta_{6} + 118\beta_{5} + 145\beta_{4} - 24\beta_{3} + 743\beta_{2} + 1613\beta _1 + 5392 \)
|
| \(\nu^{7}\) | \(=\) |
\( 169\beta_{7} + 259\beta_{6} + 1150\beta_{5} + 1115\beta_{4} - 204\beta_{3} + 3007\beta_{2} + 13265\beta _1 + 16994 \)
|
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 |
|
−5.18758 | 0 | 18.9110 | 15.0327 | 0 | 17.9029 | −56.6018 | 0 | −77.9833 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.04140 | 0 | 17.4158 | −11.8767 | 0 | −2.73922 | −47.4687 | 0 | 59.8752 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.27832 | 0 | 2.74738 | −11.5120 | 0 | 29.7005 | 17.2198 | 0 | 37.7399 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.14314 | 0 | −6.69323 | −17.4756 | 0 | −23.7195 | 16.7964 | 0 | 19.9771 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.855206 | 0 | −7.26862 | 14.7095 | 0 | −8.75474 | 13.0578 | 0 | −12.5797 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.697180 | 0 | −7.51394 | −2.48450 | 0 | 31.2447 | −10.8160 | 0 | −1.73214 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.22767 | 0 | 2.41783 | 12.2384 | 0 | −12.0289 | −18.0174 | 0 | 39.5015 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.58081 | 0 | 12.9838 | −5.63187 | 0 | 7.39420 | 22.8299 | 0 | −25.7985 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1287.4.a.h | 8 | |
| 3.b | odd | 2 | 1 | 429.4.a.g | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.4.a.g | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 1287.4.a.h | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 7T_{2}^{7} - 24T_{2}^{6} - 224T_{2}^{5} - 13T_{2}^{4} + 1641T_{2}^{3} + 1684T_{2}^{2} - 672T_{2} - 864 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1287))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 7 T^{7} + \cdots - 864 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots - 90475024 \)
|
| $7$ |
\( T^{8} - 39 T^{7} + \cdots + 840538368 \)
|
| $11$ |
\( (T + 11)^{8} \)
|
| $13$ |
\( (T + 13)^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 3739018597184 \)
|
| $19$ |
\( T^{8} + \cdots - 618788254838784 \)
|
| $23$ |
\( T^{8} + \cdots + 105103773258752 \)
|
| $29$ |
\( T^{8} + \cdots + 88475712939216 \)
|
| $31$ |
\( T^{8} + \cdots + 67\!\cdots\!36 \)
|
| $37$ |
\( T^{8} + \cdots - 39\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 90\!\cdots\!56 \)
|
| $43$ |
\( T^{8} + \cdots + 60\!\cdots\!72 \)
|
| $47$ |
\( T^{8} + \cdots - 15\!\cdots\!64 \)
|
| $53$ |
\( T^{8} + \cdots + 36\!\cdots\!56 \)
|
| $59$ |
\( T^{8} + \cdots - 11\!\cdots\!56 \)
|
| $61$ |
\( T^{8} + \cdots - 14\!\cdots\!24 \)
|
| $67$ |
\( T^{8} + \cdots + 50\!\cdots\!64 \)
|
| $71$ |
\( T^{8} + \cdots - 11\!\cdots\!48 \)
|
| $73$ |
\( T^{8} + \cdots - 17\!\cdots\!92 \)
|
| $79$ |
\( T^{8} + \cdots + 66\!\cdots\!12 \)
|
| $83$ |
\( T^{8} + \cdots + 30\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 88\!\cdots\!92 \)
|
| $97$ |
\( T^{8} + \cdots - 92\!\cdots\!16 \)
|
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