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: | \(0\) |
| 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: |
\( x^{6} - x^{5} - 26x^{4} + 16x^{3} + 158x^{2} + 20x - 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 26x^{4} + 16x^{3} + 158x^{2} + 20x - 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 16\nu^{3} - 12\nu^{2} + 10\nu - 16 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 20\nu^{3} - 12\nu^{2} + 66\nu - 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - \nu^{4} + 20\nu^{3} + 16\nu^{2} - 70\nu - 28 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{4} - 12\nu^{3} - 76\nu^{2} - 46\nu + 32 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 14\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 16\beta_{4} + 17\beta_{3} - 2\beta_{2} + 16\beta _1 + 113 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} - 4\beta_{4} - 21\beta_{3} + 22\beta_{2} + 210\beta _1 + 47 \)
|
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 |
|
−2.90224 | 0 | 0.422972 | 21.3957 | 0 | 32.6648 | 21.9903 | 0 | −62.0952 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.80939 | 0 | −0.107356 | −3.97213 | 0 | −27.6394 | 22.7767 | 0 | 11.1593 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.611207 | 0 | −7.62643 | 3.28889 | 0 | −0.135321 | −9.55098 | 0 | 2.01019 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.55137 | 0 | −5.59326 | 10.0675 | 0 | −0.474755 | −21.0882 | 0 | 15.6184 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.44961 | 0 | 3.89978 | −13.6846 | 0 | −34.8935 | −14.1442 | 0 | −47.2064 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.09944 | 0 | 18.0043 | −0.0953566 | 0 | −0.521765 | 51.0163 | 0 | −0.486265 | |||||||||||||||||||||||||||||||||||||
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.e | 6 | |
| 3.b | odd | 2 | 1 | 429.4.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.4.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 1287.4.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 5T_{2}^{5} - 16T_{2}^{4} + 78T_{2}^{3} + 55T_{2}^{2} - 281T_{2} + 136 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1287))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 5 T^{5} + \cdots + 136 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 17 T^{5} + \cdots - 3672 \)
|
| $7$ |
\( T^{6} + 31 T^{5} + \cdots - 1056 \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( (T - 13)^{6} \)
|
| $17$ |
\( T^{6} + \cdots + 6619088128 \)
|
| $19$ |
\( T^{6} + \cdots + 8631155648 \)
|
| $23$ |
\( T^{6} + \cdots + 282242277888 \)
|
| $29$ |
\( T^{6} + \cdots + 6084688020544 \)
|
| $31$ |
\( T^{6} + \cdots - 10180828348416 \)
|
| $37$ |
\( T^{6} + \cdots + 271401251151744 \)
|
| $41$ |
\( T^{6} + \cdots - 17554575478464 \)
|
| $43$ |
\( T^{6} + \cdots + 41899205407888 \)
|
| $47$ |
\( T^{6} + \cdots + 30180254507008 \)
|
| $53$ |
\( T^{6} + \cdots + 86038960934976 \)
|
| $59$ |
\( T^{6} + \cdots + 22663856195328 \)
|
| $61$ |
\( T^{6} + \cdots - 405518137993984 \)
|
| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!12 \)
|
| $71$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots - 22\!\cdots\!36 \)
|
| $79$ |
\( T^{6} + \cdots + 27\!\cdots\!88 \)
|
| $83$ |
\( T^{6} + \cdots + 51166666192896 \)
|
| $89$ |
\( T^{6} + \cdots - 71\!\cdots\!16 \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!08 \)
|
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