Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [992,2,Mod(1,992)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(992, 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("992.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 992 = 2^{5} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 992.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: | \(7.92115988051\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.66862976.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 20x^{2} - 16x - 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 20x^{2} - 16x - 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 11\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 6\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 5\nu^{3} - 13\nu^{2} - 7\nu + 7 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 5\nu^{3} - 17\nu^{2} - 3\nu + 19 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} - \beta _1 + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{5} + 9\beta_{4} - 2\beta_{3} + 4\beta_{2} - 7\beta _1 + 31 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{5} + 20\beta_{4} - 16\beta_{3} + 22\beta_{2} - 21\beta _1 + 39 ) / 2 \)
|
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 | −3.07118 | 0 | 1.24104 | 0 | 3.86022 | 0 | 6.43216 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.59078 | 0 | −2.33595 | 0 | −4.18539 | 0 | 3.71216 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.02478 | 0 | 3.57714 | 0 | −2.91074 | 0 | −1.94983 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.211803 | 0 | −4.34258 | 0 | 3.73081 | 0 | −2.95514 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.62588 | 0 | 0.534072 | 0 | 2.49282 | 0 | −0.356523 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.84906 | 0 | 3.32627 | 0 | −0.987725 | 0 | 5.11717 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(31\) | \(-1\) |
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 | 992.2.a.g | ✓ | 6 |
| 3.b | odd | 2 | 1 | 8928.2.a.br | 6 | ||
| 4.b | odd | 2 | 1 | 992.2.a.h | yes | 6 | |
| 8.b | even | 2 | 1 | 1984.2.a.bb | 6 | ||
| 8.d | odd | 2 | 1 | 1984.2.a.ba | 6 | ||
| 12.b | even | 2 | 1 | 8928.2.a.bq | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 992.2.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 992.2.a.h | yes | 6 | 4.b | odd | 2 | 1 | |
| 1984.2.a.ba | 6 | 8.d | odd | 2 | 1 | ||
| 1984.2.a.bb | 6 | 8.b | even | 2 | 1 | ||
| 8928.2.a.bq | 6 | 12.b | even | 2 | 1 | ||
| 8928.2.a.br | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(992))\):
|
\( T_{3}^{6} + 2T_{3}^{5} - 12T_{3}^{4} - 20T_{3}^{3} + 32T_{3}^{2} + 32T_{3} - 8 \)
|
|
\( T_{7}^{6} - 2T_{7}^{5} - 29T_{7}^{4} + 52T_{7}^{3} + 228T_{7}^{2} - 288T_{7} - 432 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 8 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 80 \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 432 \)
|
| $11$ |
\( T^{6} + 10 T^{5} + \cdots + 1296 \)
|
| $13$ |
\( T^{6} - 10 T^{5} + \cdots + 24 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 3296 \)
|
| $19$ |
\( T^{6} + 2 T^{5} + \cdots + 2304 \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots + 2048 \)
|
| $29$ |
\( T^{6} - 10 T^{5} + \cdots + 72 \)
|
| $31$ |
\( (T - 1)^{6} \)
|
| $37$ |
\( T^{6} - 26 T^{5} + \cdots - 67632 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 240 \)
|
| $43$ |
\( T^{6} + 10 T^{5} + \cdots + 2984 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots + 1152 \)
|
| $53$ |
\( T^{6} - 18 T^{5} + \cdots - 5184 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 124864 \)
|
| $61$ |
\( T^{6} - 10 T^{5} + \cdots - 13256 \)
|
| $67$ |
\( T^{6} - 4 T^{5} + \cdots + 5120 \)
|
| $71$ |
\( T^{6} - 30 T^{5} + \cdots + 768 \)
|
| $73$ |
\( T^{6} - 4 T^{5} + \cdots - 32192 \)
|
| $79$ |
\( T^{6} - 20 T^{5} + \cdots - 10112 \)
|
| $83$ |
\( T^{6} + 30 T^{5} + \cdots + 2959056 \)
|
| $89$ |
\( T^{6} + 20 T^{5} + \cdots - 2656 \)
|
| $97$ |
\( T^{6} - 30 T^{5} + \cdots - 75024 \)
|
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