Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [991,1,Mod(990,991)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(991, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("991.990");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 991 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 991.b (of order \(2\), degree \(1\), minimal) |
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: | \(0.494573427519\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{34})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 7x^{6} + 6x^{5} + 15x^{4} - 10x^{3} - 10x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{17}\) |
| Projective field: | Galois closure of 17.1.930227631978098127294721.1 |
$q$-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 \(\nu = \zeta_{34} + \zeta_{34}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/991\mathbb{Z}\right)^\times\).
| \(n\) | \(6\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 990.1 |
|
−1.96595 | 0 | 2.86494 | 0.891477 | 0 | 0 | −3.66638 | 1.00000 | −1.75260 | ||||||||||||||||||||||||||||||||||||||||||
| 990.2 | −1.70043 | 0 | 1.89148 | −1.96595 | 0 | 0 | −1.51590 | 1.00000 | 3.34296 | |||||||||||||||||||||||||||||||||||||||||||
| 990.3 | −1.20527 | 0 | 0.452674 | 1.47802 | 0 | 0 | 0.659675 | 1.00000 | −1.78141 | |||||||||||||||||||||||||||||||||||||||||||
| 990.4 | −0.547326 | 0 | −0.700434 | 0.184537 | 0 | 0 | 0.930692 | 1.00000 | −0.101002 | |||||||||||||||||||||||||||||||||||||||||||
| 990.5 | 0.184537 | 0 | −0.965946 | −1.70043 | 0 | 0 | −0.362789 | 1.00000 | −0.313793 | |||||||||||||||||||||||||||||||||||||||||||
| 990.6 | 0.891477 | 0 | −0.205269 | 1.86494 | 0 | 0 | −1.07447 | 1.00000 | 1.66255 | |||||||||||||||||||||||||||||||||||||||||||
| 990.7 | 1.47802 | 0 | 1.18454 | −0.547326 | 0 | 0 | 0.272749 | 1.00000 | −0.808958 | |||||||||||||||||||||||||||||||||||||||||||
| 990.8 | 1.86494 | 0 | 2.47802 | −1.20527 | 0 | 0 | 2.75642 | 1.00000 | −2.24776 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 991.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-991}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 991.1.b.a | ✓ | 8 |
| 991.b | odd | 2 | 1 | CM | 991.1.b.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 991.1.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 991.1.b.a | ✓ | 8 | 991.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(991, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $31$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $59$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $61$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $67$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $71$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} + T^{7} - 7 T^{6} + \cdots + 1 \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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