Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8993,2,Mod(1,8993)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, 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("8993.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8993 = 17 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8993.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: | \(71.8094665377\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.169.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 391) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
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 |
|
−2.65109 | 0 | 5.02830 | −0.273891 | 0 | −0.651093 | −8.02830 | −3.00000 | 0.726109 | |||||||||||||||||||||||||||
| 1.2 | 0.273891 | 0 | −1.92498 | −1.37720 | 0 | 2.27389 | −1.07502 | −3.00000 | −0.377203 | ||||||||||||||||||||||||||||
| 1.3 | 1.37720 | 0 | −0.103312 | 2.65109 | 0 | 3.37720 | −2.89669 | −3.00000 | 3.65109 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
| \(23\) | \(-1\) |
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 | 8993.2.a.d | 3 | |
| 23.b | odd | 2 | 1 | 391.2.a.c | ✓ | 3 | |
| 69.c | even | 2 | 1 | 3519.2.a.k | 3 | ||
| 92.b | even | 2 | 1 | 6256.2.a.q | 3 | ||
| 115.c | odd | 2 | 1 | 9775.2.a.k | 3 | ||
| 391.c | odd | 2 | 1 | 6647.2.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 391.2.a.c | ✓ | 3 | 23.b | odd | 2 | 1 | |
| 3519.2.a.k | 3 | 69.c | even | 2 | 1 | ||
| 6256.2.a.q | 3 | 92.b | even | 2 | 1 | ||
| 6647.2.a.c | 3 | 391.c | odd | 2 | 1 | ||
| 8993.2.a.d | 3 | 1.a | even | 1 | 1 | trivial | |
| 9775.2.a.k | 3 | 115.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
|
\( T_{2}^{3} + T_{2}^{2} - 4T_{2} + 1 \)
|
|
\( T_{5}^{3} - T_{5}^{2} - 4T_{5} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 4T + 1 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - T^{2} - 4T - 1 \)
|
| $7$ |
\( T^{3} - 5 T^{2} + \cdots + 5 \)
|
| $11$ |
\( T^{3} - 3 T^{2} + \cdots + 25 \)
|
| $13$ |
\( T^{3} + 3 T^{2} + \cdots + 1 \)
|
| $17$ |
\( (T + 1)^{3} \)
|
| $19$ |
\( T^{3} - 2 T^{2} + \cdots - 8 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} + 8 T^{2} + \cdots - 8 \)
|
| $31$ |
\( T^{3} + 8 T^{2} + \cdots - 320 \)
|
| $37$ |
\( T^{3} + 3 T^{2} + \cdots - 103 \)
|
| $41$ |
\( T^{3} + 8 T^{2} + \cdots - 8 \)
|
| $43$ |
\( T^{3} - 12 T^{2} + \cdots + 40 \)
|
| $47$ |
\( T^{3} - 5 T^{2} + \cdots - 125 \)
|
| $53$ |
\( T^{3} - 18 T^{2} + \cdots + 200 \)
|
| $59$ |
\( T^{3} + 11 T^{2} + \cdots + 31 \)
|
| $61$ |
\( T^{3} - T^{2} + \cdots + 415 \)
|
| $67$ |
\( T^{3} + 10 T^{2} + \cdots - 40 \)
|
| $71$ |
\( T^{3} - 8 T^{2} + \cdots + 320 \)
|
| $73$ |
\( T^{3} - 4 T^{2} + \cdots + 664 \)
|
| $79$ |
\( T^{3} + 5 T^{2} + \cdots - 5 \)
|
| $83$ |
\( T^{3} + 16 T^{2} + \cdots - 376 \)
|
| $89$ |
\( T^{3} - 26T^{2} + 2600 \)
|
| $97$ |
\( T^{3} + 5 T^{2} + \cdots - 941 \)
|
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