Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8619,2,Mod(1,8619)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, 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("8619.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8619.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: | \(68.8230615021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.153424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 4x^{3} + 8x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 663) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 4x^{3} + 8x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{4} + 6\beta_{3} + \beta_{2} + \beta _1 + 13 \)
|
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 |
|
−2.50524 | 1.00000 | 4.27625 | −0.870698 | −2.50524 | −0.344694 | −5.70256 | 1.00000 | 2.18131 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.20674 | 1.00000 | 2.86972 | 1.01669 | −2.20674 | −1.61344 | −1.91925 | 1.00000 | −2.24358 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.18726 | 1.00000 | −0.590417 | −3.12506 | −1.18726 | −0.707402 | 3.07549 | 1.00000 | 3.71026 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.10204 | 1.00000 | −0.785505 | 4.31838 | 1.10204 | 3.53871 | −3.06974 | 1.00000 | 4.75904 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.79721 | 1.00000 | 1.22995 | −1.33931 | 1.79721 | −2.87317 | −1.38394 | 1.00000 | −2.40702 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(13\) | \(1\) |
| \(17\) | \(1\) |
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 | 8619.2.a.w | 5 | |
| 13.b | even | 2 | 1 | 663.2.a.g | ✓ | 5 | |
| 39.d | odd | 2 | 1 | 1989.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.a.g | ✓ | 5 | 13.b | even | 2 | 1 | |
| 1989.2.a.l | 5 | 39.d | odd | 2 | 1 | ||
| 8619.2.a.w | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
|
\( T_{2}^{5} + 3T_{2}^{4} - 4T_{2}^{3} - 14T_{2}^{2} + 3T_{2} + 13 \)
|
|
\( T_{5}^{5} - 16T_{5}^{3} - 16T_{5}^{2} + 16T_{5} + 16 \)
|
|
\( T_{7}^{5} + 2T_{7}^{4} - 10T_{7}^{3} - 28T_{7}^{2} - 20T_{7} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 13 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} - 16 T^{3} + \cdots + 16 \)
|
| $7$ |
\( T^{5} + 2 T^{4} + \cdots - 4 \)
|
| $11$ |
\( T^{5} + 10 T^{4} + \cdots - 688 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( (T + 1)^{5} \)
|
| $19$ |
\( T^{5} - 8 T^{4} + \cdots - 3584 \)
|
| $23$ |
\( T^{5} - 4 T^{4} + \cdots - 976 \)
|
| $29$ |
\( T^{5} - 18 T^{4} + \cdots + 2128 \)
|
| $31$ |
\( T^{5} - 2 T^{4} + \cdots + 964 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots - 4 \)
|
| $41$ |
\( T^{5} + 8 T^{4} + \cdots + 1424 \)
|
| $43$ |
\( T^{5} - 8 T^{4} + \cdots + 976 \)
|
| $47$ |
\( T^{5} - 82 T^{3} + \cdots + 28 \)
|
| $53$ |
\( T^{5} - 14 T^{4} + \cdots - 2336 \)
|
| $59$ |
\( T^{5} + 12 T^{4} + \cdots - 788 \)
|
| $61$ |
\( T^{5} + 18 T^{4} + \cdots - 1504 \)
|
| $67$ |
\( T^{5} + 8 T^{4} + \cdots - 128 \)
|
| $71$ |
\( T^{5} - 14 T^{4} + \cdots - 190288 \)
|
| $73$ |
\( T^{5} - 154 T^{3} + \cdots + 8252 \)
|
| $79$ |
\( T^{5} - 4 T^{4} + \cdots - 60544 \)
|
| $83$ |
\( T^{5} + 8 T^{4} + \cdots + 143756 \)
|
| $89$ |
\( T^{5} - 30 T^{4} + \cdots + 54236 \)
|
| $97$ |
\( T^{5} + 12 T^{4} + \cdots - 1516 \)
|
| show more | |
| show less |