Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5577,2,Mod(1,5577)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5577, 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("5577.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5577 = 3 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5577.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: | \(44.5325692073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.233489.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + x^{2} + 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + x^{2} + 5x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{4} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 9 \)
|
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 |
|
−1.80262 | 1.00000 | 1.24943 | 2.90617 | −1.80262 | −1.40886 | 1.35298 | 1.00000 | −5.23871 | |||||||||||||||||||||||||||||||||
| 1.2 | 0.118037 | 1.00000 | −1.98607 | 2.39753 | 0.118037 | 2.97027 | −0.470505 | 1.00000 | 0.282997 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.798005 | 1.00000 | −1.36319 | −1.11866 | 0.798005 | −0.789412 | −2.68384 | 1.00000 | −0.892692 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.16000 | 1.00000 | 2.66560 | −2.38790 | 2.16000 | 2.35647 | 1.43770 | 1.00000 | −5.15787 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.72658 | 1.00000 | 5.43422 | 2.20286 | 2.72658 | −0.128461 | 9.36366 | 1.00000 | 6.00627 | ||||||||||||||||||||||||||||||||||
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 | 5577.2.a.w | 5 | |
| 13.b | even | 2 | 1 | 5577.2.a.n | 5 | ||
| 13.e | even | 6 | 2 | 429.2.i.f | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.2.i.f | ✓ | 10 | 13.e | even | 6 | 2 | |
| 5577.2.a.n | 5 | 13.b | even | 2 | 1 | ||
| 5577.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):
|
\( T_{2}^{5} - 4T_{2}^{4} + 13T_{2}^{2} - 10T_{2} + 1 \)
|
|
\( T_{5}^{5} - 4T_{5}^{4} - 5T_{5}^{3} + 30T_{5}^{2} - 4T_{5} - 41 \)
|
|
\( T_{7}^{5} - 3T_{7}^{4} - 4T_{7}^{3} + 9T_{7}^{2} + 9T_{7} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 41 \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots + 1 \)
|
| $11$ |
\( (T + 1)^{5} \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} - 9 T^{4} + \cdots + 1723 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots + 9 \)
|
| $23$ |
\( T^{5} + 9 T^{4} + \cdots + 361 \)
|
| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 171 \)
|
| $31$ |
\( T^{5} - 6 T^{4} + \cdots - 3087 \)
|
| $37$ |
\( T^{5} - 17 T^{4} + \cdots + 53 \)
|
| $41$ |
\( T^{5} - 6 T^{4} + \cdots - 81 \)
|
| $43$ |
\( T^{5} - 13 T^{4} + \cdots - 6053 \)
|
| $47$ |
\( T^{5} - 16 T^{4} + \cdots - 36279 \)
|
| $53$ |
\( T^{5} + 13 T^{4} + \cdots + 361 \)
|
| $59$ |
\( T^{5} - 8 T^{4} + \cdots - 3221 \)
|
| $61$ |
\( T^{5} - 4 T^{4} + \cdots - 353 \)
|
| $67$ |
\( T^{5} - 5 T^{4} + \cdots - 45907 \)
|
| $71$ |
\( T^{5} - 19 T^{4} + \cdots + 26117 \)
|
| $73$ |
\( T^{5} - 2 T^{4} + \cdots - 14879 \)
|
| $79$ |
\( T^{5} + 8 T^{4} + \cdots - 7457 \)
|
| $83$ |
\( T^{5} - 10 T^{4} + \cdots - 198059 \)
|
| $89$ |
\( T^{5} - 32 T^{4} + \cdots - 7 \)
|
| $97$ |
\( T^{5} + 5 T^{4} + \cdots - 5657 \)
|
| show more | |
| show less |