Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2548,2,Mod(1,2548)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, 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("2548.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2548.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: | \(20.3458824350\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.39110656.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 14\nu^{2} - 34\nu - 11 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 19\nu^{2} - 29\nu + 9 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} + 27\nu^{3} - 6\nu^{2} - 46\nu - 14 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{5} + \nu^{4} + 41\nu^{3} + 2\nu^{2} - 88\nu - 27 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} - 6\beta_{3} + 9\beta_{2} + 8\beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} - 21\beta_{4} - 2\beta_{3} + 13\beta_{2} + 42\beta _1 + 22 \)
|
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 | −2.71991 | 0 | −3.85705 | 0 | 0 | 0 | 4.39789 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.57021 | 0 | 0.599047 | 0 | 0 | 0 | −0.534430 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.05913 | 0 | −4.09294 | 0 | 0 | 0 | −1.87824 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.655042 | 0 | 0.738118 | 0 | 0 | 0 | −2.57092 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.46169 | 0 | −0.327782 | 0 | 0 | 0 | −0.863457 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.23252 | 0 | −3.05939 | 0 | 0 | 0 | 7.44916 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 2548.2.a.r | ✓ | 6 |
| 7.b | odd | 2 | 1 | 2548.2.a.s | yes | 6 | |
| 7.c | even | 3 | 2 | 2548.2.j.s | 12 | ||
| 7.d | odd | 6 | 2 | 2548.2.j.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2548.2.a.r | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2548.2.a.s | yes | 6 | 7.b | odd | 2 | 1 | |
| 2548.2.j.r | 12 | 7.d | odd | 6 | 2 | ||
| 2548.2.j.s | 12 | 7.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2548))\):
|
\( T_{3}^{6} - 12T_{3}^{4} - 4T_{3}^{3} + 28T_{3}^{2} + 8T_{3} - 14 \)
|
|
\( T_{5}^{6} + 10T_{5}^{5} + 29T_{5}^{4} + 8T_{5}^{3} - 47T_{5}^{2} + 6T_{5} + 7 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 12 T^{4} + \cdots - 14 \)
|
| $5$ |
\( T^{6} + 10 T^{5} + \cdots + 7 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots - 82 \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} + 16 T^{5} + \cdots - 254 \)
|
| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 3647 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots - 4879 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 25 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots - 89 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 53918 \)
|
| $41$ |
\( T^{6} + 8 T^{5} + \cdots - 76816 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 73 \)
|
| $47$ |
\( T^{6} + 30 T^{5} + \cdots - 14481 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots - 32719 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots - 19204 \)
|
| $61$ |
\( T^{6} + 16 T^{5} + \cdots + 34552 \)
|
| $67$ |
\( T^{6} - 176 T^{4} + \cdots + 19936 \)
|
| $71$ |
\( T^{6} - 212 T^{4} + \cdots - 15842 \)
|
| $73$ |
\( T^{6} + 14 T^{5} + \cdots - 1377 \)
|
| $79$ |
\( T^{6} - 14 T^{5} + \cdots - 839 \)
|
| $83$ |
\( T^{6} - 14 T^{5} + \cdots - 38089 \)
|
| $89$ |
\( T^{6} + 30 T^{5} + \cdots + 271111 \)
|
| $97$ |
\( T^{6} + 22 T^{5} + \cdots - 233 \)
|
| show more | |
| show less |