Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1148,2,Mod(1,1148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1148, 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("1148.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1148 = 2^{2} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1148.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: | \(9.16682615204\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.470117.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 8\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 6\nu^{2} - 9\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 3\nu^{3} + 5\nu^{2} - 13\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 5\beta_{3} + 8\beta_{2} + 7\beta _1 + 17 \)
|
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 | −2.78088 | 0 | −0.592821 | 0 | −1.00000 | 0 | 4.73330 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.94177 | 0 | 2.68629 | 0 | −1.00000 | 0 | 0.770473 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.189142 | 0 | −1.51194 | 0 | −1.00000 | 0 | −2.96423 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.475832 | 0 | 1.19617 | 0 | −1.00000 | 0 | −2.77358 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.05768 | 0 | −2.77770 | 0 | −1.00000 | 0 | 1.23404 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(41\) | \(-1\) |
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 | 1148.2.a.c | ✓ | 5 |
| 4.b | odd | 2 | 1 | 4592.2.a.be | 5 | ||
| 7.b | odd | 2 | 1 | 8036.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1148.2.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 4592.2.a.be | 5 | 4.b | odd | 2 | 1 | ||
| 8036.2.a.l | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 2T_{3}^{4} - 6T_{3}^{3} - 8T_{3}^{2} + 7T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1148))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{5} + T^{4} - 9 T^{3} + \cdots + 8 \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots - 24 \)
|
| $13$ |
\( T^{5} + T^{4} + \cdots + 103 \)
|
| $17$ |
\( T^{5} + 3 T^{4} + \cdots + 463 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 699 \)
|
| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 71 \)
|
| $29$ |
\( T^{5} + 9 T^{4} + \cdots - 2360 \)
|
| $31$ |
\( T^{5} + 11 T^{4} + \cdots + 6080 \)
|
| $37$ |
\( T^{5} + 11 T^{4} + \cdots - 6092 \)
|
| $41$ |
\( (T - 1)^{5} \)
|
| $43$ |
\( T^{5} + 27 T^{4} + \cdots + 5351 \)
|
| $47$ |
\( T^{5} + 3 T^{4} + \cdots - 732 \)
|
| $53$ |
\( T^{5} + 19 T^{4} + \cdots - 11976 \)
|
| $59$ |
\( T^{5} + 15 T^{4} + \cdots - 2440 \)
|
| $61$ |
\( T^{5} - 115 T^{3} + \cdots - 1704 \)
|
| $67$ |
\( T^{5} + 21 T^{4} + \cdots + 22184 \)
|
| $71$ |
\( T^{5} + 16 T^{4} + \cdots + 456 \)
|
| $73$ |
\( T^{5} + 10 T^{4} + \cdots + 320 \)
|
| $79$ |
\( T^{5} + 14 T^{4} + \cdots + 19616 \)
|
| $83$ |
\( T^{5} + 2 T^{4} + \cdots + 50360 \)
|
| $89$ |
\( T^{5} - 6 T^{4} + \cdots - 19931 \)
|
| $97$ |
\( T^{5} - 20 T^{4} + \cdots - 365983 \)
|
| show more | |
| show less |