Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [159,2,Mod(1,159)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(159, 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("159.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 159 = 3 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 159.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: | \(1.26962139214\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1054013.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 9x^{2} + x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{5} - x^{4} - 7x^{3} + 9x^{2} + x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 2\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 7\nu^{2} - 8\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} - 9\beta _1 + 18 \)
|
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 |
|
−2.68424 | −1.00000 | 5.20516 | 0.0297788 | 2.68424 | −2.76971 | −8.60342 | 1.00000 | −0.0799336 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.58750 | −1.00000 | 0.520161 | −4.04441 | 1.58750 | 4.18825 | 2.34925 | 1.00000 | 6.42051 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.232991 | −1.00000 | −1.94572 | 2.56042 | 0.232991 | 2.83982 | 0.919314 | 1.00000 | −0.596554 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.05991 | −1.00000 | 2.24323 | 3.37557 | −2.05991 | −3.91662 | 0.501021 | 1.00000 | 6.95337 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.44483 | −1.00000 | 3.97717 | −1.92136 | −2.44483 | 3.65826 | 4.83384 | 1.00000 | −4.69739 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(53\) | \(-1\) |
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 | 159.2.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 477.2.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 2544.2.a.w | 5 | ||
| 5.b | even | 2 | 1 | 3975.2.a.x | 5 | ||
| 7.b | odd | 2 | 1 | 7791.2.a.be | 5 | ||
| 12.b | even | 2 | 1 | 7632.2.a.bw | 5 | ||
| 53.b | even | 2 | 1 | 8427.2.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 159.2.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 477.2.a.f | 5 | 3.b | odd | 2 | 1 | ||
| 2544.2.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 3975.2.a.x | 5 | 5.b | even | 2 | 1 | ||
| 7632.2.a.bw | 5 | 12.b | even | 2 | 1 | ||
| 7791.2.a.be | 5 | 7.b | odd | 2 | 1 | ||
| 8427.2.a.j | 5 | 53.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 10T_{2}^{3} + 22T_{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(159))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 10 T^{3} + \cdots + 5 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( T^{5} - 19 T^{3} + \cdots - 2 \)
|
| $7$ |
\( T^{5} - 4 T^{4} + \cdots - 472 \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots - 64 \)
|
| $13$ |
\( T^{5} - 8 T^{4} + \cdots - 110 \)
|
| $17$ |
\( T^{5} - 40 T^{3} + \cdots - 160 \)
|
| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 64 \)
|
| $23$ |
\( T^{5} + 6 T^{4} + \cdots + 272 \)
|
| $29$ |
\( T^{5} - 20 T^{4} + \cdots + 1504 \)
|
| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 128 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots + 34 \)
|
| $41$ |
\( T^{5} - 18 T^{4} + \cdots + 3474 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 3004 \)
|
| $47$ |
\( T^{5} + 4 T^{4} + \cdots + 2048 \)
|
| $53$ |
\( (T - 1)^{5} \)
|
| $59$ |
\( T^{5} + 18 T^{4} + \cdots - 11968 \)
|
| $61$ |
\( T^{5} - 12 T^{4} + \cdots + 18272 \)
|
| $67$ |
\( T^{5} - 6 T^{4} + \cdots - 34240 \)
|
| $71$ |
\( T^{5} - 4 T^{4} + \cdots - 24992 \)
|
| $73$ |
\( T^{5} - 8 T^{4} + \cdots - 8800 \)
|
| $79$ |
\( T^{5} + 2 T^{4} + \cdots - 1408 \)
|
| $83$ |
\( T^{5} + 36 T^{4} + \cdots - 128420 \)
|
| $89$ |
\( T^{5} - 26 T^{4} + \cdots + 41504 \)
|
| $97$ |
\( T^{5} + 16 T^{4} + \cdots + 148286 \)
|
| show more | |
| show less |