Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,6,Mod(1,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.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: | \(25.9821788097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.125628.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 63x + 159 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 18) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 63x + 159 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{2} + 24\nu - 135 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -15\nu^{2} - 12\nu + 639 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta _1 + 18 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{2} - 2\beta _1 + 1143 ) / 27 \)
|
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 |
|
−4.00000 | 0 | 16.0000 | −78.4839 | 0 | 221.131 | −64.0000 | 0 | 313.936 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | −41.6029 | 0 | −203.321 | −64.0000 | 0 | 166.412 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 0 | 16.0000 | 66.0868 | 0 | 114.190 | −64.0000 | 0 | −264.347 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
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 | 162.6.a.i | 3 | |
| 3.b | odd | 2 | 1 | 162.6.a.j | 3 | ||
| 9.c | even | 3 | 2 | 54.6.c.b | 6 | ||
| 9.d | odd | 6 | 2 | 18.6.c.b | ✓ | 6 | |
| 36.f | odd | 6 | 2 | 432.6.i.b | 6 | ||
| 36.h | even | 6 | 2 | 144.6.i.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.6.c.b | ✓ | 6 | 9.d | odd | 6 | 2 | |
| 54.6.c.b | 6 | 9.c | even | 3 | 2 | ||
| 144.6.i.b | 6 | 36.h | even | 6 | 2 | ||
| 162.6.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 162.6.a.j | 3 | 3.b | odd | 2 | 1 | ||
| 432.6.i.b | 6 | 36.f | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 54T_{5}^{2} - 4671T_{5} - 215784 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 54 T^{2} + \cdots - 215784 \)
|
| $7$ |
\( T^{3} - 132 T^{2} + \cdots + 5134078 \)
|
| $11$ |
\( T^{3} + 315 T^{2} + \cdots - 41768163 \)
|
| $13$ |
\( T^{3} - 744 T^{2} + \cdots + 384824422 \)
|
| $17$ |
\( T^{3} + 1449 T^{2} + \cdots - 930192444 \)
|
| $19$ |
\( T^{3} - 1131 T^{2} + \cdots - 352455920 \)
|
| $23$ |
\( T^{3} + 3168 T^{2} + \cdots + 425600514 \)
|
| $29$ |
\( T^{3} + \cdots + 6505725654 \)
|
| $31$ |
\( T^{3} + \cdots + 59316561604 \)
|
| $37$ |
\( T^{3} + \cdots - 188019064016 \)
|
| $41$ |
\( T^{3} + \cdots + 2157363401913 \)
|
| $43$ |
\( T^{3} + \cdots + 513661035961 \)
|
| $47$ |
\( T^{3} + \cdots - 84596352750 \)
|
| $53$ |
\( T^{3} + \cdots - 1764512817552 \)
|
| $59$ |
\( T^{3} + \cdots + 5778215946243 \)
|
| $61$ |
\( T^{3} + \cdots - 5359653497960 \)
|
| $67$ |
\( T^{3} + \cdots + 3034793402875 \)
|
| $71$ |
\( T^{3} + \cdots + 139951336896 \)
|
| $73$ |
\( T^{3} + \cdots + 14322358753732 \)
|
| $79$ |
\( T^{3} + \cdots - 115303078320596 \)
|
| $83$ |
\( T^{3} + \cdots + 103421607911604 \)
|
| $89$ |
\( T^{3} + \cdots - 9104584153608 \)
|
| $97$ |
\( T^{3} + \cdots - 292114819729997 \)
|
| show more | |
| show less |