Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(50.6063741284\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.69765.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 72x - 179 \)
|
| 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
\( x^{3} - 72x - 179 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu^{2} + 3\nu - 144 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} - 15\nu - 144 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 5\beta _1 + 864 ) / 18 \)
|
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 |
|
−8.00000 | 0 | 64.0000 | −318.162 | 0 | −452.096 | −512.000 | 0 | 2545.30 | |||||||||||||||||||||||||||
| 1.2 | −8.00000 | 0 | 64.0000 | −3.41638 | 0 | −815.637 | −512.000 | 0 | 27.3311 | ||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 0 | 64.0000 | 375.578 | 0 | 1057.73 | −512.000 | 0 | −3004.63 | ||||||||||||||||||||||||||||
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 | 162.8.a.e | 3 | |
| 3.b | odd | 2 | 1 | 162.8.a.f | 3 | ||
| 9.c | even | 3 | 2 | 54.8.c.a | 6 | ||
| 9.d | odd | 6 | 2 | 18.8.c.a | ✓ | 6 | |
| 36.f | odd | 6 | 2 | 432.8.i.a | 6 | ||
| 36.h | even | 6 | 2 | 144.8.i.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.8.c.a | ✓ | 6 | 9.d | odd | 6 | 2 | |
| 54.8.c.a | 6 | 9.c | even | 3 | 2 | ||
| 144.8.i.a | 6 | 36.h | even | 6 | 2 | ||
| 162.8.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
| 162.8.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 432.8.i.a | 6 | 36.f | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 54T_{5}^{2} - 119691T_{5} - 408240 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 54 T^{2} + \cdots - 408240 \)
|
| $7$ |
\( T^{3} + 210 T^{2} + \cdots - 390034934 \)
|
| $11$ |
\( T^{3} + \cdots + 18747598095 \)
|
| $13$ |
\( T^{3} + \cdots - 1359282566258 \)
|
| $17$ |
\( T^{3} + \cdots - 2214690411708 \)
|
| $19$ |
\( T^{3} + \cdots - 22074972070832 \)
|
| $23$ |
\( T^{3} + \cdots + 12974435986590 \)
|
| $29$ |
\( T^{3} + \cdots + 14\!\cdots\!06 \)
|
| $31$ |
\( T^{3} + \cdots - 28\!\cdots\!40 \)
|
| $37$ |
\( T^{3} + \cdots - 81\!\cdots\!16 \)
|
| $41$ |
\( T^{3} + \cdots - 40\!\cdots\!31 \)
|
| $43$ |
\( T^{3} + \cdots - 37\!\cdots\!37 \)
|
| $47$ |
\( T^{3} + \cdots + 41\!\cdots\!98 \)
|
| $53$ |
\( T^{3} + \cdots - 33\!\cdots\!40 \)
|
| $59$ |
\( T^{3} + \cdots + 30\!\cdots\!09 \)
|
| $61$ |
\( T^{3} + \cdots + 80\!\cdots\!96 \)
|
| $67$ |
\( T^{3} + \cdots - 23\!\cdots\!19 \)
|
| $71$ |
\( T^{3} + \cdots - 47\!\cdots\!44 \)
|
| $73$ |
\( T^{3} + \cdots + 84\!\cdots\!08 \)
|
| $79$ |
\( T^{3} + \cdots - 29\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots - 18\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 93\!\cdots\!81 \)
|
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