Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(9.55830942093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{105}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 26 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{105})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
2.00000 | 0 | 4.00000 | −10.8704 | 0 | 24.8704 | 8.00000 | 0 | −21.7409 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 19.8704 | 0 | −5.87043 | 8.00000 | 0 | 39.7409 | |||||||||||||||||||||||||
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 | 162.4.a.g | 2 | |
| 3.b | odd | 2 | 1 | 162.4.a.f | 2 | ||
| 4.b | odd | 2 | 1 | 1296.4.a.r | 2 | ||
| 9.c | even | 3 | 2 | 54.4.c.b | 4 | ||
| 9.d | odd | 6 | 2 | 18.4.c.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1296.4.a.l | 2 | ||
| 36.f | odd | 6 | 2 | 432.4.i.b | 4 | ||
| 36.h | even | 6 | 2 | 144.4.i.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.4.c.b | ✓ | 4 | 9.d | odd | 6 | 2 | |
| 54.4.c.b | 4 | 9.c | even | 3 | 2 | ||
| 144.4.i.b | 4 | 36.h | even | 6 | 2 | ||
| 162.4.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 162.4.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 432.4.i.b | 4 | 36.f | odd | 6 | 2 | ||
| 1296.4.a.l | 2 | 12.b | even | 2 | 1 | ||
| 1296.4.a.r | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9T_{5} - 216 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 9T - 216 \)
|
| $7$ |
\( T^{2} - 19T - 146 \)
|
| $11$ |
\( T^{2} - 24T - 801 \)
|
| $13$ |
\( T^{2} - 61T + 694 \)
|
| $17$ |
\( T^{2} + 3T - 234 \)
|
| $19$ |
\( T^{2} - 133T - 1484 \)
|
| $23$ |
\( T^{2} + 69T + 954 \)
|
| $29$ |
\( T^{2} + 237T + 2466 \)
|
| $31$ |
\( T^{2} - 211T + 9004 \)
|
| $37$ |
\( T^{2} - 262T - 29144 \)
|
| $41$ |
\( T^{2} + 468T + 53811 \)
|
| $43$ |
\( T^{2} + 86T - 134231 \)
|
| $47$ |
\( T^{2} + 483T + 5166 \)
|
| $53$ |
\( T^{2} + 150T - 40680 \)
|
| $59$ |
\( T^{2} + 168T + 6111 \)
|
| $61$ |
\( T^{2} + 1049 T + 221944 \)
|
| $67$ |
\( T^{2} + 1166 T + 305869 \)
|
| $71$ |
\( T^{2} - 312T - 217584 \)
|
| $73$ |
\( T^{2} + 311T - 80006 \)
|
| $79$ |
\( T^{2} - 349T - 9476 \)
|
| $83$ |
\( T^{2} + 1221 T + 268524 \)
|
| $89$ |
\( T^{2} - 492T - 317484 \)
|
| $97$ |
\( T^{2} + 128T - 110249 \)
|
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