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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 728x^{2} - 525x + 107446 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{8} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 728x^{2} - 525x + 107446 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 11\nu^{2} + 923\nu - 3224 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 59\nu^{2} + 179\nu - 23051 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{2} - 135\nu - 9828 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 162 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 25\beta_{3} + 5\beta_{2} + 10\beta _1 + 58973 ) / 162 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -108\beta_{3} + 163\beta_{2} - 79\beta _1 + 21223 ) / 54 \)
|
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 | −494.214 | 0 | −1296.48 | 512.000 | 0 | −3953.72 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | 0 | 64.0000 | −94.3532 | 0 | 202.733 | 512.000 | 0 | −754.826 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 0 | 64.0000 | 206.504 | 0 | 1617.21 | 512.000 | 0 | 1652.03 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 0 | 64.0000 | 436.064 | 0 | −479.459 | 512.000 | 0 | 3488.51 | |||||||||||||||||||||||||||||||
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.i | 4 | |
| 3.b | odd | 2 | 1 | 162.8.a.h | 4 | ||
| 9.c | even | 3 | 2 | 54.8.c.b | 8 | ||
| 9.d | odd | 6 | 2 | 18.8.c.b | ✓ | 8 | |
| 36.f | odd | 6 | 2 | 432.8.i.b | 8 | ||
| 36.h | even | 6 | 2 | 144.8.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.8.c.b | ✓ | 8 | 9.d | odd | 6 | 2 | |
| 54.8.c.b | 8 | 9.c | even | 3 | 2 | ||
| 144.8.i.b | 8 | 36.h | even | 6 | 2 | ||
| 162.8.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 162.8.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 432.8.i.b | 8 | 36.f | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 54T_{5}^{3} - 241515T_{5}^{2} + 23036400T_{5} + 4199040000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 4199040000 \)
|
| $7$ |
\( T^{4} + \cdots + 203801922484 \)
|
| $11$ |
\( T^{4} + \cdots + 350509866172425 \)
|
| $13$ |
\( T^{4} + \cdots + 5285891390452 \)
|
| $17$ |
\( T^{4} + \cdots - 21\!\cdots\!28 \)
|
| $19$ |
\( T^{4} + \cdots - 57\!\cdots\!48 \)
|
| $23$ |
\( T^{4} + \cdots - 19\!\cdots\!60 \)
|
| $29$ |
\( T^{4} + \cdots - 13\!\cdots\!76 \)
|
| $31$ |
\( T^{4} + \cdots - 10\!\cdots\!20 \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!48 \)
|
| $41$ |
\( T^{4} + \cdots - 78\!\cdots\!19 \)
|
| $43$ |
\( T^{4} + \cdots - 38\!\cdots\!59 \)
|
| $47$ |
\( T^{4} + \cdots - 12\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 32\!\cdots\!80 \)
|
| $59$ |
\( T^{4} + \cdots + 27\!\cdots\!53 \)
|
| $61$ |
\( T^{4} + \cdots - 63\!\cdots\!68 \)
|
| $67$ |
\( T^{4} + \cdots + 27\!\cdots\!85 \)
|
| $71$ |
\( T^{4} + \cdots + 10\!\cdots\!28 \)
|
| $73$ |
\( T^{4} + \cdots + 17\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 25\!\cdots\!08 \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!08 \)
|
| $89$ |
\( T^{4} + \cdots - 25\!\cdots\!80 \)
|
| $97$ |
\( T^{4} + \cdots - 54\!\cdots\!75 \)
|
| show more | |
| show less |