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: | \(4\) |
| Coefficient field: | 4.4.43103376.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{8} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{3} + \nu^{2} - 773\nu - 960 ) / 26 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -16\nu^{3} + 23\nu^{2} + 5873\nu - 2748 ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{2} - 27\nu - 5184 ) / 26 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 4\beta _1 + 54 ) / 108 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 103\beta_{3} + \beta_{2} + 4\beta _1 + 20790 ) / 108 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -219\beta_{3} + 193\beta_{2} + 1474\beta _1 + 31158 ) / 108 \)
|
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 | −471.679 | 0 | 735.308 | −512.000 | 0 | 3773.43 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 0 | 64.0000 | −325.500 | 0 | −1450.45 | −512.000 | 0 | 2604.00 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 0 | 64.0000 | −11.2463 | 0 | 1439.47 | −512.000 | 0 | 89.9705 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 0 | 64.0000 | 280.425 | 0 | −164.323 | −512.000 | 0 | −2243.40 | |||||||||||||||||||||||||||||||
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.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 162.8.a.j | yes | 4 | |
| 9.c | even | 3 | 2 | 162.8.c.r | 8 | ||
| 9.d | odd | 6 | 2 | 162.8.c.q | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.8.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 162.8.a.j | yes | 4 | 3.b | odd | 2 | 1 | |
| 162.8.c.q | 8 | 9.d | odd | 6 | 2 | ||
| 162.8.c.r | 8 | 9.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 528T_{5}^{3} - 64206T_{5}^{2} - 43841520T_{5} - 484199775 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 528 T^{3} + \cdots - 484199775 \)
|
| $7$ |
\( T^{4} + \cdots + 252275377936 \)
|
| $11$ |
\( T^{4} + \cdots + 267055051594896 \)
|
| $13$ |
\( T^{4} + \cdots + 14\!\cdots\!09 \)
|
| $17$ |
\( T^{4} + \cdots - 16\!\cdots\!11 \)
|
| $19$ |
\( T^{4} + \cdots + 66\!\cdots\!04 \)
|
| $23$ |
\( T^{4} + \cdots - 45\!\cdots\!16 \)
|
| $29$ |
\( T^{4} + \cdots + 33\!\cdots\!81 \)
|
| $31$ |
\( T^{4} + \cdots - 83\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots - 16\!\cdots\!39 \)
|
| $41$ |
\( T^{4} + \cdots + 61\!\cdots\!16 \)
|
| $43$ |
\( T^{4} + \cdots - 19\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 41\!\cdots\!36 \)
|
| $59$ |
\( T^{4} + \cdots + 76\!\cdots\!24 \)
|
| $61$ |
\( T^{4} + \cdots - 56\!\cdots\!11 \)
|
| $67$ |
\( T^{4} + \cdots + 47\!\cdots\!16 \)
|
| $71$ |
\( T^{4} + \cdots + 48\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!49 \)
|
| $79$ |
\( T^{4} + \cdots - 64\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 11\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 33\!\cdots\!81 \)
|
| $97$ |
\( T^{4} + \cdots - 15\!\cdots\!36 \)
|
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