Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(83.4358054585\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 2x^{3} - 953x^{2} + 954x + 195702 \)
|
| 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
\( x^{4} - 2x^{3} - 953x^{2} + 954x + 195702 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -24\nu^{3} + 42\nu^{2} + 15138\nu - 10440 ) / 103 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 12\nu^{3} - 48\nu^{2} - 1980\nu + 15318 ) / 103 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -81\nu^{2} + 81\nu + 38637 ) / 103 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 6\beta_{2} + 3\beta _1 + 162 ) / 324 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -138\beta_{3} + 2\beta_{2} + \beta _1 + 51570 ) / 108 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -662\beta_{3} + 1265\beta_{2} + 169\beta _1 + 77328 ) / 108 \)
|
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 |
|
16.0000 | 0 | 256.000 | −1985.27 | 0 | 3496.33 | 4096.00 | 0 | −31764.3 | ||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | 0 | 256.000 | −1914.12 | 0 | −49.9798 | 4096.00 | 0 | −30626.0 | |||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 0 | 256.000 | 637.540 | 0 | 3781.54 | 4096.00 | 0 | 10200.6 | |||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 0 | 256.000 | 1293.86 | 0 | −2731.90 | 4096.00 | 0 | 20701.7 | |||||||||||||||||||||||||||||||
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 | 162.10.a.g | yes | 4 |
| 3.b | odd | 2 | 1 | 162.10.a.f | ✓ | 4 | |
| 9.c | even | 3 | 2 | 162.10.c.s | 8 | ||
| 9.d | odd | 6 | 2 | 162.10.c.t | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.10.a.f | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 162.10.a.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 162.10.c.s | 8 | 9.c | even | 3 | 2 | ||
| 162.10.c.t | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 1968T_{5}^{3} - 2906334T_{5}^{2} - 4122858960T_{5} + 3134607030225 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 3134607030225 \)
|
| $7$ |
\( T^{4} + \cdots + 1805264053264 \)
|
| $11$ |
\( T^{4} + \cdots + 71\!\cdots\!48 \)
|
| $13$ |
\( T^{4} + \cdots + 40\!\cdots\!17 \)
|
| $17$ |
\( T^{4} + \cdots - 66\!\cdots\!59 \)
|
| $19$ |
\( T^{4} + \cdots + 56\!\cdots\!24 \)
|
| $23$ |
\( T^{4} + \cdots - 12\!\cdots\!76 \)
|
| $29$ |
\( T^{4} + \cdots + 10\!\cdots\!97 \)
|
| $31$ |
\( T^{4} + \cdots + 68\!\cdots\!68 \)
|
| $37$ |
\( T^{4} + \cdots + 18\!\cdots\!21 \)
|
| $41$ |
\( T^{4} + \cdots - 27\!\cdots\!48 \)
|
| $43$ |
\( T^{4} + \cdots - 30\!\cdots\!88 \)
|
| $47$ |
\( T^{4} + \cdots - 15\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 13\!\cdots\!47 \)
|
| $67$ |
\( T^{4} + \cdots - 18\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 13\!\cdots\!48 \)
|
| $73$ |
\( T^{4} + \cdots - 25\!\cdots\!99 \)
|
| $79$ |
\( T^{4} + \cdots - 35\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 35\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!01 \)
|
| $97$ |
\( T^{4} + \cdots - 13\!\cdots\!32 \)
|
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