Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,8,Mod(1,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.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: | \(144.321881774\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 265159x^{3} - 2710480x^{2} + 11336017300x - 263206586000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 265159x^{3} - 2710480x^{2} + 11336017300x - 263206586000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2043\nu^{4} + 339336\nu^{3} + 354426937\nu^{2} - 46760278110\nu + 707097634900 ) / 2715929300 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 337\nu^{4} + 114946\nu^{3} - 73523973\nu^{2} - 22679384590\nu + 1160509814500 ) / 387989900 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4629\nu^{4} - 427582\nu^{3} + 1210245441\nu^{2} + 102273278730\nu - 39344829693300 ) / 2715929300 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 9\beta_{4} + 9\beta_{3} - 10\beta_{2} + 15\beta _1 + 106064 \)
|
| \(\nu^{3}\) | \(=\) |
\( 793\beta_{4} + 3063\beta_{3} + 1740\beta_{2} + 179139\beta _1 + 1873248 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1693067\beta_{4} + 2070107\beta_{3} - 2775210\beta_{2} + 9468643\beta _1 + 19057609772 \)
|
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 | 27.0000 | 64.0000 | −464.868 | −216.000 | 343.000 | −512.000 | 729.000 | 3718.94 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 27.0000 | 64.0000 | −276.131 | −216.000 | 343.000 | −512.000 | 729.000 | 2209.05 | ||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 27.0000 | 64.0000 | 3.66176 | −216.000 | 343.000 | −512.000 | 729.000 | −29.2941 | ||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 27.0000 | 64.0000 | 186.436 | −216.000 | 343.000 | −512.000 | 729.000 | −1491.49 | ||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | 27.0000 | 64.0000 | 452.901 | −216.000 | 343.000 | −512.000 | 729.000 | −3623.21 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 462.8.a.k | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.k | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 98T_{5}^{4} - 261319T_{5}^{3} - 18544820T_{5}^{2} + 10910143300T_{5} - 39688824000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{5} \)
|
| $3$ |
\( (T - 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 39688824000 \)
|
| $7$ |
\( (T - 343)^{5} \)
|
| $11$ |
\( (T + 1331)^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 14\!\cdots\!08 \)
|
| $17$ |
\( T^{5} + \cdots + 22\!\cdots\!20 \)
|
| $19$ |
\( T^{5} + \cdots + 25\!\cdots\!80 \)
|
| $23$ |
\( T^{5} + \cdots + 88\!\cdots\!04 \)
|
| $29$ |
\( T^{5} + \cdots + 33\!\cdots\!68 \)
|
| $31$ |
\( T^{5} + \cdots + 68\!\cdots\!20 \)
|
| $37$ |
\( T^{5} + \cdots + 34\!\cdots\!64 \)
|
| $41$ |
\( T^{5} + \cdots - 27\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 27\!\cdots\!92 \)
|
| $47$ |
\( T^{5} + \cdots + 47\!\cdots\!40 \)
|
| $53$ |
\( T^{5} + \cdots + 27\!\cdots\!20 \)
|
| $59$ |
\( T^{5} + \cdots + 35\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 34\!\cdots\!80 \)
|
| $67$ |
\( T^{5} + \cdots + 37\!\cdots\!08 \)
|
| $71$ |
\( T^{5} + \cdots + 39\!\cdots\!48 \)
|
| $73$ |
\( T^{5} + \cdots + 10\!\cdots\!44 \)
|
| $79$ |
\( T^{5} + \cdots - 43\!\cdots\!20 \)
|
| $83$ |
\( T^{5} + \cdots - 74\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots + 61\!\cdots\!80 \)
|
| $97$ |
\( T^{5} + \cdots - 40\!\cdots\!40 \)
|
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