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: |
\( x^{5} - 2x^{4} - 240311x^{3} - 13732152x^{2} + 10881177516x + 498918562560 \)
|
| 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
\( x^{5} - 2x^{4} - 240311x^{3} - 13732152x^{2} + 10881177516x + 498918562560 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 193\nu^{4} - 12197\nu^{3} - 9900281\nu^{2} - 3060508203\nu - 1725564403278 ) / 299849193 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 269\nu^{4} - 90982\nu^{3} - 37769017\nu^{2} + 8943292284\nu + 524520391500 ) / 142785330 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2123\nu^{4} - 134167\nu^{3} - 408752284\nu^{2} - 7578710442\nu + 9830101122540 ) / 599698386 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{4} + 11\beta_{2} + 87\beta _1 + 96086 \)
|
| \(\nu^{3}\) | \(=\) |
\( 648\beta_{4} - 1930\beta_{3} + 2085\beta_{2} + 150355\beta _1 + 8466690 \)
|
| \(\nu^{4}\) | \(=\) |
\( -61642\beta_{4} - 121970\beta_{3} + 2249653\beta_{2} + 29822345\beta _1 + 14404720318 \)
|
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 | −480.004 | 216.000 | −343.000 | 512.000 | 729.000 | −3840.04 | |||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | 27.0000 | 64.0000 | −245.457 | 216.000 | −343.000 | 512.000 | 729.000 | −1963.66 | ||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 27.0000 | 64.0000 | 29.2964 | 216.000 | −343.000 | 512.000 | 729.000 | 234.371 | ||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 27.0000 | 64.0000 | 276.595 | 216.000 | −343.000 | 512.000 | 729.000 | 2212.76 | ||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | 27.0000 | 64.0000 | 337.570 | 216.000 | −343.000 | 512.000 | 729.000 | 2700.56 | ||||||||||||||||||||||||||||||||||
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.p | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.p | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 82T_{5}^{4} - 237623T_{5}^{3} + 2241256T_{5}^{2} + 11136407980T_{5} - 322287425600 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{5} \)
|
| $3$ |
\( (T - 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 322287425600 \)
|
| $7$ |
\( (T + 343)^{5} \)
|
| $11$ |
\( (T + 1331)^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 21\!\cdots\!40 \)
|
| $17$ |
\( T^{5} + \cdots - 15\!\cdots\!16 \)
|
| $19$ |
\( T^{5} + \cdots - 21\!\cdots\!20 \)
|
| $23$ |
\( T^{5} + \cdots - 17\!\cdots\!08 \)
|
| $29$ |
\( T^{5} + \cdots - 16\!\cdots\!36 \)
|
| $31$ |
\( T^{5} + \cdots - 22\!\cdots\!56 \)
|
| $37$ |
\( T^{5} + \cdots - 44\!\cdots\!96 \)
|
| $41$ |
\( T^{5} + \cdots - 81\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 42\!\cdots\!08 \)
|
| $47$ |
\( T^{5} + \cdots - 26\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots - 77\!\cdots\!72 \)
|
| $59$ |
\( T^{5} + \cdots + 66\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 15\!\cdots\!28 \)
|
| $67$ |
\( T^{5} + \cdots - 63\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 87\!\cdots\!12 \)
|
| $79$ |
\( T^{5} + \cdots + 63\!\cdots\!60 \)
|
| $83$ |
\( T^{5} + \cdots - 39\!\cdots\!64 \)
|
| $89$ |
\( T^{5} + \cdots + 77\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!52 \)
|
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