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: | \(1\) |
| 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: |
\( x^{4} - 157085x^{2} - 578808x + 3535970020 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 157085x^{2} - 578808x + 3535970020 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 102\nu^{2} - 111835\nu - 8446358 ) / 1834 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{3} + 1324\nu^{2} + 551839\nu - 101819740 ) / 20174 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 11\beta_{3} + 5\beta_{2} + 4\beta _1 + 78545 \)
|
| \(\nu^{3}\) | \(=\) |
\( -1122\beta_{3} + 1324\beta_{2} + 111427\beta _1 + 434768 \)
|
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 | −366.130 | 216.000 | −343.000 | −512.000 | 729.000 | 2929.04 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −27.0000 | 64.0000 | −165.244 | 216.000 | −343.000 | −512.000 | 729.000 | 1321.95 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −27.0000 | 64.0000 | 164.887 | 216.000 | −343.000 | −512.000 | 729.000 | −1319.09 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | −27.0000 | 64.0000 | 354.488 | 216.000 | −343.000 | −512.000 | 729.000 | −2835.90 | |||||||||||||||||||||||||||||||
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.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 12T_{5}^{3} - 157031T_{5}^{2} - 363594T_{5} + 3536292760 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( (T + 27)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 3536292760 \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( (T - 1331)^{4} \)
|
| $13$ |
\( T^{4} + \cdots + 173840600054956 \)
|
| $17$ |
\( T^{4} + \cdots - 12\!\cdots\!64 \)
|
| $19$ |
\( T^{4} + \cdots + 61\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!60 \)
|
| $29$ |
\( T^{4} + \cdots - 16\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!08 \)
|
| $37$ |
\( T^{4} + \cdots + 36\!\cdots\!64 \)
|
| $41$ |
\( T^{4} + \cdots + 19\!\cdots\!36 \)
|
| $43$ |
\( T^{4} + \cdots - 21\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots - 42\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 55\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots + 39\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 18\!\cdots\!52 \)
|
| $73$ |
\( T^{4} + \cdots - 17\!\cdots\!60 \)
|
| $79$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 88\!\cdots\!12 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots - 15\!\cdots\!60 \)
|
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