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: |
\( x^{5} - 2x^{4} - 287051x^{3} - 30693248x^{2} + 8815737676x + 817008616000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{3} \) |
| 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
\( x^{5} - 2x^{4} - 287051x^{3} - 30693248x^{2} + 8815737676x + 817008616000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 349\nu^{4} + 482079\nu^{3} - 43361732\nu^{2} - 128791803288\nu - 13072542783800 ) / 7096893300 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -29929\nu^{4} + 3395481\nu^{3} + 8310177032\nu^{2} + 4377267408\nu - 241023094180300 ) / 21290679900 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36283\nu^{4} - 6819567\nu^{3} - 8416379984\nu^{2} + 354057539964\nu + 152382154901500 ) / 21290679900 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 11\beta_{4} + 14\beta_{3} + 19\beta_{2} + 159\beta _1 + 114757 \)
|
| \(\nu^{3}\) | \(=\) |
\( -1129\beta_{4} - 961\beta_{3} + 11654\beta_{2} + 230465\beta _1 + 18668192 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2926207\beta_{4} + 3066883\beta_{3} + 6597758\beta_{2} + 70441209\beta _1 + 25928543644 \)
|
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 | −508.887 | −216.000 | −343.000 | −512.000 | 729.000 | 4071.10 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 27.0000 | 64.0000 | −128.826 | −216.000 | −343.000 | −512.000 | 729.000 | 1030.61 | ||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 27.0000 | 64.0000 | 135.208 | −216.000 | −343.000 | −512.000 | 729.000 | −1081.66 | ||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 27.0000 | 64.0000 | 276.935 | −216.000 | −343.000 | −512.000 | 729.000 | −2215.48 | ||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | 27.0000 | 64.0000 | 463.570 | −216.000 | −343.000 | −512.000 | 729.000 | −3708.56 | ||||||||||||||||||||||||||||||||||
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.l | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.8.a.l | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 238T_{5}^{4} - 264395T_{5}^{3} + 70950320T_{5}^{2} + 3910746700T_{5} - 1137945424000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{5} \)
|
| $3$ |
\( (T - 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 1137945424000 \)
|
| $7$ |
\( (T + 343)^{5} \)
|
| $11$ |
\( (T - 1331)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 29\!\cdots\!40 \)
|
| $17$ |
\( T^{5} + \cdots + 12\!\cdots\!20 \)
|
| $19$ |
\( T^{5} + \cdots - 13\!\cdots\!52 \)
|
| $23$ |
\( T^{5} + \cdots + 76\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 27\!\cdots\!84 \)
|
| $31$ |
\( T^{5} + \cdots + 10\!\cdots\!60 \)
|
| $37$ |
\( T^{5} + \cdots + 57\!\cdots\!20 \)
|
| $41$ |
\( T^{5} + \cdots + 68\!\cdots\!72 \)
|
| $43$ |
\( T^{5} + \cdots + 36\!\cdots\!60 \)
|
| $47$ |
\( T^{5} + \cdots + 99\!\cdots\!20 \)
|
| $53$ |
\( T^{5} + \cdots - 12\!\cdots\!84 \)
|
| $59$ |
\( T^{5} + \cdots + 16\!\cdots\!64 \)
|
| $61$ |
\( T^{5} + \cdots - 49\!\cdots\!40 \)
|
| $67$ |
\( T^{5} + \cdots + 33\!\cdots\!52 \)
|
| $71$ |
\( T^{5} + \cdots + 38\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 49\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 55\!\cdots\!84 \)
|
| $89$ |
\( T^{5} + \cdots + 46\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots - 16\!\cdots\!20 \)
|
| show more | |
| show less |