Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [484,6,Mod(1,484)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(484, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("484.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 484 = 2^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 484.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: | \(77.6257687895\) |
| 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: |
\( x^{4} - 130x^{2} - 228x + 2376 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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
\( x^{4} - 130x^{2} - 228x + 2376 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 12\nu^{2} - 158\nu + 438 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 24\nu^{2} + 122\nu - 1218 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 6\beta _1 + 260 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} + 6\beta_{2} + 97\beta _1 + 342 ) / 2 \)
|
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 |
|
0 | −18.2146 | 0 | 28.1845 | 0 | 164.357 | 0 | 88.7731 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −16.0276 | 0 | −105.600 | 0 | −180.291 | 0 | 13.8826 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 5.27834 | 0 | 49.5067 | 0 | −120.404 | 0 | −215.139 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 20.9639 | 0 | −40.0908 | 0 | 96.3379 | 0 | 196.483 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 484.6.a.f | ✓ | 4 |
| 11.b | odd | 2 | 1 | 484.6.a.g | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 484.6.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 484.6.a.g | yes | 4 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(484))\):
|
\( T_{3}^{4} + 8T_{3}^{3} - 496T_{3}^{2} - 3872T_{3} + 32304 \)
|
|
\( T_{7}^{4} + 40T_{7}^{3} - 40848T_{7}^{2} - 897952T_{7} + 343715504 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{3} + \cdots + 32304 \)
|
| $5$ |
\( T^{4} + 68 T^{3} + \cdots + 5907249 \)
|
| $7$ |
\( T^{4} + 40 T^{3} + \cdots + 343715504 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 1463911119 \)
|
| $17$ |
\( T^{4} + \cdots - 915685230735 \)
|
| $19$ |
\( T^{4} + \cdots + 8830395388848 \)
|
| $23$ |
\( T^{4} + \cdots + 1653552344880 \)
|
| $29$ |
\( T^{4} + \cdots + 14662354110129 \)
|
| $31$ |
\( T^{4} + \cdots + 936158818551984 \)
|
| $37$ |
\( T^{4} + \cdots + 12\!\cdots\!93 \)
|
| $41$ |
\( T^{4} + \cdots + 35\!\cdots\!01 \)
|
| $43$ |
\( T^{4} + \cdots - 50\!\cdots\!48 \)
|
| $47$ |
\( T^{4} + \cdots - 13\!\cdots\!32 \)
|
| $53$ |
\( T^{4} + \cdots - 30\!\cdots\!23 \)
|
| $59$ |
\( T^{4} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 43\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots - 12\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots - 66\!\cdots\!32 \)
|
| $73$ |
\( T^{4} + \cdots - 55\!\cdots\!28 \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots - 24\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots - 96\!\cdots\!11 \)
|
| $97$ |
\( T^{4} + \cdots - 40\!\cdots\!75 \)
|
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