Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [666,4,Mod(1,666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(666, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("666.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 666.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: | \(39.2952720638\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 96x^{2} - 287x + 330 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 74) |
| 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
\( x^{4} - 96x^{2} - 287x + 330 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} + 23\nu^{2} + 364\nu - 15 ) / 51 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 8\nu^{2} - 49\nu + 173 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -5\beta_{3} - 3\beta_{2} + 7\beta _1 + 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( -23\beta_{3} - 24\beta_{2} + 105\beta _1 + 227 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | −11.3318 | 0 | 3.28728 | −8.00000 | 0 | 22.6637 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −8.82739 | 0 | −33.6633 | −8.00000 | 0 | 17.6548 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −8.55405 | 0 | 30.9585 | −8.00000 | 0 | 17.1081 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 7.71328 | 0 | 22.4175 | −8.00000 | 0 | −15.4266 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 666.4.a.q | 4 | |
| 3.b | odd | 2 | 1 | 74.4.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 592.4.a.d | 4 | ||
| 15.d | odd | 2 | 1 | 1850.4.a.j | 4 | ||
| 24.f | even | 2 | 1 | 2368.4.a.k | 4 | ||
| 24.h | odd | 2 | 1 | 2368.4.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.4.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 592.4.a.d | 4 | 12.b | even | 2 | 1 | ||
| 666.4.a.q | 4 | 1.a | even | 1 | 1 | trivial | |
| 1850.4.a.j | 4 | 15.d | odd | 2 | 1 | ||
| 2368.4.a.h | 4 | 24.h | odd | 2 | 1 | ||
| 2368.4.a.k | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 21T_{5}^{3} + 51T_{5}^{2} - 1246T_{5} - 6600 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(666))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 21 T^{3} + \cdots - 6600 \)
|
| $7$ |
\( T^{4} - 23 T^{3} + \cdots - 76800 \)
|
| $11$ |
\( T^{4} - 66 T^{3} + \cdots - 232956 \)
|
| $13$ |
\( T^{4} - 53 T^{3} + \cdots - 243816 \)
|
| $17$ |
\( T^{4} + 12 T^{3} + \cdots + 2259504 \)
|
| $19$ |
\( T^{4} + 34 T^{3} + \cdots + 11080000 \)
|
| $23$ |
\( T^{4} + 45 T^{3} + \cdots - 43345368 \)
|
| $29$ |
\( T^{4} - 21 T^{3} + \cdots - 59893020 \)
|
| $31$ |
\( T^{4} - 17 T^{3} + \cdots - 34964464 \)
|
| $37$ |
\( (T - 37)^{4} \)
|
| $41$ |
\( T^{4} - 174 T^{3} + \cdots - 206445798 \)
|
| $43$ |
\( T^{4} + \cdots - 4525901824 \)
|
| $47$ |
\( T^{4} + \cdots + 3502908000 \)
|
| $53$ |
\( T^{4} + \cdots - 8943617448 \)
|
| $59$ |
\( T^{4} + 354 T^{3} + \cdots + 31770624 \)
|
| $61$ |
\( T^{4} + \cdots - 13276304296 \)
|
| $67$ |
\( T^{4} + \cdots - 4730048496 \)
|
| $71$ |
\( T^{4} + 27 T^{3} + \cdots + 112341888 \)
|
| $73$ |
\( T^{4} + \cdots + 61284714458 \)
|
| $79$ |
\( T^{4} + \cdots + 10190211672 \)
|
| $83$ |
\( T^{4} + \cdots - 1337884224 \)
|
| $89$ |
\( T^{4} + \cdots + 261101734464 \)
|
| $97$ |
\( T^{4} + \cdots + 5575773984 \)
|
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