|
The intrinsics below require that the base ring of M has
characteristic 0. To compute mod p eigenforms, use the
Reduction intrinsic (see
Section Reductions and Embeddings).
List of the Eisenstein series associated to the modular forms
space M. By "associated to" we mean that the Eisenstein
series lies in M tensor C.
Returns true if the modular form f was created using EisensteinSeries.
The data <χ, ψ, t, χ', ψ'> that defines the
Eisenstein series (modular form) f.
Here χ is a primitive character of
conductor S, ψ is primitive of conductor M,
and MSt divides N, where N is the level of f.
(The additional characters χ' and ψ' are
equal to χ and ψ respectively, except
they take values in the big field
Q(ζφ(N)) * instead
of Q(ζn) * , where n is the order of χ
or ψ.)
The Eisenstein series associated to (χ, ψ, t)
has q-expansion
c0 + ∑m≥1 (∑n|mψ(n)nk - 1χ(m/n))qmt,
where
c0=0 if S>1
and c0=L(1 - k, ψ)/2 if S=1.
We illustrate the above intrinsics by computing
the Eisenstein series in M 3(Γ 1(12)).
> M := ModularForms(Gamma1(12),3); M;
Space of modular forms on Gamma_1(12) of weight 3 and dimension 13
over Integer Ring.
> E := EisensteinSubspace(M); E;
Space of modular forms on Gamma_1(12) of weight 3 and dimension 10
over Integer Ring.
> s := EisensteinSeries(E); s;
[*
-1/9 + q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 + O(q^8),
-1/9 + q^2 - 3*q^4 + q^6 + O(q^8),
-1/9 + q^4 + O(q^8),
-1/4 + q + q^2 - 8*q^3 + q^4 + 26*q^5 - 8*q^6 - 48*q^7 + O(q^8),
-1/4 + q^3 + q^6 + O(q^8),
q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + O(q^8),
q^2 + 3*q^4 + 9*q^6 + O(q^8),
q^4 + O(q^8),
q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + O(q^8),
q^3 + 4*q^6 + O(q^8)
*]
> a := EisensteinData(s[1]); a;
<1, $.1, 1, 1, $.2>
> Parent(a[2]);
Group of Dirichlet characters of modulus 3 over Rational Field
> Order(a[2]);
2
> Parent(a[5]);
Group of Dirichlet characters of modulus 12 over Cyclotomic Field of
order 4 and degree 2
> Parent(s[1]);
Space of modular forms on Gamma_1(12) of weight 3 and dimension 10
over Rational Field.
> IsEisensteinSeries(s[1]);
true
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|