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.
> 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