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 f was created using EisensteinSeries.
The data <chi, psi, t, chi', psi'> that defines the Eisenstein series f. Here chi is a primitive character of conductor S, psiis primitive of conductor M, and MSt divides N, where N is the level of f. (The additional characters chi' and psi' are equal to chi and psirespectively, except they take values in the big field Q(zeta_(varphi(N)))^ * instead of Q(zeta_(n))^ *, where n is the order of chi or psi.) The Eisenstein series associated to (chi, psi, t) has q-expansion c_0 + sum_(m >= 1) (sum_(n|m)psi(n)n^(k - 1)chi(m/n))q^(mt), where c_0=0 if S>1 and c_0=L(1 - k, psi)/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