Function: mfeisenstein
Section: modular_forms
C-Name: mfeisenstein
Prototype: LDGDG
Help: mfeisenstein(k,{CHI1},{CHI2}): create the Eisenstein
E_k(CHI1,CHI2), where an omitted character is considered as trivial.
Doc: create the Eisenstein series $E_k(\chi_1,\chi_2)$, where $k \geq 1$,
$\chi_i$ are Dirichlet characters and an omitted character is considered as
trivial.
\bprog
? CHI = Mod(3,4);
? E = mfeisenstein(3, CHI);
? mfcoefs(E, 6)
%2 = [-1/4, 1, 1, -8, 1, 26, -8]
? CHI2 = Mod(4,5);
? mfcoefs(mfeisenstein(3,CHI,CHI2), 6)
%3 = [0, 1, -1, -10, 1, 25, 10]
? mfcoefs(mfeisenstein(4,CHI,CHI), 6)
%4 = [0, 1, 0, -28, 0, 126, 0]
? mfcoefs(mfeisenstein(4), 6)
%5 = [1/240, 1, 9, 28, 73, 126, 252]
@eprog\noindent Note that \kbd{meisenstein}$(k)$ is 0 for $k$ odd and
$-B_{k}/(2k) \cdot E_k$ for $k$ even, where
$$E_k(q) = 1 - (2k/B_k)\sum_{n\geq 1} \sigma_{k-1}(n) q^n$$
is the standard Eisenstein series. In other words it is normalized so that its
linear coefficient is $1$.