  We compute the Fourier expansion of vector valued Eisenstein series for the
Weil representation associated to an even lattice. To this end, we define
certain twists by Dirichlet characters of the usual Eisenstein series
associated to isotropic elements in the discriminant form of the underlying
lattice. These twisted functions still form a generating system for the space
of Eisenstein series but have better multiplicative properties than the
individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain
algebraic formulas for the Fourier coefficients of the twisted Eisenstein
series in terms of special values of Dirichlet $L$-functions and representation
numbers modulo prime powers of the underlying lattice. In particular, we obtain
that the Fourier coefficients of the individual Eisenstein series are rational
numbers. Additionally, we show that the twisted Eisenstein series are
eigenforms of the Hecke operators on vector valued modular forms introduced by
Bruinier and Stein.
