On a formal analogue of the Bernoulli numbers (Q798354)

For integral \(k\geq 2\) and \(Im(z)>0\) the modular form of weight 2k is given by \(G_ k(z)=\sum (r+sz)^{-2k},\) where the summation is over all pairs of nonzero integers r and s. When z is an integer in an imaginary quadratic field, say \(z=f\omega,\) where f is an integer \(\geq 1\) and [1,\(\omega]\) is a basis for the integers of the field, it is shown that \(G_ k(z)=\alpha_ k\omega^{2k}\), where the \(\alpha_ k\) are algebraic numbers. They can be computed recursively and involve analogues of the Bernoulli numbers.