Periods of second kind differentials of (n,s)-curves

We discuss the following problem, which is solved in particular cases: Consider a curve C of genus g>1 and its a- and b-periods of holomorphic differentials 2 \omega, 2 \omega'. Let 2 \eta, 2 \eta' be the periods of the differentials of the second kind conjugated to 2 \omega, 2 \omega', according to the generalized Legendre relations \eta^T \omega = \omega^T \eta, \eta^T \omega' - \omega^T \eta' = i\pi/2, {\eta'}^T \omega' = {\omega'}^T \eta'. Express the periods of the differentials of the second kind in terms of the data, including a-periods, 2 \omega, \theta-constants, depending on the Riemann period matrix \tau = \omega'/\omega, and coefficients of the polynomial defining the curve C.