Let $f \in S_2(\Gamma_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ takes algebraic values.

If $\chi$ is a Dirichlet character, one can form the twist $f_\chi$ of $f$.

How are the periods $\Omega^\pm_f$ and $\Omega^\pm_{f_\chi}$ related? In particular, are they algebraic multiples of each other?