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?



Periods of Twists of Modular Forms
By a famous theorem of Manin, one can define $\Omega^{\pm}$ such that $L(f\otimes\chi,j)\in \Omega^{\epsilon}\_{f}\mathbb Q$ with $\chi(-1)(-1)^{j}=\epsilon$. So the period depends on $\chi$ only insofar as you need to know $\chi(-1)$ to determine if you should choose $\Omega\_f^{+}$ or $\Omega\_f^{-}$.


This result is the key property allowing one to construct the $p$-adic $L$-function of a modular form.


