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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?

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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.

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  • $\begingroup$ Or maybe it's already in Shimura, I don't remember now. $\endgroup$ – Olivier Sep 7 '13 at 19:21

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