Twist (awaiting review)

Associated to each newform $f$ and primitive Dirichlet character $\psi$, there is a unique newform $g:=f\otimes\psi$, the twist of $f$ by $\psi$, that satisfies \[ a_n(g)=\psi(n)a_n(f) \] for all integers $n\ge 1$ coprime to $N$ and the conductor of $\psi$. If the newform $g$ is a Galois conjugate of $f$, it is said to be an inner twist.

The newform orbit $[g]$ is a twist of the newform orbit $[f]$ by the character orbit $[\psi]$ if some $g\in [g]$ is a twist of $f$ by some $\psi$ in $[\psi]$. This may occur with multiplicity.