Inner twist group (reviewed)

For a given newform $f$, the set of pairs $(\chi,\sigma)$ giving rise to an inner twist of $f$ form a group $G$; here $\chi$ is a primitive Dirichlet character and $\sigma$, is a $\Q$-automorphism of the coefficient field $K$ of $f$.

The group operation is given by \[ (\chi,\sigma)\cdot (\chi',\sigma') = (\chi\sigma(\chi'),\sigma\sigma'). \] The projection map $(\chi,\sigma)\mapsto \sigma$ is a group homomorphism $G\to \Aut_\Q(K)$ with kernel equal to the subgroup of self_twists and abelian image.

The projection map $(\chi,\sigma)\mapsto \chi$ is injective, but it is not a group homomorphism, in general, although its restriction to the subgroup of self twists is.