Galois representation (reviewed)

As shown by Deligne and Serre [MR:0379379], every newform of weight one has an associated Galois representation $\rho\colon \Gal(\overline\Q/\Q)\to \GL_2(\C)$.

This representation corresponds to an Artin representation of dimension two whose conductor is the level $N$ of the modular form.

Conversely, every odd irreducible two-dimensional Artin representation of conductor $N$ gives rise to a modular form of weight one and level $N$.

Composing the representation $\rho$ with the natural map $\GL_2(\C)\to \PGL_2(\C)$ yields the projective Galois representation $\bar\rho\colon \Gal(\overline\Q/\Q)\to \PGL_2(\C)$.