In "Modular Forms and Fermat's Last Theorem" Chp.1, p.7 Glenn Stevens talks about the character of a weight 2 newform $\epsilon$ in relation to the characteristic polynomial of a Frobenius element. However he does not define or show where the value comes from.
Also Ken Ribet in the following paper, http://modular.math.washington.edu/edu/Fall2003/252/lectures/11-12-03/11-12-03.pdf on newforms introduces $\epsilon$ on p.2 but again without any reference to its origin.
I would like to understand what the character of the newform represents.
See Chapter 9 of William Stein's book, in particular Definition 9.3.