I was reading this paper, and on the first page they define a cusp form as
$$ f(z) = \sum_{n > -\alpha} a(n) e^{2\pi i (n + \alpha)z}. $$
Is this equivalent to the usual definition of a cusp form
$$ f(z) = \sum_{n = 1}^\infty a(n) q^n. $$ where $q = e^{2\pi iz}$?
Also what is a cusp parameter?