# What is a cusp parameter?

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?

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The link to the paper seems to require a password. What is the paper? Otherwise can you tell us what is $\alpha$ and over which set we are summing? –  Giovanni De Gaetano May 26 '12 at 16:42
The paper is "Rankin-Selberg method for real analytic cusp forms of arbitrary real weight" by Matthes. It just says that $f$ is a holomorphic cusp form with weight $r$ and cusp parameter $0 \leq \alpha < 1$. –  Eugene May 26 '12 at 16:53
I'll just edit it into the question. –  Dylan Moreland May 26 '12 at 18:58
@DylanMoreland Oh wow. Thanks again! –  Eugene May 26 '12 at 19:04