Fourier coefficients of powers of the Dedekind eta function

In this paper, we study the vanishing properties of Fourier coefficients of rth powers of the Dedekind eta function. We give a certain type of classification of this property based on r. Further we extend the results of Atkin, Cohen, and Newman for odd powers. This leads to a partial answer to a question raised by van Lint. We also indicate possible generalization of the Lehmer conjecture.