  For a finite group $G$, there is a map $RO(G) \to {\rm Pic}(Sp^G)$ from the
real representation ring of $G$ to the Picard group of $G$-spectra. This map is
not known to be surjective in general, but we prove that when $G$ is cyclic
this map is indeed surjective and in that case we describe ${\rm Pic}(Sp^G)$
explicitly. We also show that for an arbitrary finite group $G$ homology and
cohomology with coefficients in a cohomological Mackey functor do not see the
part of ${\rm Pic}(Sp^G)$ coming from the Picard group of the Burnside ring.
Hence these homology and cohomology calculations can be graded on a smaller
group.
