  Let $\pi$ be a unitary automorphic cuspidal representation of
$GL_2(\mathbb{Q}_\mathbb{A})$ with Fourier coefficients $\lambda_\pi(n)$.
Asymptotic expansions of certain sums of $\lambda_\pi(n)$ are proved using
known functorial liftings from $GL_2$, including symmetric powers, isobaric
sums, exterior square from $GL_4$ and base change. These asymptotic expansions
are manifestation of the underlying functoriality and reflect value
distribution of $\lambda_\pi(n)$ on integers, squares, cubes and fourth powers.
