Asymptotics for cuspidal representations by functoriality from GL(2)

Let $π$ be a unitary automorphic cuspidal representation of $GL_2(\mathbb{Q}_\mathbb{A})$ with Fourier coefficients $λ_π(n)$. Asymptotic expansions of certain sums of $λ_π(n)$ 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 $λ_π(n)$ on integers, squares, cubes and fourth powers.

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