Let $$\pi _i$$ , $$i=1,2,3$$ , be unitary automorphic cuspidal representations of $$GL_2({\mathbb {Q}}_{\mathbb {A}})$$ with Fourier coefficients $$\lambda _{\pi _i}(n)$$ . Consider an automorphic representation $$\Pi $$ which is equivalent to $$\wedge ^2(\mathrm{Sym}^3\pi _1)$$ , $$\pi _1\boxtimes \pi _2$$ , $$\pi _1\boxtimes \mathrm{Sym}^2\pi _2$$ , $$\wedge ^2(\pi _1\boxtimes \pi _2)$$ , or $$\pi _1\times \pi _2\times \pi _3$$ . Since the Dirichlet series of $$L(s,\Pi \times \widetilde{\Pi })$$ is known to be complicated, a simpler Dirichlet series $$\sum \lambda (n)n^{-s}$$ is defined and analytically continued in each case, which is closely related to $$L(s,\Pi \times \widetilde{\Pi })$$ and catches the essence of the underlying functoriality. Asymptotics of $$\sum _{n\le x}\lambda (n)$$ are proved. As applications, certain means, variance, and covariances of $$|\lambda _{\pi _i}(n)|^{k}$$ for $$k=2,4,6$$ and $$|\lambda _{\pi _i}(n^j)|^2$$ for $$j=2,3,4$$ are computed. These statistics provide a deep insight of the distribution of the GL(2) Fourier coefficients $$\lambda _{\pi _i}(n)$$ .