The distributions of coefficients of triple product L-functions over arithmetic progressions

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights k₁ and k₂ for the full modular group SL (2, Z), respectively. And let ₅ ₅ ₅ (n) and ₅ ₅ ₆ (n) denote the nth coefficients of triple product L-functions L (f f f, s) and L (f f g, s) associated to f and f, g, respectively. In this paper, we establish the asymptotic formulae for\₍ ₗ \\ ₍ ₋ (ₐ) ₅ ₅ ₅^2 (n) and ₍ ₗ \\ ₍ ₋ (ₐ) ₅ ₅ ₆^2 (n) q is a prime with (l, q) =1. And we also consider other similar results of the above types.