A note on the cancellations of sums involving Hecke eigenvalues

Let f and g be distinct primitive holomorphic cusp forms of even integral weights $$k_{1}$$ k 1 and $$k_{2}$$ k 2 for the full modular group $$\Gamma =SL(2,{\mathbb {Z}})$$ Γ = S L ( 2 , Z ) , respectively. Denote by $$\lambda _{f}(n)$$ λ f ( n ) and $$\lambda _{g}(n)$$ λ g ( n ) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function $$\lambda _{f}(n^{i})\lambda _{f}(n^{j})$$ λ f ( n i ) λ f ( n j ) and $$\lambda _{f}(n^{i})\lambda _{g}(n^{j})$$ λ f ( n i ) λ g ( n j ) for any integers $$i,j\ge 1$$ i , j ≥ 1 . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least $$q^{\frac{1}{2}+\varepsilon }$$ q 1 2 + ε for arbitrarily small $$\varepsilon >0$$ ε > 0 . In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients $$\lambda _{f\times f\times f}(n)$$ λ f × f × f ( n ) and $$\lambda _{f\times f\times g}(n)$$ λ f × f × g ( n ) of triple product L-functions $$L(f\times f\times f,s)$$ L ( f × f × f , s ) and $$L(f\times f\times g,s)$$ L ( f × f × g , s ) , respectively.