Let ${{S}_{k}}\left( \Gamma \right)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL\left( 2,\mathbb{Z} \right)$. Let ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{g}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{h}}\left( n \right)$ be the $n$-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f\left( z \right)\in {{S}_{{{k}_{1}}}}\left( \Gamma \right),g\left( z \right)\in {{S}_{{{k}_{2}}}}\left( \Gamma \right)$, and $h\left( z \right)\in {{S}_{{{k}_{3}}}}\left( \Gamma \right)$, respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\text{ }{{\lambda }_{f}}{{\left( n \right)}^{4}}{{\lambda }_{g}}{{\left( n \right)}^{2}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{6}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{2}}{{\lambda }_{h}}{{\left( n \right)}^{4}}$, and ${{\text{ }\!\!\lambda\!\!\text{ }}_{g}}{{\left( {{n}^{3}} \right)}^{2}}$ twisted by the arithmetic function ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$.

We study the first occurrence of a cuspidal representation $\pi$ of $\mathrm{Sp}_{2n}(\mathbb{A})$ in the tower of split orthogonal groups under symplectic-orthogonal theta lift. We give upper and lower bounds for the first occurrence in terms of the wave front of $\pi$. Besides we describe a class of small cuspidal representations whose first occurrence is the maximal possible.