Shifted convolution sums of $GL_3$ cusp forms with $θ$-series

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $SL_3(\mathbb{Z})$ and $r_3(n)=\#\left\{(n_1,n_2,n_3)\in \mathbb{Z}^3:n_1^2+n_2^2+n_3^2=n\right\}.$ Let $1\leq h\leq X$ and $ϕ(x)$ be a smooth function compactly supported on $[1/2,1]$. It is shown that $\sum_{n\geq 1}A_f(1,n+h)r_3(n)ϕ\left(\frac{n}{X}\right) \ll_{f,\varepsilon} X^{\frac{3}{2}-\frac{1}{8}+\varepsilon}$ uniformly with respect to the shift $h$.

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