Sums of Cusp Form Coefficients Along Quadratic Sequences

Let $f(z)=\sum A(n)n^{(k-1)\mathrm{/}2}e(nz)$ be a cusp form of weight $k\ge 3$ on ${\mathrm{\Gamma }}_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that ${\sum }_{n\le X}A(n^2+h)=c_{f,h}X+O_{f,h,\epsilon }(X^{\frac{3}{4}+\epsilon })$ for $\epsilon >0$, which improves a result of Blomer from 2008 with error $X^{\frac{6}{7}+\epsilon }$. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.