Spectral fourth moments of Hecke--Maaß cusp forms

In this note, we establish essentially optimal bounds for certain spectral moments of automorphic forms for $\mathrm{G}\mathrm{L}(2)$. More precisely, we consider the family of Hecke–Maaß cusp forms with spectral parameter in a dyadic interval and study the fourth moment of these forms evaluated at a Heegner point. We additionally present applications of our main result to the shifted convolution problem involving the sum of two squares function $r(n)$ as well as to pointwise Diophantine exponents.