2 problems

Let $\psi\colon \mathbb{R}^+ \to \mathbb{R}$ be a smooth compactly supported function, and let $f(z)$ run over $L^2$-normalized holomorphic Hecke cusp forms of weight $k$. Young's…

Let $U(z)=y^{k/2}f(z)$, where $f$ is a holomorphic Hecke cusp form of weight $k$, with Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$…