2 problems

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 $0<a\le y\le b$…

Let $U(z)$ be either $y^{k/2}f(z)$ or $u_j(z)$, and let $\psi:\mathbb{Z}\backslash\mathbb{R}\to\mathbb{R}$ be smooth. For fixed $y>0$, Young's horocycle QUE conjecture. … as the we…