Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k\ge 2$ on ${\mathrm{\Gamma }}_0(q)$. If $f\mathrm{\ne }g$, then assume that $q$ is squarefree. For a nice test function $\psi$ supported on ${\mathrm{\Gamma }}_0(1)\mathrm{\backslash }\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $\psi$) for $${\int }_{{\mathrm{\Gamma }}_0(q)\mathrm{\backslash }\mathbb{H}}\psi (z)f(z)\overline{g(z)}{y}^k\frac{dxdy}{y^2}-{\mathbf{1}}_{f=g}\frac{3}{\pi }{\int }_{{\mathrm{\Gamma }}_0(1)\mathrm{\backslash }\mathbb{H}}\psi (z)\frac{dxdy}{y^2}\mathrm{.}$$ When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f\mathrm{\ne }g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan’s weak subconvexity bound for Rankin-Selberg $L$-functions.