Large values of modular forms

Abstract

We show that there are primitive holomorphic modular forms $f$ of weight two and arbitrary large level $N$ such that $|f(z)| \gg N^{1/4}$ for some $z\in \mathfrak{H}$. Thereby we disprove a folklore conjecture that the $L^\infty$-norm of such forms would be as small as $N^\epsilon$ for any $\epsilon>0$.