Pointwise bounds for Eisenstein series on ${\mathrm{\Gamma }}_0(q)\setminus S{L}_2(\mathbb{R})$

We construct pointwise bounds in the weight aspect for Eisenstein series on $X_0(q)={\mathrm{\Gamma }}_0(q)\setminus S{L}_2(\mathbb{R})$, with squarefree level $q$, using a Sobolev technique. More specifically, we show that for an Eisenstein series $E$ on $X_0(q)$ of weight parameter $n$ and type $t$, one has for all $x\in X_0(q)$: $|E(x,1/2 + it)| \ll_{\epsilon} q^{\epsilon}(1 + |n|^{1/2 + \epsilon} + |t|^{1/2 + \epsilon})\sqrt{y(x) + y(x)^{-1}}$,where$y(x)$ is the Iwasawa $y$-coordinate of the point $x$.