Pointwise bounds for Eisenstein series on $Γ_0(q)\setminus SL_2(\mathbb{R})$

We construct pointwise bounds in the weight aspect for Eisenstein series on $X_0(q) = Γ_0(q)\setminus SL_2(\mathbb{R})$, 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 $x\in X_0(q)$: $|E(x,1/2 + it)| \ll_ε q^ε(1 + |n|^{1/2 + ε} + |t|^{1/2 + ε})\sqrt{y(x) + y(x)^{-1}}$, where $y(x)$ is the Iwasawa $y$-coordinate of the point $x$.

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