On the \(U_p\) operator acting on \(p\)-adic overconvergent modular forms when \(X_{0}(p)\) has genus 1 (Q2268796)

Let \(p\in\{11,17,19\}\), \(\alpha\) and \(\beta\) the two distinct supersingular \(j\)-invariants modulo \(p\), \(z:=\frac{j-\alpha}{j-\beta}\) a parameter on the closed unit disc, \(q\in{\mathbb Q}_p\) such that \(0<w=\nu_p(q)<t\frac{p}{p+1}\), where \(t=\frac 13\) if \(p\in\{11,17\}\) and \(t=\frac 12\) if \(p=19\). The author proves that \[ \left\{1,z,z^2,z^3,\ldots\right\}\cup\left\{\frac{q}{z},\left(\frac{q}{z}\right)^2,\left(\frac{q}{z}\right)^3,\ldots\right\} \] is an orthonormal basis for the space of \(p\)-adic \(w\)-overconvergent modular forms of weight \(0\).