We construct a family of isometric immersions of $\mathbf{R}^{2}$ into $\mathbf{R}^{4}$ with vanishing normal curvature.

For each prime $p$, we give an upper bound in $m$ for Poincaré series $P_{k}^{+}(z,m)$ of weight $k$ for $\Gamma_{0}^{+}(p)$ to be non-vanishing.