1 \pm i\sqrt{4{\theta}^2-5}] \label{eqn:c}.\end{aligned}$$ We then check that there are no solutions to the characterizing equations for any choice of integer ${\theta}$. For small ${\theta}$ we can check explicitly. When ${\theta}$ is large, we examine $\arg(ce^{i2{\theta}^2 t})$ in two different ways: once from combining and and once from . We use asymptotics with error bounds to show the values are not equal. We need one preliminary observation: when ${\theta}$ is large, $e^{i {\theta}t}$ is close to $\pm i$ and so the argument of $e^{i{\theta}t}$ is approximately $\pm (\pi/2 - {\mathrm{Re}}(e^{i{\theta}t}))$.

\[lem:argbound\] For any $x$ on the unit circle, $\arg(x) = \pm (\pi/2 - {\mathrm{Re}}(x)) + {\epsilon}$, where ${\left|{\epsilon}\right|} \leq {\mathrm{Re}}(x)^2$.

[*Proof.* ]{}Assume ${\mathrm{Im}}(x) > 0$ and ${\mathrm{Re}}(x) < 0$: the other cases are similar. The part of $\arg(x)$ past $\pi/2$ is the portion of the unit circle arc in the upper left quadrant, which is contained in a rectangle with horizontal width ${\mathrm{Re}}(x)$ and vertical height $1 - \sqrt{1-{\mathrm{Re}}(x)^2}$. Since $\sqrt{z} \geq z$ for any $z \in [0,1]$, we have $$\begin{aligned}
 \arg(x) - \pi/2 & \leq -{\mathrm{Re}}(x) + 1 - \sqrt{1-{\mathrm{Re}}(x)^2} \\
& \leq -{\mathrm{Re}}(x) + 1 - (1-{\mathrm{Re}}(x)^2) \\
& = -{\mathrm{Re}}(x) + {\mathrm{Re}}(x)^2.\end{aligned}$$ On the the other hand 