Proving a state is a KMS state

Question

Define an *-algebra generated by the symbols $W(u)$ for $u\in \mathbb{C}$ subject to the relation $$W(u)W(v) := e^{\frac{1}{2}i\Im(\overline{v}u)}W(u+v), W(u)^* = W(-u)\quad u,v \in \mathbb{C}.$$ Define also a group of automorphisms $\alpha_t$ given by $$\alpha_t(W(u)) = W(e^{it}u)$$ then show that the state defined by $$\tau(W(u)) = \exp\left(-\frac{1}{2}|u|^2\coth\left(\frac{\beta}{2}\right)\right), \beta>0$$ obeys the KMS condition. That is $$\tau(W(u)\alpha_{t+i\beta}(W(v))) = \tau(\alpha_t(W(v))W(u)), \forall u,v\in \mathbb{C}.$$ So far I have expanded $$\tau(W(u)W(v)) = \tau(W(u))\tau(W(v))\exp\left(-\Re(u\overline{v})\coth\left(\frac{\beta}{2}\right)\right),$$ but then plugging in the KMS condition with the action of $\alpha_{t+i\beta}$ I get a lot of $e^{-\beta}$ factors that don't seem to cancel out or help at all. I thought maybe they would somehow interfere with the $\coth\left(\frac{\beta}{2}\right)$ term but in my calculations nothing nice seems to come out of it.