showing a processes is martingale using ito's lemma

Question

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt + t^2 d W_t$, integrating and noting $Z(0) = 0$ gives $Z(t) = 2\int_0^t sW_s \ dt + \int_0^t s^2 \ dW_s$ subbing htis back into $Y(t)$ I get $Y(t) = \int_0^t s^2 \ dW_s$. How do I show it has 0 drift from here?