Let $f: Y=Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong \mathbb{P}(1,1,2)$ are singular at the same point. If I consider the log resolution $g$ of this singularity with respect to the pair $(Y,E)$, how to compute $g^*E$?

I think locally, the germ of singularity in $Y$ is isomorphic to a cone over Veronese surface, living in $\mathbb{C}^6$, and $E$ is the cone over a quadratic curve.

$g$ could be the blow up of $\mathbb{C}^6$ at 0, but I don't know what the number $d$ is in the following:

$$g^*E\sim_{\mathbb{Q}} \tilde{E}+dE'$$ where $E'$ is the exceptional divisor of $g$.