Hilbert polynomial of algebraic variety with 4 generators

Question

I wanted to compute the Hilbert polynomial of the algebraic variety $Z(f,f_1, f_2, f_3) \subset \mathbb{P}^5$. Here $f = a_0^3 + 4a_0^2a_5 + 2a_0a_3a_2 + 4a_0a_1a_2 + 4a_3a_5a_2 + a_1a_2^2 \in K[a_0,a_1,a_2,a_3,a_4,a_5]$ and \begin{align*} f_1 = a_0a_1 - a_3^2 \\ f_2 = a_0a_2 - a_4^2 \\ f_3 = a_1a_2 - a_5^2 \end{align*} I know how to compute the Hilbert polynomial when the algebraic variety would just be $Z(f)$, but for a variety generated by more then one polynomial I have no idea. Can someone help me?