Let $f \colon X \to \mathcal{Q}_7$ be a branched covering of degree $3$ of a $7$-dimensional smooth projective quadric $\mathcal{Q}_7 \subset \mathbb{P}_8$, where $X$ is a smooth connected projective variety. We can define the associated vector bundle $\mathcal{E}$ of rank 2 by $$f_*\mathcal{O}_X = \mathcal{O}_{\mathcal{Q}_7} \oplus \mathcal{E}^*.$$
I want to compute the Chern classes $c_1$ and $c_2$ of the associated bundle $\mathcal{E}$, but I am not very familiar with that. Here is, what I get so far.
Since $(f_*\mathcal{O}_X)^*$ is a direct sum, the Chern classes $$c_1((f_*\mathcal{O}_X)^*) = c_1(\mathcal{O}_{\mathcal{Q}_7}) + c_1(\mathcal{E})$$ and $$c_2((f_*\mathcal{O}_X)^*) = c_1(\mathcal{O}_{\mathcal{Q}_7}).c_1(\mathcal{E}) + c_2(\mathcal{E}),$$ where $c_1(\mathcal{O}_{\mathcal{Q}_7}).c_1(\mathcal{E})$ is the cup-product in $H^4(\mathcal{Q}_7, \mathbb{Z})$.
On the other hand, I know that $(f_*\mathcal{O}_X)^*$ is the relative canonical bundle $f_*\omega_{X|\mathcal{Q}_7} = f_*K_X \otimes K_{\mathcal{Q}_7}^*$ and $K_{\mathcal{Q}_7} = \mathcal{O}_{\mathcal{Q}_7}(-7)$. Hence we get $$c_1((f_*\mathcal{O}_X)^*) = c_1(f_*K_X) + 3\cdot c_1(K_{\mathcal{Q}_7}^*)$$ and $$c_2((f_*\mathcal{O}_X)^*) = c_2(f_*K_X) + 2\cdot c_1(f_*K_X).c_1(K_{\mathcal{Q}_7}^*) + c_1^2(K_{\mathcal{Q}_7}^*).$$
Now - in case these results are correct - I can solve the equations for $c_1(\mathcal{E})$ resp. $c_2(\mathcal{E})$. But, as I said before I'm not familiar with it, I don't know how to handle the other objects, which take part.
I hope the question is not too extensive. Thanks in advance.
