Grothendieck Riemann Roch, universal bundle on $SU_C(2,L)$

Question

I've problems with the use of Grothendieck Riemann Roch Theorem. Let $U$ be the the universal bundle on the moduli space $n:=SU_C(2,L)$, $C$ a smooth projective curve of genus $g$ and $L$ a line bundle of odd degree $d$. I'd like to understand why, if a name $\pi: N:=SU_C(2,L) \times C \to SU_C(2,L)$ the projection, then I have $$c_1(\pi_*(U))= -\omega+(d+2-2g)\phi,$$ where $\omega$ and $\phi$ are such as: $$c_2(U)= \eta \otimes \omega + \psi + 1 \otimes \chi$$ ($\omega \in H^2(N), \psi \in H^0(N) \otimes H^1(C), \chi \in H^4(N)$). Futhemore, we know that $$c_1(U) = c_1(L) \otimes 1+ 1 \otimes \phi$$ where $\phi=c_1(\Phi) \in H^2(N)$ with $\Phi$ a certain line bundle. I can't deduce it from GRR because I don't know how to compute $ch(R^1 \pi_* U)$