Fricke involution’s effect on character

Question

I’m using the definition $W_Nf(\tau)=i^kN^{-k/2}\tau^{-k}f(-1/N\tau)$. Now suppose $f\in M_k(\Gamma_1(N),\chi)$, show $W_Nf\in M_k(\Gamma_1(N),\chi^{-1})$.

I know this is pure calculation but I’m stuck on expressing $\langle p\rangle(\tau^{-k}f(-1/N\tau) )$ and relating it with $\langle p\rangle f$…Please let me know if this is trivial.