Show that $\sum_{j}\chi_{\pi}^3(j)\zeta^{qj} \equiv \chi_{\pi}(q)g(\overline{\chi_{\pi}}) \pmod q.$

Question

Hi I'm currently working on the proof of the Law of Biquadratic reciprocity from Ireland & Rosen's book on 'A Classical Introduction to Modern Number theory' and was stuck on the proof of Proposition 9.9.6, which states that if $\pi$ is a primary irreducible, and $q\equiv 3 \pmod 4$ a rational prime then $\chi_{\pi}(-q) = \chi_{q}(\pi).$ I understand the first 2 equivalences, however I'm stuck on how the last one holds.