PDFs were fixed to ZEUS-13p. These values for the couplings are compatible to those obtained by ZEUS-EW-Z. This cross-check confirms that the determination of the $Z$ couplings is essentially decoupled from the QCD part of the fit.

Another fit, HPDF1, was performed with the $Z$ couplings free and the PDFs fixed to HERAPDF2.0 NLO. The results, also listed in Table \[tab:Z\], are in agreement with ZEUS-EW-Z. It should be noted that HERAPDF2.0 was extracted using a different value of $\sin^2\theta_W$. Therefore, a fit HPDF2 using this $\sin^2\theta_W$ value was also performed. The result is also listed in Table \[tab:Z\]. The values agree well within uncertainties with those from the fit using the on-shell value.

The correlations between the four couplings obtained in ZEUS-EW-Z are listed as part of Table \[tab:matrix\]. Two-dimensional scans were performed to obtain so-called profile likelihood contours. The two parameters under investigation were modified in small steps. For each point, a fit was performed to minimise $\chi^2$ with respect to all other parameters. The $\chi^2$-values thus calculated were used to obtain 68%C.L. contours. The results for the couplings $a_u,v_u$ and $a_d,v_d$ are shown in Fig. \[fig:av-us\] [^5]. Figure \[fig:aavv\] shows the 68%C.L. contour plots for $a_u,a_d$ and 