Let $$\alpha=\frac{40169}{3}+\frac{50\sqrt{1993741}}{3} \cos\left( \frac{1}{3}\arccos\left(\frac{2731019453\sqrt{1993741}}{1993741^2}\right)-\frac{4\pi}{3} \right)$$ and $$\begin{aligned} h(u)=u^6\cdot 2054570000 & + u^5\cdot 4109140000\\ & + u^4\cdot\left(16\alpha^2+9768064\alpha-5227993936\right)\\ & + u^3\cdot\left(1956\alpha^2+64132324\alpha-12170223176\right)\\ & + u^2\cdot\left(11393\alpha^2+427075897\alpha+1676016222\right)\\ & + u \cdot\left(4644\alpha^2+2280446676\alpha+8524444776\right) - \left(17074\alpha^2-3269963754\alpha-2727593304\right). \end{aligned}$$ The six real roots of $h(u): u_1 < u_2 < u_3 < u_4 < u_5 < u_6$ describe the real part of six unimodular numbers $ z_k = u_k/2 + i \sqrt{1-u_k^2/4}$ with $k\in\{1,2,3,4,5,6\}$. The circulant matrix ${\rm circ}\big[z_1, z_3, \bar{z}_2, z_4, \bar{z}_5, \bar{z}_6\big]$ constitutes the core of the dephased complex $Q_7^{(0)}$ Hadamard matrix of dimension $7$. More details in [37].