The only maximal affine Hadamard families stemming from the Fourier matrix $$F_{14} = \left[\begin{array}{lllllllllllllll} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& w& w^2& w^3& w^4& w^5& w^6& w^7& w^8& w^9& w^{10}& w^{11}& w^{12}& w^{13}\\ 1& w^2& w^4& w^6& w^8& w^{10}& w^{12}& 1& w^2& w^4& w^6& w^8& w^{10}& w^{12}\\ 1& w^3& w^6& w^9& w^{12}& w& w^4& w^7& w^{10}& w^{13}& w^2& w^5& w^8& w^{11}\\ 1& w^4& w^8& w^{12}& w^2& w^6& w^{10}& 1& w^4& w^8& w^{12}& w^2& w^6& w^{10}\\ 1& w^5& w^{10}& w& w^6& w^{11}& w^2& w^7& w^{12}& w^3& w^8& w^{13}& w^4& w^9\\ 1& w^6& w^{12}& w^4& w^{10}& w^2& w^8& 1& w^6& w^{12}& w^4& w^{10}& w^2& w^8\\ 1& w^7& 1& w^7& 1& w^7& 1& w^7& 1& w^7& 1& w^7& 1& w^7\\ 1& w^8& w^2& w^{10}& w^4& w^{12}& w^6& 1& w^8& w^2& w^{10}& w^4& w^{12}& w^6\\ 1& w^9& w^4& w^{13}& w^8& w^3& w^{12}& w^7& w^2& w^{11}& w^6& w& w^{10}& w^5\\ 1& w^{10}& w^6& w^2& w^{12}& w^8& w^4& 1& w^{10}& w^6& w^2& w^{12}& w^8& w^4\\ 1& w^{11}& w^8& w^5& w^2& w^{13}& w^{10}& w^7& w^4& w& w^{12}& w^9& w^6& w^3\\ 1& w^{12}& w^{10}& w^8& w^6& w^4& w^2& 1& w^{12}& w^{10}& w^8& w^6& w^4& w^2\\ 1& w^{13}& w^{12}& w^{11}& w^{10}& w^9& w^8& w^7& w^6& w^5& w^4& w^3& w^2& w \end{array}\right] : \quad w=\exp\frac{2i\pi}{14}$$ are $$\begin{aligned} F_{14}^{(6)}(a,b,c,d,e,f) & = F_{14}\circ{\rm EXP}\left(i R_{F_{14}^{(6)}}(a,b,c,d,e,f)\right),\\ \left(F_{14}^{(6)}(a,b,c,d,e,f)\right)^{\rm T} & = F_{14}\circ{\rm EXP}\left(i \left(R_{F_{14}^{(6)}}(a,b,c,d,e,f)\right)^{\rm T}\right) \end{aligned}$$ with $$\begin{aligned} R_{F_{14}^{(6)}}& (a,b,c,d,e,f)=\\ & \left[\begin{array}{cccccccccccccc} 0& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f\\ \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet& \bullet\\ \bullet& a& b& c& d& e& f& \bullet& a& b& c& d& e& f \end{array}\right]. \end{aligned}$$