$M_{13A}^{(0)}$
A set of three Butson type matrices found
by R. Nicoară, A. LaClair, N. Geist and A. Wintenberg in May 2016 (UT, Knoxville).
$${\rm LOG}\left(M_{13A}\right)=\frac{1}{3}\pi
\left[\begin{array}{rrrrrrrrrrrrr}
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\
\bullet & 5 & 5 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 1 & 1 & 1\\
\bullet & 5 & 5 & 1 & 2 & 2 & 2 & 4 & 4 & 4 & 1 & 1 & 4\\
\bullet & 1 & 1 & 2 & 4 & 4 & 1 & 4 & 2 & 2 & 5 & 5 & 4\\
\bullet & 1 & 1 & 4 & 2 & 2 & 4 & 1 & 4 & 4 & 5 & 5 & 2\\
\bullet & 1 & 4 & 4 & 5 & 1 & 2 & 2 & 5 & 1 & 4 & 2 & 4\\
\bullet & 2 & 4 & 1 & 4 & 1 & 5 & 1 & 2 & 4 & 2 & 4 & 5\\
\bullet & 2 & 4 & 2 & 1 & 5 & 4 & 4 & 1 & 5 & 4 & 1 & 2\\
\bullet & 2 & 4 & 5 & 2 & 4 & 1 & 5 & 4 & 1 & 2 & 4 & 1\\
\bullet & 4 & 1 & 4 & 1 & 5 & 2 & 2 & 1 & 5 & 2 & 4 & 4\\
\bullet & 4 & 2 & 1 & 1 & 4 & 5 & 1 & 4 & 2 & 4 & 2 & 5\\
\bullet & 4 & 2 & 2 & 5 & 1 & 4 & 4 & 5 & 1 & 1 & 4 & 2\\
\bullet & 4 & 2 & 5 & 4 & 2 & 1 & 5 & 1 & 4 & 4 & 2 & 1\\
\end{array}\right].$$