  We classify all the cocyclic Butson Hadamard matrices $\mathrm{BH}(n,p)$ of
order $n$ over the $p$th roots of unity for an odd prime $p$ and $np\leq 100$.
That is, we compile a list of matrices such that any cocyclic
$\mathrm{BH}(n,p)$ for these $n$, $p$ is equivalent to exactly one element in
the list. Our approach encompasses non-existence results and computational
machinery for Butson and generalized Hadamard matrices that are of independent
interest.
