  We give a systematic method of providing numerical evidence for higher order
Stark-type conjectures such as (in chronological order) Stark's conjecture over
$\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to
Burns that constitutes a generalization of Brumer's classical conjecture on
annihilation of class groups. Our approach is general and could be used for any
abelian extension of number fields, independent of the signature and type of
places (finite or infinite) that split completely in the extension.
  We then employ our techniques in the situation where $K$ is a totally real,
abelian, ramified cubic extension of a real quadratic field. We numerically
verify the conjectures listed above for all fields $K$ of this type with
absolute discriminant less than $10^{12}$, for a total of $19197$ examples. The
places that split completely in these extensions are always taken to be the two
real archimedean places of $k$ and we are in a situation where all the
$S$-truncated $L$-functions have order of vanishing at least two.
