Let
$K$
be a real quadratic field, and
$p$
a rational prime which is inert in
$K$
. Let
$\alpha $
be a modular unit on
${{\Gamma }_{0}}(N)$
. In an earlier joint article with Henri Darmon, we presented the definition of an element
$u\left( \alpha ,\,\text{ }\!\!\tau\!\!\text{ } \right)\,\in \,K_{P}^{\times }$
attached to
$\alpha $
and each
$\tau \,\in \,K$
. We conjectured that the
$p$
-adic number
$u(\alpha ,\,\tau )$
lies in a specific ring class extension of
$K$
depending on
$\tau $
, and proposed a “Shimura reciprocity law” describing the permutation action of Galois on the set of
$u(\alpha ,\,\tau )$
. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing
$u(\alpha ,\,\tau )$
, and implement this algorithm with the modular unit
$\alpha (z)\,=\,\Delta {{(z)}^{2}}\,\Delta (4z)\,/\,\Delta {{(2z)}^{3}}$
. Using
$p\,=\,3,\,5,\,7\,and\,11$
, and all real quadratic fields
$K$
with discriminant
$D\,<\,500$
such that 2 splits in
$K$
and
$K$
contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define
$u(\alpha ,\,\tau )$
is shown to be
$\mathbf{Z}$
-valued rather than only
${{\mathbf{Z}}_{P}}\,\cap \,\mathbf{Q}-$
valued; this is an improvement over our previous result and allows for a precise definition of
$u(\alpha ,\,\tau )$
, instead of only up to a root of unity.