  In this paper we investigate invariant distributions on $p$-adic
$\mathfrak{sp}_{2n}$ defined by Waldspurger in his 2001 tome and find the
Fourier eigenspaces in their span. We prove that there is a single eigenvalue
if $n$ can be represented as a sum of triangular numbers or is triangular
itself and that none exist otherwise. We determine that the dimension of this
lone eigenspace is equal to the number of noncommuting representations of $n$
as the sum of at most two triangular numbers. Each such representation
corresponds to what we call a Lusztig distribution. These distributions belong
to the generating set defined by Waldspurger and form a basis for the
eigenspace. Finally, we show that the eigenspace contains a 1-dimensional
subspace consisting of stable distributions when $n=2\Delta$, $\Delta$ a
triangular number, but otherwise consists of distributions that are not stable.
