  Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd
squarefree level $N$. We undertake a detailed study of the analytic properties
of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with
$-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies
along the equivalence classes of fundamental matrices with $det(S) \asymp X$,
the sequence $a(F,S)$ has at least $X^{1-\epsilon}$ sign changes, and takes at
least $X^{1-\epsilon}$ "large values". Furthermore, assuming the Generalized
Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we
prove the bound $|a(F,S)| \ll_{F, \epsilon} \frac{\det(S)^{\frac{k}2 -
\frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - \epsilon}}$ for fundamental matrices
$S$.
