AbstractLetFbe a Siegel cusp form of degree$2$, even weight$k \ge 2$, and odd square-free levelN. We undertake a detailed study of the analytic properties of Fourier coefficients$a(F,S)$ofFat fundamental matricesS(i.e., with$-4\det (S)$equal to a fundamental discriminant). We prove that asSvaries along the equivalence classes of fundamental matrices with$\det (S) \asymp X$, the sequence$a(F,S)$has at least$X^{1-\varepsilon }$sign changes and takes at least$X^{1-\varepsilon }$‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound$\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$for fundamental matricesS.