On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Let F be a Siegel cusp form of degree 2, even weight k ≥ 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) X, the sequence $a(F,S)$ has at least X1-ε sign changes, and takes at least X1-ε "large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound |a(F,S)| F, ε (S)^k2 - 1/2 (log |(S)|)18 - ε for fundamental matrices S.