  Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic
open image theorem for the compatible system of Galois representations
associated to $f$, generalising the results of Ribet and Momose for elliptic
modular forms. Using this result, we investigate the distribution of the Hecke
eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets
$\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the
Lang--Trotter conjecture for elliptic curves.
