On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2

Abstract We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ degree 2, weight $k,$ level $N$. First, assuming that is a eigenform not Saito–Kurokawa type, we prove an improved bound in the $k$-aspect for smallest prime at which its eigenvalue negative. Secondly, show there are infinitely many sign changes among primes lying arithmetic progression. Third, positive as well negative any “radial” sequence comprising multiples fixed fundamental matrix. Finally, consider case when this (essentially sharp) $| a(T) | ~\ll _{F, \epsilon }~ \big ( \det T )^{\frac {k-1}{2}+\epsilon }$ whenever $\gcd (4 (T), N)$ squarefree, confirming conjecture made (in $N=1$) by Das Kohnen.