On Hecke eigenvalues of Siegel modular forms in the Maass space

In this article, we prove an Omega result for the Hecke eigenvalues {\lambda_{F}(n)} of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k, genus two for the Siegel modular group {Sp_{2}({\mathbb{Z}})}. In particular, we prove\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\exp\biggl{(}c\frac{\sqrt{\log n}}{\log% \log n}\biggr{)}\biggr{)},when {c>0} is an absolute constant. This improves the earlier result\lambda_{F}(n)=\Omega\biggl{(}n^{k-1}\biggl{(}\frac{\sqrt{\log n}}{\log\log n}% \biggr{)}\biggr{)}of Das and the third author. We also show that for any {n\geq 3}, one has\lambda_{F}(n)\leq n^{k-1}\exp\biggl{(}c_{1}\sqrt{\frac{\log n}{\log\log n}}% \biggr{)},where {c_{1}>0} is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {\{\lambda_{F}(n)/n^{k-1}\}_{n\in{\mathbb{N}}}} and show that it has infinitely many limit points. Finally, we show that {\lambda_{F}(n)>0} for all n, a result proved earlier by Breulmann by a different technique.