## Abstract

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 $S{p}_{2}(\mathbb{Z})$.
In particular, we prove

${\lambda}_{F}(n)=\mathrm{\Omega}\left({n}^{k-1}\mathrm{exp}\left(c\frac{\sqrt{\mathrm{log}n}}{\mathrm{log}\mathrm{log}n}\right)\right),$

when $c>0$ is an absolute constant. This improves the earlier result

${\lambda}_{F}(n)=\mathrm{\Omega}\left({n}^{k-1}\left(\frac{\sqrt{\mathrm{log}n}}{\mathrm{log}\mathrm{log}n}\right)\right)$

of Das and the third author. We also show that for any $n\ge 3$, one has

${\lambda}_{F}(n)\le {n}^{k-1}\mathrm{exp}\left({c}_{1}\sqrt{\frac{\mathrm{log}n}{\mathrm{log}\mathrm{log}n}}\right),$

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.

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