  We present here simple trace formulas for Hecke operators $T_k(p)$ for all
$p>3$ on $S_k(\Gamma_0(3))$ and $S_k(\Gamma_0(9))$, the spaces of cusp forms of
weight $k$ and levels 3 and 9. These formulas can be expressed in terms of
special values of Gaussian hypergeometric series and lend themselves to simple
recursive expressions in terms of traces of Hecke operators on spaces of lower
weight. Along the way, we show how to express the traces of Frobenius of a
family of elliptic curves with 3-torsion as special values of a Gaussian
hypergeometric series over $\mathbb{F}_q$, when $q\equiv 1 \pmod{3}$. We also
use these formulas to provide a simple expression for the Fourier coefficients
of $\eta(3z)^8$, the unique normalized cusp form of weight 4 and level 9, and
then show that the number of points on a certain threefold is expressible in
terms of these coefficients.
