  A partition of $n$ is $l$-regular if none of its parts is divisible by $l$.
Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper,
using the theory of Hecke eigenforms explored by J.-P. Serre, we establish
exact criteria for the $3$-divisibility of $b_9(n)$ and $b_{27}(n)$.
