Some new congruences on biregular overpartitions

Recently, Nadji, Ahmia and Ramírez \cite{Nadji2025} investigate the arithmetic properties of ${\bar{B}}_{{\mathrm{\ell }}_1,{\mathrm{\ell }}_2}(n)$, the number of overpartitions where no part is divisible by ${\mathrm{\ell }}_1$ or ${\mathrm{\ell }}_2$ with $\gcd ({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$$=1$ and ${\mathrm{\ell }}_1$,${\mathrm{\ell }}_2$$>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs of $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$$\in$$\{(4,3),(4,9),(8,3),(8,9)\}$, using the concept of generating functions, dissection formulas and Smoot’s implementation of Radu’s Ramanujan-Kolberg algorithm. After that, Alanazi, Munagi and Saikia \cite{Alanazi2024} studied and found some congruences for the pairs of $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)\in \{(2,3),(4,3),(2,5),(3,5),(4,9),(8,27),(16,81)\}$ using the theory of modular forms and Radu’s algorithm. Recently Paudel, Sellers and Wang \cite{Paudel2025} extended several of their results and established infinitely many families of new congruences. In this paper, we find infinitely many families of congruences modulo $3$ and powers of $2$ for the pairs $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ $\in$ $\{(2,9),(5,2),(5,4),(8,3)\}$ and in general for $(5,2^t)$ $\mathrm{\forall }t\ge 3$ and for $(3,2^t)$,$(4,3^t)$ $\mathrm{\forall }t\ge 2$, using the theory of Hecke eigenform, an identity due to Newman and the concept of dissection formulas and generating functions.