Explicit congruences modulo 2048 for overpartitions

Let $${\bar{p}}(n)$$ denote the number of overpartitions of n. Recently, a number of congruences modulo powers of 2, 3 and 5 have been discovered. The moduli for these congruences modulo powers of 2 discovered before ranged as high as 1024. In this paper, we establish explicit congruences modulo 2048 for $${\bar{p}}(n)$$. In particular, we deduce some strange congruences modulo 2048 for $${\bar{p}}(n)$$. For instance, we prove that for $$n\ge 0$$, $$\begin{aligned} {\bar{p}}(1820n^2+1820n+455)\equiv 0\!\!\!\!\pmod {2048}. \end{aligned}$$