Explicit congruences modulo 2048 for overpartitions

Let <overline>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 <overline>p(n). In particular, we deduce some strange congruences modulo 2048 for <overline>p(n). For instance, we prove that for n = 0, <overline>p(1820n2 + 1820n