Andrews-Beck type congruences modulo powers of 5

Let NT(m, k, n) denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for NT(m, k, n) modulo 5 and 7. Generalizing Andrews' results, Chern obtained congruences for NT(m, k, n) modulo 11 and 13. More recently, the second author used the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes l >= 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ge 7$$\end{document}. In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.