A cubic partition is an integer partition in which the even parts can occur in two distinct colors. In this paper, we denote the number of cubic partitions of n by a(n) and define \(a_m(n)\) as the number of cubic partitions of n in which only even parts that are not congruent to 0 modulo \(2^m\) can appear in two colors. By leveraging classical results on truncated theta series, we introduce a collection of linear relations associated with the cubic partition functions a(n) and \(a_m(n)\) . This context reveals intriguing connections among ordinary partitions, overpartitions, \(2^m\) -regular partitions, pod-partitions and cubic partitions. A complete characterization of the congruences of \(a_2(n)\) modulo 2 and 4 is provided.
