Abstract Extending our previous work we construct weakly holomorphic Hecke eigenforms whose period polynomials correspond to elements in a basis consisting of odd and even Hecke eigenpolynomials induced by only cusp forms. As an application of our results, we give an explicit construction of the holomorphic parts of harmonic weak Maass forms that are good for Hecke eigenforms. Moreover, we give an explicit construction of the Hecke-equivariant map between the space of weakly holomorphic cusp forms and two copies of the spaces of cusp forms, and show that the map is compatible with the corresponding map on the spaces of period polynomials.