The ﬁltration of the Virasoro minimal series representations M__r,s(p, p') induced by the (1, 3)-primary ﬁeld φ1,3(z) is studied. For 1 < p'/p < 2, a conjectural basis of M__r,s(p, p') compatible with the ﬁltration is given by using monomial vectors in terms of the Fourier coeffcients of φ1,3(z). In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series (p' = p + 1), we establish for each m the equality between the character of the degree m monomial basis and the character of the degree m component in the associated graded module gr(M__r,s(p, p+1)) with respect to the ﬁltration deﬁned by φ1,3(z).