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The aim of this paper is to construct a natural mapping ${\check{C}}_k$, $k=1,2,3,\cdots$, from the multiplicative $K$-theory $K\left(X\right)$ of a differential manifold $X$, associated to the trivial filtration of the de Rham complex, as defined by in [C. R. Acad. Sci., Paris, Sér. I 302, 321-324 (1986; Zbl 0593.55004)] to the odd cohomology $H_s^{2k-1}(X;C^{*})$. By using this mapping, the author associates to any flat complex vector bundle $E$ on $X$ characteristic classes ${\check{C}}_k\left(E\right)\in H_{dR}^{2k-1}(X;C^{*})$ analogous to the classes studied by , and in [Differential characters and geometric invariants,...