The Ehrhart polynomial \(\text{ehr}_P(n)\) of a lattice polytope \(P\) counts the number of integer points in the \(n\)-th integral dilate of \(P\). The \(f^*\)-vector of \(P\), introduced by Felix Breuer in 2012, is the vector of coefficients of \(\text{ehr}_P(n)\) with respect to the binomial coefficient basis \( \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}\), where \(d = \dim P\). Similarly to \(h/h^*\)-vectors, the \(f^*\)-vector of \(P\) coincides with the \(f\)-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of \(f^*\)-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of \(f\)-vectors of simplicial polytopes; e.g., the first half of the \(f^*\)-coefficients increases and the last quarter decreases. Even though \(f^*\)-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart \(h^*\)-vector, there is a polytope with the same \(h^*\)-vector whose \(f^*\)-vector is unimodal.