We study the depth filtration on multiple zeta values, motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and Grothendieck–Teichmüller group, its relation to modular forms. Using period polynomials for cusp forms $\mathrm {SL} _2(\mathbb {Z})$ , we construct an explicit Lie algebra solutions linearized double shuffle equations, which gives a conjectural description all identities bet...