  Brown showed that the affine ring of the motivic path torsor
$\pi_1^{\text{mot}}(\mathbb{P}^1 \backslash \left\{0,1,\infty\right\},
\vec{1}_0, -\vec{1}_1)$, whose periods are multiple zeta values, generates the
Tannakian category $\mathsf{MT}(\mathbb{Z})$ of mixed Tate motives over
$\mathbb{Z}$. Brown also introduced multiple modular values, which are periods
of the relative completion of the fundamental group of the moduli stack
$\mathcal{M}_{1,1}$ of elliptic curves. We prove that all motivic multiple zeta
values may be expressed as $\mathbb{Q}[2 \pi i]$-linear combinations of motivic
iterated Eisenstein integrals along elements of $\pi_1 (\mathcal{M}_{1,1})
\cong SL_2(\mathbb{Z})$, which are examples of motivic multiple modular values.
This provides a new modular generator for $\mathsf{MT}(\mathbb{Z})$. We also
explain how the coefficients in this linear combination may be partially
determined using the motivic coaction.
