  Deligne and Goncharov constructed a neutral tannakian category of mixed Tate
motives unramified over $\mathbb{Z}[\mu_N,1/N]$. Brown and Hain--Matsumoto
computed the depth 2 quadratic relations of the motivic Galois group of this
category for $N = 1$. We take the first steps in generalizing their results to
all $N \ge 1$ by realizing the generators of the motivic Galois group by
derivations on the Lie algebra of the unipotent fundamental group of a
restriction of the Tate elliptic curve.
  This representation is compatible with a natural identification of the odd
rational $K$-groups of the rings $\mathbb{Z}[\mu_N,1/N]$ with spaces of
$\Gamma_1(N)$ Eisenstein series, thus inducing a natural action of the prime to
$N$ part of the Hecke algebra on the $K$-groups.
  We establish these results by first showing the inclusion of $\mathbb{P}^1 -
\{0,\mu_N,\infty\}$ into the nodal elliptic curve with a cyclic subgroup of
order $N$ removed induces a morphism of mixed Tate motives on unipotent
fundamental groups and then by computing the periods of the limit mixed Hodge
structure of an elliptic polylogarithm variation of MHS over the universal
elliptic curve of $Y_1(N)$.
