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Dedekind sums arising from newform Eisenstein series

For primitive non-trivial Dirichlet characters $χ_1$ and $χ_2$, we study the weight zero newform Eisenstein series $E_{χ_1,χ_2}(z,s)$ at $s=1$. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of $H^1(Γ_0(N), \mathbb{C})$. We also give a short proof of the reciprocity formula for this Dedekind sum.