Dedekind sums arising from newform Eisenstein series

For primitive non-trivial Dirichlet characters \(\chi_1\) and \(\chi_2\), we study the weight zero newform Eisenstein series \(E_{\chi_1,\chi_2}(z,s)\) 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 \(H^1(\Gamma_0(N), \mathbb{C})\). We also give a short proof of the reciprocity formula for this Dedekind sum.