# Dedekind sums arising from newform Eisenstein series

## TL;DR

This paper explores new Dedekind sums derived from Eisenstein series associated with primitive Dirichlet characters, providing explicit constructions and a proof of their reciprocity formula.

## Contribution

It introduces a finite-term expression for Dedekind sums from Eisenstein series and constructs elements of cohomology groups explicitly.

## Key findings

- Explicit finite-term formulas for Dedekind sums
- Construction of cohomology elements from Eisenstein series
- Proof of reciprocity formula for these Dedekind sums

## Abstract

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.