Computing with Jacobi-sum motives and Hecke Grossencharacters

In 1952, Weil wrote a short paper describing how to interpret Jacobi sums in terms of Hecke Grossencharacters, which in modern terms might be termed a reciprocity law. Grossencharacters are easier to work with computationally, as one knows how to compute coefficients of the L-series  in polynomial time, and moreover the information about bad primes (like the conductor) is much more accessible. In conjunction with David Roberts, the latest version of Magma has an implementation of Jacobi-sum motives that in particular has the correspondence to Grossencharacters as its main tool. We shall describe its workings in more detail.