Binary Hermitian forms over a cyclotomic field

Let ζ be a primitive fifth root of unity and let F be the cyclotomic field F=Q(ζ). Let O⊂F be the ring of integers. We compute the Voronoï polyhedron of binary Hermitian forms over F and classify GL2(O)-conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group GL2(O) and Hecke eigenforms.