Periods of Cusp forms and elliptic curves over imaginary quadratic fields

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups , where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series at and compare with the value of which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that whenever has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.