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 ${\Gamma _0}(\mathfrak{n})$, 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 $L(F,s)$ at $s = 1$ and compare with the value of $L(E,1)$ which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that $L(F,1) = 0$ whenever $E(K)$ has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.