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
Γ
0
(
n
)
{\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
)
L(F,s)
at
s
=
1
s = 1
and compare with the value of
L
(
E
,
1
)
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
L(F,1) = 0
whenever
E
(
K
)
E(K)
has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.