One-level densities in families of Gr\"ossencharakters associated to CM elliptic curves

We study the low-lying zeros of a family of L-functions attached to the CM elliptic curves Ed \;: \; y² = x³ - dx, for each odd and square-free integer d. Writing the L-function of Ed as L (s-12, d) for the appropriate Gr\"ossencharakter d of conductor fd, the family Fd is defined as the family of L-functions attached to the Gr\"ossencharakters ₃, ₊, where for each integer k 1, ₃, ₊ denotes the primitive character inducing dᵏ. We observe that the average root number over the family Fd is 14, which makes the symmetry type of the family (unitary, symplectic or orthogonal) somehow mysterious, as none of the symmetry types would lead to this average value. By computing the one-level density, we find that Fd breaks down into two natural subfamilies, namely a symplectic family (L (s, ₃, ₊) for k even) and an orthogonal family (L (s, ₃, ₊) for k odd). For k odd, Fd is in fact a subfamily of the automorphic forms of fixed level 4 N (fd), and even weight k+1, and this larger family also has orthogonal symmetry. The main term of the one-level density gives the symmetry and we also compute explicit lower order terms for each case.