  We study the low-lying zeros of a family of $L$-functions attached to the CM
elliptic curves $E_d \;:\; y^2 = x^3 - dx$, for each odd and square-free
integer $d$. Writing the $L$-function of $E_d$ as $L(s-\frac12, \xi_d)$ for the
appropriate Gr\"ossencharakter $\xi_d$ of conductor $\mathfrak{f}_d$, the
family $\mathcal{F}_d$ is defined as the family of $L$-functions attached to
the Gr\"ossencharakters $\xi_{d,k}$, where for each integer $k \geq 1$,
$\xi_{d, k}$ denotes the primitive character inducing $\xi_d^k$. We observe
that $25\%$ of the functions in $\mathcal{F}_d$ have negative root number. This
makes the symmetry type of the family (unitary, symplectic or orthogonal)
somewhat mysterious, as none of the symmetry types would lead to this
proportion. We give an asymptotic expression for the one-level density in the
family of $L$-functions in $\mathcal{F}_{d}$ with conductor at most $K^2
\mathrm{N} (\mathfrak{f}_d)$, and find that $\mathcal{F}_d$ breaks down into
two natural subfamilies; namely, a symplectic family ($L(s, \xi_{d,k})$ for $k$
even) and an orthogonal family ($L(s, \xi_{d,k})$ for $k$ odd). For $k$ odd,
$\mathcal{F}_d$ is in fact a subfamily of the automorphic forms of fixed level
$4 \mathrm{N} (\mathfrak{f}_d )$, and even weight $k+1$, and this larger family
also has orthogonal symmetry. Finally, we compute explicit lower order terms in
decreasing powers of $\log (K^2 \mathrm{N} (\mathfrak{f}_d) )$ for each case.
