  Using Hecke characters, we construct two infinite families of newforms with
complex multiplication, one by $\mathbb{Q}(\sqrt{-3})$ and the other by
$\mathbb{Q}(\sqrt{-2})$. The values of the $p$-th Fourier coefficients of all
the forms in each family can be described by a single formula, which we provide
explicitly. This allows us to establish a formula relating the $p$-th Fourier
coefficients of forms of different weights, within each family. We then prove
congruence relations between the $p$-th Fourier coefficients of these newforms
at all odd weights and values coming from two of Zagier's sporadic Ap\'ery-like
sequences.
