  Ahlgren and Samart relate three cusp forms with complex multiplication to
certain weakly holomorphic modular forms using $p$-adic bounds related to their
Fourier coefficients. In these three examples, their result strengthens a
theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly
holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the
theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono
use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's
work to all cases where the cusp form space is one-dimensional and has trivial
Nebentypus. Along the way, we obtain a duality result relating two families of
modular forms that arise naturally in each case.
