  Griffin, the second author, and Molnar studied coefficient duality for
canonical bases for a broad range of spaces of weakly holomorphic modular
forms, showing that the Fourier coefficients of canonical basis elements appear
as negatives of Fourier coefficients for elements of a canonical basis of a
related space of forms. We investigate the effect of the trace operator on this
duality for modular forms for $\Gamma_0(N)$ of genus zero and show exactly when
duality still holds after applying the trace operator.
