The effect of the trace operator on the duality of modular grids in genus zero levels

Abstract: 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.