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In this note, we generalize the isomorphisms to the case when thediscriminant form is not necessarily induced from real quadratic fields. Inparticular, this general setting includes all the subspaces withepsilon-conditions, only two spacial cases of which were treated before. Withthis established, we shall prove the Zagier duality for canonical bases.Finally, we raise a question on the integrality of the Fourier coefficients ofthese bases elements, or equivalently we concern the existence of a Miller-likebasis for vector valued modular forms.

Zagier duality and integrality for Fourier coefficients for weakly holomorphic modular forms