Let \(M_k^{\prime}(p)\) (resp. \(M_k^{\prime\, +}(p)\)) be the space of weakly holomrorphic modular forms of weight \(k\) for the Hecke group \(\Gamma_0(p)\) (resp. \(\Gamma_0^+(p)=\) where \(W_p\) is the Fricke involution). Let \(M_k^{\prime\, -}(p)\) denote the minus subspace of \(M_k^{\prime}(p)\) consisting of all eigenfunctions of \(W_p\) with eigenvalue \(-1\). Then \(M_k^{\prime}(p)=M_k^{\prime\, +}(p)\oplus M_k^{\prime\,-}(p)\), and a canonical basis for the plus subspace \(M_k^{+}(p)\) has been constructed for even weight \(k\) by the first two authors in the article quoted below. A goal of this paper is to find a canonical basis for the minus subspace \(M_k^{\prime\,-}(p)\) and study its arithmetic properties. A canonical basis for \(M_k^{\prime\,-}(p)\) consists of the form \(f_{k,m}^{-}\) with FGourier expansion (\(q=e^{2\pi iz}\)): \(f_{k,m}^{-}=q^{-m}+\sum_{n>m_k^{-}} a_k^{-}(m,n)q^n\) for some integer \(m_k^{-1}\). (Indded, \(m_k^{-}\) is the maximal vanishing order at the cusp \(\infty\) for a nonzero \(f\in M_k^{\prime\,-}(p)\).) It is shown that \(a_k^{-}(m,n)\) are integers and satisfy the duality relation \(a_k^{-}(m,n)=- a_{2-k}^{-}(n,m)\). Then a generalization of the results of the first two authors [J. Number Theory 133, No. 4, 1300--1311 (2013; Zbl 1282.11027)] to genus zero groups \(\Gamma_0^*(N)\) (for \(N\) square-free) is presented.