ARITHMETIC OF WEAKLY HOLOMORPHIC MODULAR FORMS FOR HECKE GROUPS
Duke and Jenkins [4] constructed a nice canonical basis for the space of weakly holomorphic modular forms of weight k for that are holomorphic away from the cusp at infinity. They investigated the arithmetic of these basis elements. Let be the space of weakly holomorphic modular forms of weight k for that are holomorphic away from the cusp at infinity. We generalize the results of Duke and Jenkins to the space when the genus of is zero. As applications, first, we find a basis which consists of Poincaré series, for the space of cusp forms for Second, we show that the algebraicity of coefficients of the holomorphic part of a harmonic weak Maass form in is determined by its first few coefficients.
weakly holomorphic modular forms, Poincaré series.