Bounds for the coefficients of cusp forms for Γ0(3)

We give bounds of the coefficients of cusp forms f for Γ0(3) in terms of its first dk numbers of coefficients of f and f|kW3, where W3 is the Fricke involution of level 3, and dk is the dimension of the space Sk(Γ0(3)) of weight k cusp forms for Γ0(3). In particular, we find bounds of the coefficients of cusp forms for Γ0+(3).