The vanishing of Poincaré series

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ (1) is a linear combination over the complex field C of the Poincaré series. Here m is any positive integer, z ∈ H =z ∈ C: Im z>0 and The summation is over all matrices with different second rows in the (homogeneous) modular group, i. e. in SL (2, Z). The factor ½ is introducted for convenience.