Group cohomology for modular forms with singularities

For a nonzero divisor D:=∑_t=1ⁿp_DtD_t of X₀(1) with p_Dt>0, let M_k^!,D(SL₂(Z)) be the space of meromorphic modular forms f of integral weight k on SL₂(Z) such that f is holomorphic except at {D₁,…,D_n} and that the order of pole of f at each Q∈{D₁,…,D_n} is less than or equal to p_Q. In this paper, we give an isomorphism between M_k^!,D(SL₂(Z)) and the first cohomology group with a certain coefficient module P_D when k is a negative even integer. More generally, by considering another coefficient module P_k^weak, we prove that there exists an isomorphism between M_k^!(SL₂(Z)) and H¹(SL₂(Z),P_k^weak), where M_k^!(SL₂(Z)) denotes the space of weakly holomorphic modular forms of integral weight k on SL₂(Z).