Mackay functions and exact cutting in spaces of modular forms

In the article we consider structure problems in the theory of modular forms. The phenomenon of the exact cutting for the spaces $S_k(\Gamma_0(N),\chi),$ where $\chi$ is a quadratic character with the condition $\chi(- 1) = ( - 1)^k$. We prove that for the levels $N \ne 3,~17,~19$ the cutting function is a multiplicative eta-product of an integral weight. In the article we give the table of the cutting functions. We prove that the space of an cutting function is one-dimensional. Dimensions of the spaces are calculated by the Cohen–Oesterle formula, the orders in cusps are calculated by the Biagioli formula.