ON THE NATURE OF ADDITIONAL SPACE AT CUTTING OF SPACES OF CUSP FORMS

In the article we study a space of cusp forms by the method of cutting. This space is a direct sum of the subspace of forms divided by the fixed cusp form named the cutting function and the additional space. If the additional space is zero we have the situation of exact cutting. In common case the cutting is not exact and it is important to research the nature of the additional space. We prove that the basis of the additional space can be described by the space of cusp forms of small weight. This weight is not more than 14 and often is equal to 4. We give examples of all cutting functions for all levels. We prove the theorem about the basis of the additional space to the space of cusp forms in the space of modular forms of the same level, weight and character. We use properties of eta-products, Biagioli formula for orders in cusps and Cohen Oesterle formula for dimensions.