SHIMURA INTEGRALS OF CUSP FORMS

Abstract

This paper studies integrals of the form on the upper half-plane, where is a rational number, is integral, and is a cusp form of weight with respect to some modular group . The main result is that if is a congruence subgroup and is an eigenvector of all the Hecke operators, then all these integrals are representable as linear combinations of two complex numbers with coefficients in some field of algebraic numbers. Bibliography: 13 titles. Figures: 11.