Modular forms are complex functions that are defined on the upper half of the complex plane and exhibit specific symmetry properties. They are important in number theory and have applications in various areas of mathematics, including the study of elliptic curves and automorphic forms. These functions are often studied in relation to modular arithmetic and can be represented as a series expansion. Modular forms can be classified into different types, such as cusp forms and newforms, each with unique characteristics and significance in mathematical research.