Petersson Inner Product and Hecke Operators

This lecture introduces the Petersson inner product, a fundamental concept in the theory of modular forms, and explores the definition and properties of Hecke operators, crucial tools in the study of modular forms as functions on lattices. The lecture also covers the translation of Hecke operators to the language of matrices and their normalization. Additionally, it discusses the equivalence of different definitions of Hecke operators and their implications on Fourier coefficients, providing insights into the structure and completeness of certain collections of matrices.