Aygin Z.S. Introduction to Applications of Modular Forms: Computational Aspects

Springer, 2023. — 175 p.Preface
Acknowledgements
Dirichlet Characters
The Definition and Basic Properties
Primitive Dirichlet Characters
Orthogonal Relations
Gauss Sums Associated with Dirichlet Characters
Dirichlet L-Functions and Bernoulli Numbers
Modular Forms: Definition and Some Properties
The Definition of Modular Forms
Modular Groups Γ(N) and the Transformation Property
Definition of Modular Forms
Some Properties of the Modular Forms
Newforms
Eisenstein Series
Definition of Eisenstein Series
Constant Terms of Eisenstein Series
The Sum of Divisors Function
Projections of Modular Forms on the Eisenstein Series
The Size of the q-Series Coefficients of the Eisenstein Part of a Modular Form
Application: Quadratic Forms
The Modularity and the Constant Terms
Gauss Sums Associated with Quadratic Forms and the Eisenstein Part of θ(Q;z)
Separable Quadratic Forms
A Family of Binary Quadratic Forms
Diagonal Quadratic Forms
Generalized Kronecker Symbol
Constant Terms of Theta Functions Corresponding to the Diagonal Quadratic Forms
An Explicit Formula for Representation Numbers of Diagonal Forms
Diagonal Quadratic Forms Examples
Application: Eta Quotients
The Definition, the Modularity and the Constant Terms
Finding the Complete Sets of Eta Quotients in Mk(Γ(N),χ)
The Eisenstein Part of Eta Quotients
Eta Quotients in Ek(Γ(N),χ)
The Level Examples
The Level Examples
The Level Examples
Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series
Eta Quotients and Newforms
Bases for Cusp Form Spaces
Various Applications
The Convolution Sums
Klein Forms
The Character Analog of Dedekind's Eta Function
Cusp Forms and the Character Analog of Dedekind's Eta Function
Elliptic Curves
References
Index
Index of Notation
Index of SAGE Functions