Some studies on Klein forms and Eisenstein series

Abstract

In this thesis we discuss some modularity criterion for a product of Klein forms of the congruence subgroup $\Gamma_1(N)$ and some applications of Eisenstein series to the theta functions associated with certain quadratic forms. As an application of the Klein forms, we construct a basis of the space of modular forms for $\Gamma_1(13)$ of weight $2$. In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form. We extend it to more general cases and prove it by applying Fricke involutions and Hecke operators to Eisenstein series and theta functions.