On Sato-Tate like problems on half-integral weight of modular forms

Modular forms are an important subject of mathematics, especially number theory, and they are studied intensively. Because it brings together many branches of science, 'modular forms are everywhere' for many mathematicians. In the first part of this six-part study, modular forms will be defined and their basic properties will be examined, thus creating the background needed in the thesis. In Chapter 2, which is the first of the original parts of the thesis, the solution of the systematic selection problem of half-integral weight Hecke eigenforms will be given. In Chapter 3, Sato-Tate Conjecture, one of the most important achievements in mathematics of the 21st century, will be introduced and an application on the Bruinier-Kohnen Conjecture will be given. In the second part of the original part, Chapter 4, the distribution of Fourier coefficients of half-integral weight modular forms normalized by the Ramanujan-Petersson Conjecture will be discussed, an open question will be raised and the claim will be supported with all possible data. The last part of the original part, In Chapter 5, the Bruinier-Kohnen Conjecture will be strengthened and expressed. The sixth and last part consists of discussion, conclusion and observations.