模形式的Galois表示是现代数论研究的热点之一。然而，关于模形式伽罗瓦表示的计算方面的研究是近年来刚刚发展起来的，是当代数论研究的前沿问题之一。. 本项目的主要内容包括：1.研究如何计算level为任意正整数的模形式的伽罗瓦表示，提出可实现计算模形式伽罗瓦表示的算法。2. 对模形式伽罗瓦表示的计算，可以用来计算模形式傅里叶系数。本项目将进一步研究如何有效计算该系数，并把关于Ramanujan Tau函数的相关计算，推广到计算一般的level为1的模形式的系数上，从而讨论其傅里叶系数的分布情况。3.本项目还将讨论当素数l在模形式的系数数域中分歧时，关于mod l 伽罗瓦表示的计算算法，及其在伽罗瓦反问题上的应用，即对有限域上二阶一般线性群的某些子群，判断其是否同构于某个数域的伽罗瓦群。. 本项目的研究包含理论创新和具体应用，研究内容是目前迫切待解决的问题

The research on Galois representations associated to modular forms is one of the focuses in modern number theory. However, the research on the computational aspects of modular Galois representations is still not developed and is one of the frontier disciplines...In this project, the content consists of .1. Discuss how to compute Galois representations arising from modular forms with any level. Propose implementable algorithms to explicitly compute the modular Galois representations. .2. The Fourier coefficients of modular forms are accessible if the modular Galois representation is computed and this has been applied for Ramanujan's Tau function. In this project, we will do further work on how to efficiently compute the Fourier coefficients. Moreover, we will do the computations on the modular forms with level 1 and then discuss the distribution of the Fourier coefficients. .3. We will discuss the computations of mod l modular Galois representations for prime l which is ramified in the coefficient number fields of modular forms. Then we apply the results to solve the inverse Galois problem for certain finite groups...The researches in the project contain theoretical and computational aspects of modular Galois representations. Moreover the results will be applied for solving many interesting problems. Therefore, our researches in this project are worth carrying out.