這篇論文的目標是要去改善斯特姆界限。我們從模型式空間Mk(Γ)和Sk(Γ)的維度公式的證明中可以知道它們都是有限維的複數向量空間。然後我們定義出Mk(Γ)和Sk(Γ)的「真界限」與其斯特姆界限相比較，最後透過在sage中計算他們的差，我們找到了一個改善斯特姆界限的方法。

The aim of the paper is to improve the original Sturm bound. First,
we recall the statement and proof of the dimension formulas for Mk(Γ) and Sk(Γ) which shows that Mk(Γ) and Sk(Γ) are finite dimensional vector spaces over C. Then, we define the "true bounds" of Mk(Γ) and Sk(Γ), and compare them with the Sturm bound. Finally, according to our computation of their differences on sage, we find a way to improve the Sturm bound.

1. introduction 5
2. Riemann surface 7
3. Modular forms 9
4. Modular curves 12
5. Coordinate neighborhoods 13
6. Dimension formula 15
7. Sturm bound 25
Appendix 31
A. computational result 31
B. The proof of the Riemann-Roch theorem 39
Bibliography 47

1. Fred Diamond and Jerry Shurman: A First Course in Modular Forms. Springer, New York (2005)
2. Rick Miranda: Algebraic Curves and Riemann Surfaces. American Mathematical Society, Providence, United States (1995)
3. William A.Stein: Modular Forms:A Computational Approach. Americam Mathematical Society, Providence, United States (2007)