In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields $$\mathbb {Q}(\sqrt{D})$$Q(D) and $$\mathbb {Q}$$Q.

中文翻译：

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关于实二次域上 L 值的线性关系

在本文中，我们给出了一种从希尔伯特模形式构造经典模形式的方法。通过应用这种方法，我们可以得到与希尔伯特和经典模形式的傅立叶系数相关的线性公式。本文重点讨论实二次域上的希尔伯特模形式。我们将说明 L 函数的特殊值（尤其是在 0 处）与算术函数之间关系的构造。我们还将给出具有基础字段 $$\mathbb {Q}(\sqrt{D})$$Q(D) 和 $$\mathbb {Q}$$Q 的平方和函数之间的关系。