For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

中文翻译：

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无平方水平的 Eichler 阶和 Jacobi 形式

对于在有理数域上的定四元数代数中具有相同平方自由水平的所有 Eichler 阶，我们证明与这些阶相关的 Jacobi theta 级数的加权和是 Jacobi Eisenstein 级数。将后者的傅立叶系数乘以 2 的幂，得到我们修改后的 Hurwitz 类数。作为其推论，修改后的 Hurwitz 类数可用于计算 Brandt 矩阵的迹和 Eichler 阶的类型数。