We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.

中文翻译：

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亚纯模形式傅里叶系数的算术性质

我们研究与正定积分二元二次形式相关的权重为 $2k$ 的亚纯模形式的傅立叶系数的完整性和可分性。例如，我们证明如果不存在重量为 $2k$ 的非平凡尖点形式，那么对于每个自然数，这些亚纯模形式的 $n$-th 系数可被 $n^{k-1}$ 整除$n$。此外，我们证明了它们的系数是不为零的，并且具有恒定或交替的符号。最后，我们得到了亚纯模形式的傅里叶系数、$j$-函数的系数和配分函数之间的关系。