We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

中文翻译：

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具有非酉扭曲的向量值自守函数的傅立叶展开

我们提供双曲拉普拉斯算子的向量值本征函数的傅里叶展开式，该函数在全周方向上是扭曲周期的。扭曲可以由有限维向量空间的任何自同态给出；没有关于可逆性或单一性的假设。这种特征函数的例子包括 Fuchsian 群的向量值扭曲自守形式。我们进一步提供了傅立叶系数的详细描述，并明确地识别了它们的每个成分，这些成分与扭曲自同态的特征值及其约旦块的大小密切相关。此外，我们确定了傅里叶系数的增长特性。