We twist Zagier’s function f_k,D by a sign function and a genus character. Assuming weight 0 < k ≡ 2 (mod 4), and letting D be a positive non-square discriminant, we prove that the obstruction to modularity caused by the sign function can be corrected obtaining a locally harmonic Maaß form or a local cusp form of the same weight. In addition, we provide an alternative representation of our new function in terms of a twisted trace of modular cycle integrals of a Poincaré series due to Petersson.

中文翻译：

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正偶权重的局部调和马斯形式

我们通过符号函数和属符来扭曲 Zagier 的函数f _{k,D 。}假设权重 0 < k ≡ 2 (mod 4)，并令D为正非平方判别式，我们证明由符号函数引起的模块化障碍可以通过获得局部调和 Maaß 形式或局部尖点形式来纠正相同的重量。此外，我们还根据 Petersson 提出的庞加莱级数模循环积分的扭曲迹来提供新函数的替代表示。