On interpretation of Fourier coefficients of Zagier type lifts

Jeon, Kang, and Kim defined the Zagier lifts between harmonic weak Maass forms of negative integral weights and half integral weights. These lifts were defined by establishing that traces related to cycle integrals of harmonic weak Maass forms of integral weights appear as Fourier coefficients of harmonic weak Maass forms of half integral weights. For fundamental discriminants d and \(\delta ,\) $\delta ,$ they studied \(\delta \) $\delta$ -th Fourier coefficients of the d-th Zagier lift with respect to the condition that \(d\delta \) $d\delta$ is not a perfect square. For \(d\delta \) $d\delta$ being a perfect square, the interpretation of coefficients in terms of traces is not possible due to the divergence of cycle integrals. In this paper, we provide an alternate definition of traces called modified trace in the condition that \(d\delta \) $d\delta$ is a perfect square and interpret such coefficients in terms of the modified trace.