nt.number theory – Hecke convergence factor

I was reading a paper here. There the author define an infinite series


where $k$ is an even integer bigger than 2 and $2leqslant jleqslant k-2$. Then this series is a cusp form. This series represents the $(j-1)$th period linear map of cusp forms. It still make sense for 0th and $(k-2)$th period, but the series defined in this way not converge. Then the author said we should introduce a Hecke convergence factor. But I could not find too much about what this thing is. Does anyone know anything about it? How will it help to define the series?