Saito–Kurokawa lifts of square-free level

Abstract

Let $f\in S_{2\kappa -2}\left({\Gamma }_0\right(M\left)\right)$ be a Hecke eigenform with $\kappa \ge 2$ even and $M\ge 1$ and odd and square-free. In this paper we survey the construction of the Saito–Kurokawa lifting from the classical point of view. We also provide some arithmetic results on the Fourier coefficients of Saito–Kurokawa liftings. We then calculate the norm of the Saito–Kurokawa lift.