Saito-Kurokawa lifts of square-free level

Let f ∈ S2κ−2(Γ0(M)) be a Hecke eigenform with κ ≥ 2 even and M ≥ 1 and odd and square-free. In this paper we survey the construction of the Saito-Kurokawa lifting from the classical and representation theoretic 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.