Saito–Kurokawa liftings of level N and practical construction of Jacobi forms

Saito–Kurokawa liftings from Jacobi forms of degree one of weight k of level N to Siegel modular forms of degree two of weight k of level N with or without character are explicitly given by describing their Fourier expansions. Their L-functions are also given as well as the action of Hecke operators including some operators at bad primes. Also, a practical way of constructing Jacobi forms of given level is explained. This is explicitly executed for Jacobi forms of level up to 5, and their explicit structure theorems are given.