Trace formula for Jacobi forms of odd squarefree level

We study the traces of Hecke operators acting on the space of Jacobi forms and give the trace formula in the case of general odd squarefree level N by using results of Skoruppa and Zagier. As an application, we obtain some trace relations of them, and construct an isomorphism map from the space of all Jacobi cusp new forms of level N and index 1 having the eigenvalue +1 with respect to any Atkin–Lehner operator on the level to the space of all Jacobi cusp new forms of level 1 and index N whose eigenvalue with respect to any Atkin–Lehner operator on the index is equal to +1.