Let $N$ be an odd square-free integer. We give a Hecke isomorphism map from the space of Jacobi cusp new forms of level $1$ and index $N$ satisfying a certain condition into the space of Jacobi cusp new forms of level $N$, index $1$ and the character after reviewing about some known properties of level-index changing operators.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/7600