Hecke operator on Jacobi forms

Question

I study Jacobi forms and met the following problem:
I need to prove that Hecke operator correclty defined
$U_d: J_{k, m} \to J_{k, md^2}$ - correctly defined map, $d \in \mathbb{N}$
$\phi(\tau, z) \mapsto \phi(\tau, dz)$
(so if we have Jacobi form $\phi \in J_{k, m}$, then $U_d(\phi) \in J_{k, md^2}$)
I want to prove it with fourier expansion of Jacobi form:
$\phi(\tau, z) = \sum\limits_{n, l \in \mathbb{Z}}a(n, l) e^{2\pi i(n\tau + lz)}$.
Since we deal with holomorphic Jacobi forms, we actually have the sum: (equivalent def of holomorphic Jacobi form)
$\phi(\tau, z) = \sum\limits_{n \geq 0} \sum\limits_{l^2 \leq 4mn} a(n, l) e^{2\pi in\tau}e^{2\pi i lz}$
But $U_d(\phi)(\tau, z) = \phi(\tau, dz) = \sum\limits_{n \geq 0} \sum\limits_{l^2 \leq 4mn} a(n, l) e^{2\pi in\tau}e^{d(2\pi i lz)}$
And since $4mn \leq 4mnd$, $U_d(\phi)$ "obviously" in $ J_{k, md^2}$.
But my proof also imply that $U_d(\phi)$ in $J_{k, m}$, which is stronger conclusion.
And I feel like somewhere I made a mistake, but cannot find it