Let $\tau \in \mathbf{H}$ be in the complex upper half-plane and let $\wp(\tau, z)$ be the Weierstrauss $\wp$-function attached to the lattice $\langle1, \tau \rangle$. It is known that if we evaluate $\wp(\tau, z)$ at $N$-torsion points of the lattice $\mathbf{C}/(\mathbf{Z} + \mathbf{Z}\tau)$, we obtain a modular form of level $\Gamma(N)$. Precisely, if we define the function

$$f_{\frac{a}{N}, \frac{b}{N}}(\tau) = \wp \left(\tau, \dfrac{a}{N}\tau + \dfrac{b}{N} \right)$$

then $f_{\frac{a}{N}, \frac{b}{N}}$ is a weight $2$ modular form of level $\Gamma(N)$: $$f_{\frac{a}{N}, \frac{b}{N}}(\tau) \in M_2(\Gamma(N)).$$

My question is: Now suppose $p$ is a prime. Can we define a "Hecke operator" $T_p$ acting on the Weierstrauss $\wp$-function such that

$$T_p(\wp)\left(\tau, \dfrac{a}{N}\tau + \dfrac{b}{N} \right) = T_p f_{\frac{a}{N}, \frac{b}{N}}?$$

The $T_p$ on the right is the usual Hecke operator acting on $M_2(\Gamma(N))$. That is, can we define a function $T_p(\wp)(\tau, z)$ such that if we evaluate $T_p(\wp)(\tau, z)$ at the $N$-torsion point $z = \frac{a}{N}\tau + \frac{b}{N}$, we recover the usual Hecke operator $T_p$ acting on the modular form $f_{\frac{a}{N}, \frac{b}{N}}$?

What I've thought of so far: I came across the book "The Theory of Jacobi Forms" by Eichler and Zagier where they define Hecke operators on Jacobi forms. But their definition of Hecke operators does not seem to commute with the usual definition of Hecke operators when I specialize the $\wp$ function to torsion points, so I think another definition would be needed. Any insights / references would be appreciated :)