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I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

In more concrete terms, maps between functions with $q$-expansions, $$q^{1/N}(\sum_{m=-n}^{\infty} a_m q^m) \to q^{1/M}(\sum_{m=-n'}^{\infty} a'_m q^m)$$

I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

In more concrete terms, maps between modular functions with $q$-expansions, $$q^{1/N}(\sum_{m=-n}^{\infty} a_m q^m) \to q^{1/M}(\sum_{m=-n'}^{\infty} a'_m q^m)$$

I am working with modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

In more concrete terms, maps between functions with $q$-expansions, $$q^{1/N}(\sum_{m=-n}^{\infty} a_m q^m) \to q^{1/M}(\sum_{m=-n'}^{\infty} a'_m q^m)$$

I am working with modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(N)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious way is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

I am working with modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?