Hi,

I am interested in the following fact:

Suppose that $f(z)$ is a modular function over $\text{SL}_2(\mathbb{Z})$ such that it has the $q$-expansion

$f(z) = q^{-m} + \displaystyle \sum_{n=1}^\infty a_m(n) q^n$

It is claimed in Ono, K, "The partition function and Hecke operators", Advances in Mathematics 228 (2011), 527-534 that this function (which he denotes $j_m(z)$) is unique.

I know that modularity imposes strict restrictions on what modular functions must look like, but I have a problem seeing this claim. Can anyone offer a short proof or some ideas as to why this is the case?

Thanks in advance.