Generating functions of many multiplicative arithmetic functions of longstanding interest (e.g., sum-of-divisors function, number-of-partitions function) turn out to be Fourier expansions of modular forms.

Is this true of all multiplicative arithmetic functions? If $\gamma : \mathbb{N} \rightarrow \mathbb{C}$ is a multiplicative arithmetic function (i.e., $\gamma(n m) = \gamma (n) \gamma (m)$ whenever $n$ and $m$ are relatively prime), is the formal sum $\sum_{n=1}^\infty \gamma(n) q^n$ (related to) Fourier expansion of some modular/automorphic form?

A different way of putting the question is: is there a direct relation between the multiplicative property of arithmetic functions and the symmetry of modular/automorphic forms under the action of associated modular/reductive group, or is it the case that some arithmetic functions just happen to have modular forms as generating functions without there being any larger conceptual relationship between the two?