Given any nonzero modular form $f$ (of any weight, any level, any character), consider its $q$-expansion $f(z) = \sum_n a(n) q^n$, where $q=\exp(2\pi iz)$.

Proposition: infinitely many of the coefficients $a(n)$ are nonzero.

Which of the following statements holds?

(i) The proposition is true for trivial reasons

(ii) The proposition is false due to simple examples

(iii) The Proposition is true, but needs some nontrivial arguments

(iv) The Proposition is false, but only for sophisticated examples

Is eventually some refinement of the proposition true?

Would be glad for an answer to this question! Thanks