Fourier coefficients of modular forms

Question

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

Answer 1

If we allow weight zero then the claim is false; indeed the only nonzero modular forms of weight zero are nonzero constants, which have $a(n) \neq 0$ iff $n=0$.

For nonzero weight $k$, the proposition is true and easily proved: if finitely many $a(n)$ were nonzero then $f(z)$ would be a nonconstant polynomial in $q$, and such a function cannot satisfy a functional equation $f(z) = (cz+d)^{-k} f\bigl(\!\frac{az+b}{cz+d}\!\bigr)$ with $c \neq 0$ (for instance, because the right-hand side has an essential singularity at $z = -d/c$ while the left-hand side is an entire function).

Postscript: MathOverflow's auto-generated list of "Related" questions reminds me that seven years ago I showed, in my answer to another MO question (#244808: Can something finite over $\mathbb{C}(q)$ be a modular form?), that a nonconstant modular form cannot even be algebraic over ${\bf C}(q)$.

Answer 2

Among the McKay-Thompson series [Find "McKay-Thompson sequences or series for Monster simple group, sequences related to" in the OEIS index HERE ] are two degenerate cases of replicable functions with finite Fourier series...

A154272 $\qquad j_{1a} = q^{-1}+q = 2\cos(2\pi z)$

A154272$\qquad j_{2b} = q^{-1} - q = -2i\sin(2\pi z)$

Replicable series can sometimes be viewed as "weight zero modular forms".