Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 137 / 3 / Solution

Use the valence formula for the modular group: for a nonzero modular form of weight $k$ on $S{L}_2(\mathbb{Z})$,The sum contains one representative of each other modular group orbit of zeros. The orders are nonnegative because $f$ is holomorphic on the complex upper half-plane and at infinity. The half and third weights account for the elliptic stabilizers of the modular group.

In weight two, the transformation under $S=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ at its fixed point gives $f(i)=i^{2}f(i)=-f(i)$. Thus any nonzero $f$ would have $v_i(f)\ge 1$. The valence formula for the modular group would give a left side at least $1/2$ and a right side $1/6$, which is impossible. Therefore the unheaded request is answered by vanishing of weight-two level-one modular forms: