= Solution
Use the <valence formula for the modular group>: for a nonzero <modular form> of weight $k$ on $SL_2(\mathbb Z)$,
$$
v_\infty(f)+\frac12v_i(f)+\frac13v_\rho(f)
+\sum_{z\ne i,\rho}v_z(f)=\frac{k}{12},\qquad\rho=e^{2\pi i/3}.
$$
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 $S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ at its fixed point gives $f(i)=i^2f(i)=-f(i)$. Thus any nonzero $f$ would have $v_i(f)\geq1$. 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>:
$$
\boxed{M_2(\Gamma(1))=\{0\}.}
$$
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