Vanishing of weight-two level-one modular forms
= Vanishing of weight-two level-one modular forms
{title2=$M_2(SL_2(\mathbb Z))=0$}
A weight-two <modular form> obeys $f(i)=i^2f(i)=-f(i)$, so it vanishes at $i$. The <valence formula for the modular group> would then have left side at least $1/2$, but right side $2/12=1/6$. All orders are nonnegative by holomorphy, giving a contradiction for nonzero $f$.