Modular forms and representations of symmetric groups

We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of the symmetric group $S_n$ over a field of characteristic $p$ for $n\to\infty$. For $p\leqslant7$ we give simple explicit formulas for the number of blocks of defect zero. The study of the modular forms leads to interesting identities involving the Dedekind $n$-function.