Hecke multiplication relations ($T_m{T}_n={\sum }_{d\mid \gcd (m,n)}{d}^{k-1}{T}_{mn/d^2}$)

For level-one weight-$k$ modular forms, $T_m{T}_n={\sum }_{d\mid \gcd (m,n)}{d}^{k-1}{T}_{mn/d^2}$. Thus coprime indices multiply directly, and $T_p{T}_{p^r}=T_{p^{r+1}}+p^{k-1}{T}_{p^{r-1}}$. The Fourier coefficients of a composite-index Hecke operator prove these relations by regrouping divisors. They show that the Hecke algebra of modular forms is commutative and generated by prime operators.