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.

On a Fourier series put $U_p(\sum a_n{q}^n)=\sum a_{pn}{q}^n$ and $V_{p}f(q)=f(q^p)$. These auxiliary operators need not individually preserve the level-one modular form space. Nevertheless $U_p{V}_p=1$, and the Fourier coefficients of a composite-index Hecke operator give $T_{p^r}={\sum }_{j=0}^r{p}^{j(k-1)}{V}_p^j{U}_p^{r-j}$. Multiplication by $T_p=U_p+p^{k-1}{V}_p$, using $U_p{V}_p=1$, gives the displayed recurrence. In particular the distinction between $U_p{V}_p=1$ and $V_p{U}_p\ne 1$ is essential.