= Hecke multiplication relations
{c}
{title2=$T_mT_n=\sum_{d\mid\gcd(m,n)}d^{k-1}T_{mn/d^2}$}
For level-one weight-$k$ <modular forms>, $T_mT_n=\sum_{d\mid\gcd(m,n)}d^{k-1}T_{mn/d^2}$. Thus coprime indices multiply directly, and $T_pT_{p^r}=T_{p^{r+1}}+p^{k-1}T_{p^{r-1}}$. 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.
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