 Hecke multiplication relations

ID: hecke-multiplication-relations

For level-one weight-

k

T_{m} T_{n} = \sum_{d ∣ g c d (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.

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