Perfect integral Hecke pairing ($\mathbb{T}\times S_k(\Gamma (1),\mathbb{Z})⟶\mathbb{Z}$)

The pairing $(T,f)\mapsto a_1(Tf)$ identifies the integral Hecke algebra of level-one cusp forms with the integral dual of the cusp-form lattice. An integral basis beginning $q,q^2,\dots ,q^m$ gives a unitriangular first-$m$ coefficient matrix, so $a_1,\dots ,a_m$ form a dual basis. Commutativity of the Hecke operators proves nondegeneracy on the algebra, and $T_1,\dots ,T_m$ then form its integral basis.