Integral Hecke algebra of level-one cusp forms ($\mathbb{T}\subseteq \mathrm{End}(S_k(\Gamma (1)))$)

The integral Hecke algebra of level-one cusp forms is the subring generated by the Hecke operators $T_n$ and the integer scalars acting on a fixed weight-$k$ space of level-one cusp forms. It preserves the free abelian group of cusp forms with integral Fourier coefficients, by the Fourier coefficients of a composite-index Hecke operator formula. Thus one can study Hecke operator eigenvalues through integral matrices, including reduction modulo prime numbers.

This is a particular modular Hecke algebra, with weight, level, and coefficient ring specified. It is distinct from the generic Hecke algebra attached to a Coxeter system.

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.