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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.