The Hecke algebra associated to a level U_1(S) structure is commutative

[kbuzzard]

The Hecke algebra is generated by T_v and U_{v,alpha}. For different v's these things are supported in different places so they commute; for the same v and different alpha it's TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_comm.

[Ruben-VandeVelde]

This is already proved