In number theory, the Hecke algebra corresponding to a congruence subgroup of the modular group is spanned by elements of the double coset space formula_29; the algebra structure is that acquired from the multiplication of double cosets described above. Of particular importance are the Hecke operators formula_30 corresponding to the double cosets formula_31 or formula_32, where formula_33 (these have different properties depending on whether and are coprime or not), and the diamond operators formula_34 given by the double cosets formula_35 where formula_36 and we require formula_37 (the choice of does not affect the answer).