Editing Hecke operator (section)

== Hecke algebras ==
{{Main|Hecke algebra}}
Algebras of Hecke operators are called "Hecke algebras", and are [[commutative ring]]s. In the classical [[elliptic modular form]] theory,  the Hecke operators {{math|''T''<sub>''n''</sub>}} with {{math|''n''}} coprime to the level acting on the space of cusp forms of a given weight are [[self-adjoint operator|self-adjoint]] with respect to the [[Petersson inner product]]. Therefore, the [[spectral theorem]] implies that there is a basis of modular forms that are [[eigenfunction]]s for these Hecke operators. Each of these basic forms possesses an [[Euler product]]. More precisely, its [[Mellin transform]] is the [[Dirichlet series]] that has [[Euler product]]s with the local factor for each prime {{math|''p''}} is the inverse{{clarify|reason=Syntax error|date=April 2014}} of the '''Hecke polynomial''', a quadratic polynomial in {{math|''p''<sup>&minus;''s''</sup>}}. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of {{math|''&tau;''(''n'')}}.

[[Hecke algebra (disambiguation)|Other related mathematical rings]] are also called "Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the [[group ring|group algebra]]s of [[braid group]]s. The presence of this commutative operator algebra plays a significant role in the [[harmonic analysis]] of modular forms and generalisations.