In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes satisfying a certain kind of functional equation. Let F be a totally real number field of degree m over rational field. Let be the real embeddings of F. Through themwe have a map → Let be the ring of integers of F. The group is called the full Hilbert modular group.For every element , there is a group action of defined by For , define A Hilbert modular form of weight is an analytic function on such that for every