In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight 2k, discovered by Goro Shimura (1973). It has the property that the eigenvalue of a Hecke operator Tn2 on F is equal to the eigenvalue of Tn on f.
Let be a holomorphic cusp form with weight and character . For any prime number p, let
where 's are the eigenvalues of the Hecke operators determined by p.
is a holomorphic modular function with weight 2k and character .