Shimura correspondence

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 T_n² on F is equal to the eigenvalue of T_n on f.

Let ${\displaystyle f}$ be a holomorphic cusp form with weight ${\displaystyle (2k+1)/2}$ and character ${\displaystyle \chi }$ . For any prime number p, let

where ${\displaystyle \omega _{p}}$'s are the eigenvalues of the Hecke operators ${\displaystyle T(p^{2})}$ determined by p.

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