Ternary quadratic forms and Shimura’s correspondence

In his paper [11] Shimura defined a correspondence between modular forms of half integral weight and modular forms of integral weight. To each pair (t, f(z)), consisting of a square-free integert≥ 1 and a cusp formof weightk/2 (kodd, ≥ 3), levelN(divisible by 4) and characterϰ, he associated a certain functionf(t)(z) (Ft(z) in Shimura’s notation).