Rankin-Selberg method for real analytic cusp forms of arbitrary real weight

1.2 Indeed, Rankin and Selberg applied their method to the more general situation of holomorphic cusp forms of type (F', r, v), where F ' is of finite index in F, r > 0 is the weight and v is a compatible multiplier system. Let f ( z ) be such a cusp form, [F 'F ' ] = m and M1 . . . . . Mm be representatives of distinct right cosets. Then the vector-valued function F ( z ) = ((clz + d l ) ~ f ( M l z ) . . . . . (cruz + d,,)-rf(Mmz)) ~ (T means transpose, (c~,dl) is the second row of M~) is of type (F, r, E) where E is a unitary m-dimensional multiplier system. So, the concept of vector-valued automorphic forms for F is intimately related to the investigation of automorphic forms for subgroups of finite index in V (cf. [13]).