Lifting newforms to vector-valued modular forms for the Weil representation

Abstract

Given a discriminant form D of level N, there is a natural lifting which maps elliptic modular forms of level N to vector-valued modular forms for the Weil representation associated to D. We show that in some cases the zero component of a lifting of a newform f is just a scalar multiple of f. In order to do so, we split the lifting map into certain partial liftings corresponding to the prime powers exactly dividing N and then proceed to compute the zero components of these partial maps explicitly. As an application, we show that the L-function L𝒜(f, s) of a newform f and an ideal class 𝒜 as defined by Gross and Zagier can be written as a certain L-series of the lifting of f.