Computing Central Values of Twisted L-series the Case of Composite Levels Ariel Pacetti and Gonzalo Tornaría

We will assume that the twisted L-series are primitive (i.e. the corresponding twisted modular forms are newforms). There is no loss of generality in making this assumption: if this is not the case, then f would be a quadratic twist of a newform of smaller level, which we can choose instead. The question of efficiently computing the family of central values L(f,D, 1), for fundamental discriminants D, has been considered by several authors. By Waldspurger’s formula [Wa] these values are related to the Fourier coefficients of certain modular forms of weight 3/2. In [Gr], Gross gives a method to construct, for the case of prime level p, and provided L(f, 1) 6= 0, a weight 3/2 modular form of level 4p, and gives an explicit version of Waldspurger’s formula for the imaginary quadratic twists. In [Bö-SP] Böcherer and Schulze-Pillot extend Gross’s method to the case of square free level, but their method works only for a fraction of imaginary quadratic twists (determined by quadratic residue conditions). Later in [Pa-To1] the case of level p (p a prime) is considered, and this is used in [Pa-To2], provided p ≡ 3 (mod 4), to compute central values for real quadratic twists. In [MRVT] the non-vanishing condition is removed, and two modular forms of weight 3/2 (one giving the imaginary quadratic twists and another one giving the real quadratic twists) are constructed, in the case of prime level. The aim of this paper is to show how some of these ideas can be combined to handle the case of composite levels. In the case of odd squarefree level N , for instance, this method constructs 2 modular forms, where t is the number of prime factors of N , whose coefficients give the central values of all the quadratic twists. We will focus on examples for levels N = 27, N = 15, and N = 75, which exhibit most aspects of our methods.