Explicit application of Waldspurger's Theorem

Abstract

For a given cusp form $\phi$ of even integral weight satisfying certain hypotheses, Waldspurger's Theorem relates the critical value of the $\mathrm{L}$-function of the $n$-th quadratic twist of $\phi$ to the $n$-th coefficient of a certain modular form of half-integral weight. Waldspurger's recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and (adelic) Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our "simplified Waldspurger" by giving several examples.