Rational eigenvectors in spaces of ternary forms

Abstract

We describe the explicit computation of linear combinations of ternary quadratic forms which are eigenvectors, with rational eigenvalues, under all Hecke operators. We use this process to construct, for each elliptic curve E E of rank zero and conductor N > 2000 N > 2000 for which N N or N / 4 N/4 is squarefree, a weight 3/2 cusp form which is (potentially) a preimage of the weight two newform ϕ E \phi _{E} under the Shimura correspondence.