On theta functions associated to indefinite quadratic forms

We introduce a family of theta functions associated to an indefinite quadratic form, and prove a modular transformation formulas by regarding each such function as a specialization of a symplectic theta function. An eighth rott of unity arises in these formulas, and it is expressly given in all cases. The theta functions feature many “translation variables,” which are useful for the study of the liftings of modular forms.