title: Theta functions with harmonic coefficients over number fields
abstract: We investigate theta functions attached to quadratic forms over a number field K. We establish a functional equation by regarding the theta functions as specializations of symplectic theta functions. By applying a differential operator to the functional equation, we show how theta functions with harmonic coefficients over K behave under modular transformations
keyphrases: harmonic coefficients number fields quadratic forms functional equation symplectic theta functions differential operator modular transformations