Theta Functionswith Harmonic Coe⁄cients over Number Fields

is a modular form of weight n=2þ n on G0ðN Þ, where G 1⁄4 SL2ðZÞ and N is the level of Q, i.e., NQ 1 is integral and NQ 1 has even diagonal entries. This was proved by Schoeneberg [13] for even n and by Pfetzer [9] for odd n. Shimura [14] generalizes their results for arbitrary n and also computes the theta multiplier explicitly. Andrianov and Maloletkin [1, 2] generalize (1) and define theta series of higher degree. In [1], they construct Siegel modular forms by regarding theta series corresponding to positive-definite quadratic forms as specializations of symplectic theta functions. In addition, they apply a differential operator to the functional equation of the theta functions to show that theta series of higher degree with harmonic coefficients are also Siegel modular forms. In [2], they obtain analogous results for theta functions corresponding to indefinite quadratic forms. Stark [15] computes the theta multiplier for the symplectic theta function. As an application, he explicitly determines the theta multiplier of Andrianov’s and Maloletkin’s theta functions.