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Theta Series for Quadratic Forms of Signature $(n-1,1)$ with (Spherical) Polynomials

We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature $(n-1,1)$. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a construction of the second author) to obtain holomorphic, almost holomorphic and modular theta series. We give a criterion for these series to coincide, enabling us to construct almost holomorphic and holomorphic cusp forms on congruence subgroups of the modular group. Further, we provide numerous explicit examples.