Modular forms arising from spherical polynomials and positive definite quadratic forms

The dimension over the complex numbers of the vector space of θ-series

$\text{Φ(τ;P,Q)=∑n?Z}{}^{\mathrm{r}}\text{P(n)e}{}^{\mathrm{2\pi iQ(n)\tau }}\text{,}$

where P(X) is a “spherical polynomial” with respect to the positive definite quadratic form Q(X) is computed for binary forms. A sufficient condition that θ(τ; P, Q) be identically zero is proved, and its necessity is also proved for binary forms.