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An important variant of theta functions are modular forms coming from certain Hecke characters of a quadratic number field $K$. In this case the definition is given in terms of a sum over integral ideals but this can be reformulated as a linear combination of theta functions of the form $\Theta_{L;Q;h}(z)$ defined above. In this case the lattice $L$ is an ideal in $K$, the quadratic form $Q$ is the norm form, that is, $Q(x)=\mathcal{N}_{K/\Q}(x)$, and the function $h$ is given by the Hecke character. There are several reasons for the importance of these functions: one is that even though they are cusp forms, their Fourier coefficients can be easily computed, contrary to functions like $\Delta$. A second is that they are all CM forms, and conversely all CM forms are obtained from Hecke characters. In addition, a CM form is easily seen to be lacunary (i.e., the proportion of nonzero Fourier coefficients tends to zero), and, conversely, Serre has shown that for integral weight $k\ge1$, a nonzero form $f$ is lacunary if and only either $k=1$, or $k\ge2$ and $f$ is a linear combination of CM forms.

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  • Review status: beta
  • Last edited by Andrew Sutherland on 2019-07-31 15:30:22
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