= Covolume-one theta function of a fractional ideal
{title2=$\Theta(y,\mathfrak b)$}
Let $C=\sqrt{|D_K|}N(\mathfrak b)$ and $n=[K:\mathbb Q]$. Normalize the <Minkowski embedding of a number field> to the <covolume>-one <Euclidean lattice> $\Lambda_{\mathfrak b}=C^{-1/n}j(\mathfrak b)$, using the metric with complex coordinates weighted by $2$. Its theta function is
$$
\Theta(y,\mathfrak b)=\sum_{a\in\mathfrak b}\exp\left(-\pi C^{-2/n}\sum_v[K_v:\mathbb R]y_v|\sigma_v(a)|^2\right).
$$
Writing $\|y\|=\prod_v y_v^{[K_v:\mathbb R]}$ and using the <trace dual of a fractional ideal> gives $\Theta(y,\mathfrak b)=\|y\|^{-1/2}\Theta(y^{-1},\mathfrak b^\vee)$. Its <small-parameter asymptotic of a lattice theta sum> has leading coefficient one. For the unscaled ideal lattice, the leading coefficient is instead $C^{-1}$; one must specify the normalization when quoting this limit.
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