Denote by \(H_ g\) the Siegel upper-half space of Riemann \(g\times g\)- matrices. Then the classical theta map \(\vartheta:\mathbb{C}^ g\times\mathbb{R}^ g\times\mathbb{R}^ g\times H_ g\to\mathbb{C}\) gives via \(\vartheta(z,a,b,\tau)=\vartheta{a \brack b}(z,\tau)\) the classical theta functions with real characteristics \((a,b)\in\mathbb{R}^ g\times\mathbb{R}^ g\). In the present short communication, the author constructs a different theta map \(\vartheta:\mathbb{C}^ g\times\mathbb{C}^ g\times H_ g\to\mathbb{C}\), which is obtained from the classical one by identifying the real theta characteristics \((a,b)\in\mathbb{R}^ g\times\mathbb{R}^ g\) with complex vectors \(w\in\mathbb{C}^ g\) via the substitution formula \(w=a\tau+b\), \(\tau\in H_ g\), and by defining then \(\vartheta(z,w,\tau)\) as \(\vartheta{a \brack b}(z,\tau)\) multiplied by some exponential factor depending on \(z\), \(w\) and \(\tau\). The advantage of this modified theta series is that it depends holomorphically on all variables \(z\), \(w\) and \(\tau\). Using this analyticity property and some identities among these ``new'' theta values, the author derives an explicit description of the Poincaré divisor on the product of any principally polarized abelian variety and its dual variety, namely just as the zero divisor \(\{\vartheta(z,w,\tau)=0\}\) for fixed \(\tau\in H_ g\).