Grössencharacter of the square-lattice elliptic curve (${\psi }_E(\mathfrak{a})=\text{the primary generator of }\mathfrak{a}$)

For $E:y^2=x^3-x$ over $\mathbb{Q}(i)$, the Hecke character has conductor $(1+i{)}^3$ and ideal infinity type $(1,0)$. Its value on an ideal prime to $2$ is its unique primary Gaussian integer generator. Rational $(1+i{)}^3$-torsion fixes the unit ambiguity of the Frobenius value. At a split prime $\mathfrak{p}$, its value reduces to the degree-$p$ Frobenius isogeny; at an inert rational prime $p$, its value is $-p$, reducing to the $p^2$-power Frobenius isogeny.