I need to do some numerical computation on special values of a Hecke L-function $L(s,\chi)$. To do this, I want to construct a Hecke character in MAGMA, given that I know its infinity type.

In other words, suppose we are working on a totally real field $K$. My Hecke character $\chi$ is first defined over the principal ideals by

$$\chi((\alpha))=\prod sgn(\sigma_i(\alpha))^{m_i}|\sigma_i(\alpha)|^{n_i}$$

where $\sigma_i$'s are the real embeddings, $m_i=0$ or $1$, and $n_i\in \mathbb{C}$. For some good values of $m,n$ this can be extended to all ideals, thus becomes a Hecke character.

I tried to read https://magma.maths.usyd.edu.au/magma/handbook/text/410 for related information. It looks like the functions related to Hecke Grössencharacters are close to what I need, but it requires CM field to work, while I want to deal, say with $K=\mathbb{Q}(\sqrt{3})$.