A space of weight one modular forms attached to totally real cubic number fields

Abstract

Let $d$ be a positive fundamental discriminant, and let ${\mathcal{C}}_d$ be the set of isomorphism classes of cubic number fields of discriminant $d$. For each $K\in {\mathcal{C}}_d$, we construct a weight 1 modular form $f_K$ with level $3^{\pm 1}d$ and nebentypus $\left(\frac{-3^{\pm 1}d}{\cdot }\right)$. We show that the form $f_K$ completely determines the field $K$. Moreover, we show that $\{f_K:K\in {\mathcal{C}}_d\}$ is a linearly independent set.