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.
Categories: math.NT