Stark unit of a newform of weight one (awaiting review)

Stark's conjecture applied to the associated Galois representation of a newform $f(z)=\sum a_n q^n$ of weight one [MR:0782485] states the following. Let $E=\mathbb{Q}((a_n)_{n \in \mathbb{N}})$, $\Delta=\text{Gal}(E/\mathbb{Q})$ and $f^\alpha(z)=\sum \alpha(a_n) q^n$ for $\alpha \in \Delta$. Let $L(s, f)$ be the L-function of $f$. Then, for all $b \in E^*$ there exists an integer $m \geq 1$ and a unit $\varepsilon$ in the Artin field of $f$, called the Stark unit, such that

\[ e^{m \sum_{\alpha \in \Delta} \alpha(b)L'(0, f^\alpha)} = \varepsilon \]

In the case where the coefficients of $\text{Tr}(bf)$ are in $\mathbb{Z}$, Chinburg further conjectured that there exists a Stark unit for $m=1$ [10.1016/0001-8708(83)90006-3]. Notice that if we choose $b = 1$, the preceding condition always holds. Here, we compute the Stark unit of the newform for $b=1$ and $m=1$.