  In this article we present an algorithm that uses the graded algebra
structure of Hilbert modular forms to compute the adelic $q$-expansion of
Hilbert modular forms of weight one as the quotient of Hilbert modular forms of
higher weight. The main improvement to existing methods is that our algorithm
can be applied in weight $1$, which fills a gap left by standard computational
methods. Additionally, the algorithm can be used to compute Hilbert modular
forms over finite fields in all characteristic simultaneously. We use this
algorithm to compute a first candidate of a Hilbert modular form of parallel
weight $1$ that is non-liftable and specify the exact conditions under which
our candidate $q$-expansion corresponds to a non-liftable Hilbert Modular form.
