The database of Bianchi modular forms was mostly computed by John Cremona using modular symbol algorithms developed in his 1981 PhD thesis for the five fields $\mathbb{Q}(\sqrt{-1})$, $\mathbb{Q}(\sqrt{-2})$, $\mathbb{Q}(\sqrt{-3})$, $\mathbb{Q}(\sqrt{-7})$, and $\mathbb{Q}(\sqrt{-11})$. The code implementing the algorithm is available on GitHub. It is currently limited to these five Euclidean imaginary quadratic fields and only computes cuspidal weight 2 newforms with trivial character and 'dimension 1' (that is, having rational coefficients).
Forms of dimension 2 over $\mathbb{Q}(\sqrt{-1})$ were computed by Ciaran Schembri.
The database contains newform data for levels of the form $\Gamma_0(\mathfrak{n})\le GL_2(\mathcal{O}_K)$ and not for the larger spaces forms of level $\Gamma_0(\mathfrak{n})\cap SL_2(\mathcal{O}_K)$. However, dimension data for full cuspidal and new spaces for a range of weights and $SL_2$ levels over $\mathbb{Q}(\sqrt{-d})$ for $d=2,11,19,43,67,163$ are included, computed by Alexander Rahm. $SL_2$ dimension data for the remaining imaginary quadratic fields of class number $1$ ($d=1,2,7$) is in preparation.