The database of Bianchi modular forms was mostly computed by John Cremona using modular symbol algorithms developed in his 1981 DPhil thesis (see also J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields Compositio Mathematica, tome 51, no 3 (1984), p. 275-324) 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 open source code implementing the algorithm is available on GitHub; it depends also on the C++ library eclib. The code 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 using the Magma package written by Dan Yasaki.
Dimension data for full cuspidal and new spaces for a range of weights and $SL_2$ levels over $\mathbb{Q}(\sqrt{-d})$ for $d=2,11,19,43,67,163$ were computed by Alexander Rahm using his own code written in PARI/GP.