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 $\mathbb{Q}(\sqrt{-11})$. This algorithm was extended to handle the other four fields of class number one ($\mathbb{Q}(\sqrt{-19})$, $\mathbb{Q}(\sqrt{-43})$, $\mathbb{Q}(\sqrt{-67})$, $\mathbb{Q}(\sqrt{-163})$) by Elise Whitley in her 1990 PhD thesis, and to the fields $\mathbb{Q}(\sqrt{-23})$ and $\mathbb{Q}(\sqrt{-31})$ of class number $3$ by Mark Lingham in his 2005 PhD thesis. These extended algorithms were reimplemented by John Cremona in 2021. The open source code bianchi-progs implementing the algorithm in C++ is available on GitHub; it depends on the C++ library eclib. The code is currently (as of September 2021) limited to the nine imaginary quadratic fields of class number one, and the first two fields of class number $3$, and only computes cuspidal weight 2 newforms with trivial character and dimension $$1$$.
Dimension data for full cuspidal and new spaces for a range of weights and $\mathrm{SL}_2$ 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.