  Bianchi modular forms are automorphic forms over an imaginary quadratic
field, associated to a Bianchi group. Even though modern studies of Bianchi
modular forms go back to the mid 1960's, most of the fundamental problems
surrounding their theory are still wide open. Only for certain types of Bianchi
modular forms, which we will call non-genuine, it is possible at present to
develop dimension formulas: They are (twists of) those forms which arise from
elliptic cuspidal modular forms via the Langlands Base-Change procedure, or
arise from a quadratic extension of the imaginary quadratic field via
automorphic induction (so-called CM-forms). The remaining Bianchi modular forms
are what we call genuine, and they are of interest for an extension of the
modularity theorem (formerly the Taniyama-Shimura conjecture, crucial in the
proof of Femat's Last Theorem) to imaginary quadratic fields. In a preceding
paper by Rahm and Sengun, an extreme paucity of genuine cuspidal Bianchi
modular forms has been reported, but those and other computations were
restricted to level One. In this paper, we are extending the formulas for the
non-genuine Bianchi modular forms to deeper levels, and we are able to spot the
first, rare instances of genuine forms at deeper level and heavier weight.
