  With the formulation of the quantum spectral curve for the AdS5/CFT4
integrable system, it became potentially possible to compute its full spectrum
with high efficiency. This is the first paper in a series devoted to the
explicit design of such computations, with no restrictions to particular
subsectors being imposed.
  We revisit the representation theoretical classification of possible states
in the spectrum and map the symmetry multiplets to solutions of the quantum
spectral curve at zero coupling. To this end it is practical to introduce a
generalisation of Young diagrams to the case of non-compact representations and
define algebraic Q-systems directly on these diagrams. Furthermore, we propose
an algorithm to explicitly solve such Q-systems that circumvents the
traditional usage of Bethe equations and simplifies the computation effort.
  For example, our algorithm quickly obtains explicit analytic results for all
495 multiplets that accommodate single-trace operators in N=4 SYM with
classical conformal dimension up to 13/2. We plan to use these results as the
seed for solving the quantum spectral curve perturbatively to high loop orders
in the next paper of the series.
