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With the formulation of the quantum spectral curve for the AdS5/CFT4integrable system, it became potentially possible to compute its full spectrumwith high efficiency. This is the first paper in a series devoted to theexplicit design of such computations, with no restrictions to particularsubsectors being imposed.  We revisit the representation theoretical classification of possible statesin the spectrum and map the symmetry multiplets to solutions of the quantumspectral curve at zero coupling. To this end it is practical to introduce ageneralisation of Young diagrams to the case of non-compact representations anddefine algebraic Q-systems directly on these diagrams. Furthermore, we proposean algorithm to explicitly solve such Q-systems that circumvents thetraditional usage of Bethe equations and simplifies the computation effort.  For example, our algorithm quickly obtains explicit analytic results for all495 multiplets that accommodate single-trace operators in N=4 SYM withclassical conformal dimension up to 13/2. We plan to use these results as theseed for solving the quantum spectral curve perturbatively to high loop ordersin the next paper of the series.

The full spectrum of AdS5/CFT4 I: Representation theory and one-loop Q-system