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We prove classification results for the cuspidal automorphic algebraicrepresentations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) of smallprime conductor and small motivic weight, in the spirit of the works ofChenevier, Lannes and Ta\"ibi in conductor $1$. The main result is an explicitlist of all such representations with motivic weight up to $17$ and conductor$2$. For this, we develop the analytical method based on the Riemann-Weilexplicit formulas, and use Arthur's work to relate those representations toclassical objects. A key ingredient is a special case of Gross' conjectureregarding paramodular invariants of representations of a split ${\rmSO}_{2n+1}(\mathbb{Q}_p)$, which we prove as well.

Autour de l'\'enum\'eration des repr\'esentations automorphes cuspidales alg\'ebriques de ${\rm GL}_n$ sur $\mathbb{Q}$ de conducteur $>1$