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