  We show that for any integer $N$, there are only finitely many cuspidal
algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with
$n$ varying, whose conductor is $N$ and whose weights are in the interval
$\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq
0}$ such that for any integer $w$, any number field $E$ whose root-discriminant
is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there
are only finitely many cuspidal algebraic automorphic representations of
general linear groups over $E$ whose conductor is $N$ and whose weights are in
the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we
may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where
$\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number.
  The proofs are based on some new positivity properties of certain real
quadratic forms which occur in the study of the Weil explicit formula for
Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the
methods are discussed.
