A higher weight generalization of the Hermite-Minkowski theorem

Let $E$ be a number field, $N$ an ideal of its ring of integers, and $w \geq 0$ an integer. Consider the set of  cuspidal algebraic automorphic representations of $GL_n$ over $E$ whose conductor is $N$, and whose  ''weights'' are in the interval $\{0,\dots,w\}$ (with $n$ varying). If the root-discriminant of $E$ is less than a  certain explicit function $f$ of $w$, then I show that this set is finite. For instance, we have $f(w)>1$ if, and  only if, $w<24$. Under a suitable form of GRH, we may replace $f(w)$ by $8*\Pi*e^{-\psi(1+w)}$, where  $\psi$ is the classical digamma function.