The aim is to study properties of number fields, Galois representations, and automorphic forms for which local and residual properties force a precise global behaviour. On the one hand, this relates to the theory of average class group behaviour, higher Rédei symbols and Rédei reciprocity, and various Pell equations. On the other hand, it relates to dimensions and dimension jumps for spaces of modular forms for which we simultaneously fix the Atkin-Lehner sign and the residual representation at a prime.