Unramifiedness of weight 1 Hilbert Hecke algebras

We prove that the Galois pseudorepresentation valued in the mod $p^n$ cuspidal Hecke algebra for $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ over a totally real number field $F$, of parallel weight $1$ and level prime to $p$, is unramified at any place above $p$. The same is true for the noncuspidal Hecke algebra at places above $p$ whose ramification index is not divisible by $p-1$. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when $p$ ramifies in $F$, of generalised $\Theta$-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.