  We prove that the Galois pseudo-representation valued in the mod $p^n$
cuspidal Hecke algebra for GL(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 non-cuspidal Hecke algebra at places above $p$
whose ramification index is not divisible by $p-1$. A novel geometric
ingredient, which is also of an independent interest, is the construction and
study, in the case when $p$ ramifies in $F$, of generalised $\Theta$-operators
using Reduzzi--Xiao's generalised Hasse invariants, including especially an
injectivity criterion in terms of minimal weights.
