Obstruction criteria for modular deformation problems

For a newform $f=\sum a_n q^n$ of weight $k \geq 3$ and a prime $\lambda$ of $\mathbf{Q}(a_n)$, the deformation problem for its associated mod $\lambda$ Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for $f$ of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the $\lambda$-adic deformation problem for $f$ is unobstructed, then $f$ is not congruent mod $\lambda$ to a newform of lower level.