Obstruction criteria for modular deformation problems

For a newform $f=\sum a_n q^n$ of weight $k \geq 3$ and a prime $λ$ of $\mathbf{Q}(a_n)$, the deformation problem for its associated mod $λ$ 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 $λ$-adic deformation problem for $f$ is unobstructed, then $f$ is not congruent mod $λ$ to a newform of lower level.

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