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Given a prime $p \ge 5$ and an abstract odd representation $\rho_n$ withcoefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove theexistence of a lift of $\rho_n$ to characteristic $0$ whenever local liftsexist (under some technical conditions). Moreover, we can chose the inertialtype of our lift at all primes but finitely many (where the lift is ofSteinberg type). We apply this result to the realm of modular forms, proving alevel lowering theorem modulo prime powers and providing examples of levelraising. In particular, our method shows that given a modular eigenform $f$without Complex Multiplication or inner twists, for all primes $p$ but finitelymany, and for all positive integers $n$, there exists another eigenform $g\neqf$, which is congruent to $f$ modulo $p^n$.

Congruences between modular forms modulo prime powers