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We derive the trace and diffeomorphism anomalies of the Schr\"odinger fieldminimally coupled to the Newton-Cartan background using Fujikawa's pathintegral approach. This approach in particular enables us to calculate theone-loop contributions due to all the fields of the Newton-Cartan structure. Wedetermine the coefficients and demonstrate that gravitational anomalies forthis theory always arise in odd dimensions. Due to the gauge field contributionof the background we find that in $2+1$ dimensions the trace anomaly containsterms which have a form similar to that of the $1+1$ and $3+1$ dimensionalrelativistic trace anomalies. This result reveals that the Newton-Cartanbackground which satisfies the Frobenius condition possesses a Type A traceanomaly in contrast with the result of Lishitz spacetimes. As an application wedemonstrate that the coefficient of the term similar to the $1+1$ dimensionalrelativistic trace anomaly satisfies a c-theorem condition.

Gravitational anomalies on the Newton-Cartan background