Characteristic classes in TMF of level Γ₁(3)

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Let TMF₁(n) be the spectrum of topological modular forms equipped with a Γ₁(n)-structure. We compute the K(2)-local TMF₁(3)-cohomology of BString and BSpin: both are power series rings freely generated by classes that we explicitly construct and which generalize the classical Pontryagin classes. As a first application of this computation, we show how to construct TMF(3n)-cohomology classes from stable positive energy representations of the loop groups LSpin.