Serre's conjectures for \Q(\sqrt{13})

The present article proves (assuming GRH and a minor technical result) the veracity of Serre's conjectures for the field \Q(\sqrt{13}). In doing so, we prove a ``reduction to level 1'' result that holds for any real quadratic field with narrow class number one. However we can only establish the needed base cases for \Q(\sqrt{13}).