The principal congruence subgroup \(\Gamma\) (N) has an Eisenstein series associated to it at each cusp, which admits a Fourier expansion at each other cusp. This results in a matrix, whose determinant plays a key role in the theory, mostly due to its appearance in the trace formula. This paper introduces C(N), the Cartan subgroup of \(GL_ 2({\mathbb{Z}}/N {\mathbb{Z}})\) into the study of these Eisenstein series, and to use it in order to give a precise formula for this determinant. Our main theorem shows that it can be naturally expressed in terms of L-functions on C(N).