Invariants of an augmentation of Γ0(n)

Let G n × be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ 0 ( n ) based on G n × , paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G n × , we then define a family of functions based on η -functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.