Spectral conditions implying the existence of doubly chorded cycles without or with constraints

What spectral conditions imply a graph contains a chorded cycle? This question was asked by R.J. Gould in 2022. We answer three modified versions of Gould's question by giving tight spectral conditions that imply the existence of doubly chorded cycle, a doubly chorded cycle with two chords incident to a vertex, and a doubly chorded cycle on five vertices with two chords incident to one vertex, in a graph, respectively.