A Mod Five Approach to Modularity of Icosahedral Galois Representations

We give eight new examples of icosahedral Galois representations that satisfy Artin’s conjecture on holomorphicity of their L-function. We give in detail one example of an icosahedral representation of conductor 1376 = 2 · 43 that satisfies Artin’s conjecture. We briefly explain the computations behind seven additional examples of conductors 2416 = 2 · 151, 3184 = 2 · 199, 3556 = 2 · 7 · 127, 3756 = 2 · 3 · 313, 4108 = 2 · 13 · 79, 4288 = 2 · 67, and 5373 = 3 · 199. We also generalize a result of Sturm on computing congruences between eigenforms.