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(5) . 43 that satisfies Artin's conjecture. We briefly explain the computations behind seven additional examples of conductors 2416 = 2(4) . 151, 3184 = 2(4) . 199, 3556 = 2(2) . 7 . 127, 3756 = 2(2) . 3 . 313, 4108 = 2(2) . 13 . 79, 4288 = 2(6) . 67, and 5373 = 3(3) . 199. We also generalize a result of Sturm on computing congruences between eigenforms.