Failure of Converse Theorems of Gauss Sums Modulo ℓ

This paper investigates the failure of the converse theorem for Gauss sums modulo a prime ℓ, where distinct multiplicative characters over 𝔽*_q² produce identical twisted Gauss sums in 𝔽̅_ℓ. We construct a SageMath-based computational framework from the ground up to generate Gauss sum tables, identify row collisions, and verify violations of the classical uniqueness result in the modular setting. Our implementation confirms previously known counterexamples and discovers new ones, including explicit violations for (ℓ, q) pairs not of the form q = 2ℓ^i + 1. In particular, we exhibit a counterexample at (ℓ, q) = (2, 49), disproving the conjecture that failures occur only for such q. These results reveal that the breakdown of the converse theorem modulo ℓ is more widespread and structurally intricate than previously suggested.