Learning Fricke signs from Maass form Coefficients

TL;DR

This study uses machine learning to predict Fricke signs of Maass forms from their Fourier coefficients, revealing patterns and achieving high accuracy, thus advancing understanding of Maass form symmetries.

Contribution

The paper introduces a machine learning approach to classify Fricke signs of Maass forms, demonstrating high accuracy and validating predictions with heuristic methods.

Findings

LDA achieves 96% accuracy for even parity forms

Neural networks produce comparable classification results

Predicted Fricke signs align well with heuristic guesses

Abstract

In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered "murmuration" phenomenon, and that these patterns become more pronounced when parity is incorporated as an additional feature. Approximately 43% of the forms in our dataset have an unknown Fricke sign. For the remaining forms, we employ Linear Discriminant Analysis (LDA) to machine learn their Fricke sign, achieving 96% (resp. 94%) accuracy for forms with even (resp. odd) parity. We apply the trained LDA model to forms with unknown Fricke signs to make predictions. The average values based on the predicted Fricke signs are computed and compared to those for forms with known signs to verify the reasonableness of the predictions. Additionally, a subset of these predictions is evaluated against heuristic guesses provided by Hejhal's algorithm, showing a match approximately 95% of the time. We also use neural networks to obtain results comparable to those from the LDA model.