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Learning Fricke signs from Maass form Coefficients

Joanna Bieri, Giorgi Butbaia, Edgar Costa, Alyson Deines, Kyu-Hwan Lee, David Lowry-Duda, Thomas Oliver, Yidi Qi, Tamara Veenstra

TL;DR

The paper tackles the challenge of determining the Fricke sign $w_N$ of Maass newforms from finite Fourier data, exploiting murmurations between root numbers and coefficients. It develops and compares supervised (Linear Discriminant Analysis) and neural-network approaches on Fourier data from the LMFDB, revealing that parity and the product structure $w_N=\prod_{p|N} w_p$ encode robust predictive signals, achieving around $96\%$ accuracy even when $w_N$ is unknown. The results demonstrate strong agreement with heuristic Hejhal-type guesses for many cases and show that global invariants can be inferred from local Fourier coefficients, offering a new tool for investigating arithmetic objects where direct computation is difficult. The work suggests pathways for interpretability and extension to other number-theoretic quantities informed by machine learning.

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.

Learning Fricke signs from Maass form Coefficients

TL;DR

The paper tackles the challenge of determining the Fricke sign of Maass newforms from finite Fourier data, exploiting murmurations between root numbers and coefficients. It develops and compares supervised (Linear Discriminant Analysis) and neural-network approaches on Fourier data from the LMFDB, revealing that parity and the product structure encode robust predictive signals, achieving around accuracy even when is unknown. The results demonstrate strong agreement with heuristic Hejhal-type guesses for many cases and show that global invariants can be inferred from local Fourier coefficients, offering a new tool for investigating arithmetic objects where direct computation is difficult. The work suggests pathways for interpretability and extension to other number-theoretic quantities informed by machine learning.

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.
Paper Structure (9 sections, 13 equations, 14 figures, 6 tables)

This paper contains 9 sections, 13 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Experiments with Maass forms
  3. Definitions
  4. Averaging
  5. Predicting the Fricke sign with LDA
  6. Embedding the Fricke signs into the Fourier coefficients
  7. Other LDA analysis
  8. Predicting the Fricke sign with a neural network
  9. Comparison of Hejhal's algorithm, LDA, and neural networks

Figures (14)

  • Figure 2.1: Average value of $a_p$ over Maass forms with given Fricke sign, with and without separating by symmetry
  • Figure 2.2: Average value of $(-1)^{\sigma(f)} a_p$ over Maass Forms separated by Fricke sign.
  • Figure 2.3: Comparing the distribution of $a_7$ for Maass forms in $\mathcal{L}_1$ and $\mathcal{L}_{-1}$. In the left (resp. right) frame, we consider all levels (resp. only levels co-prime to $7$). The brown represents areas where the histograms overlap, and the green arc in the right frame is the semi-circle $y=\frac{1}{2\pi}\sqrt{4-x^2}$, given by the (vertical) Sato--Tate distribution Sar.
  • Figure 2.4: Average value of $a_p(-1)^{\sigma(f)}$ for Maass forms with odd parity, separated by Fricke sign.
  • Figure 2.5: Average value of $a_p(-1)^{\sigma(f)}$ for Maass forms with even parity, separated by Fricke sign.
  • ...and 9 more figures

Theorems & Definitions (1)

  • Remark 2.1