# Learning Fricke signs from Maass form Coefficients

arXiv: https://arxiv.org/abs/2501.02105 · HTML: https://mlsanity.com/paper/2501.02105/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 3 Jan 2025.

## Builds on (most cited first, 5 of 5)

- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)
- [Hardness of Learning Fixed Parities with Neural Networks](https://mlsanity.com/paper/2501.00817/) (2025)

## Built on by (most cited first, 1 of 1)

- [Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves](https://mlsanity.com/paper/2607.24251/) (2026)

## Abstract (arXiv)

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.