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.

翻译：本文对Maass形式进行了数据科学探究。我们发现，对具有相同Fricke符号的Maass形式的傅里叶系数进行平均后，会呈现出与近期发现的"鸟群扰动"现象类似的模式，且当加入宇称作为附加特征时，这些模式会变得更加显著。数据集中约43%的Maass形式具有未知的Fricke符号。对于其余形式，我们采用线性判别分析（LDA）对其Fricke符号进行机器学习，在偶宇称（对应奇宇称）形式中达到了96%（对应94%）的准确率。我们将训练好的LDA模型应用于Fricke符号未知的形式进行预测。基于预测的Fricke符号计算平均值，并与已知符号形式的平均值进行比较，以验证预测的合理性。此外，选取部分预测结果与Hejhal算法提供的启发式猜测进行对比评估，显示匹配率约为95%。我们还使用神经网络获得了与LDA模型相当的结果。