Learning Fricke signs from Maass form Coefficients

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

REVIEW 4 major objections 6 minor 16 references

Learning Fricke signs from Maass form Coefficients

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that supervised machine learning on the first 1,000 Fourier coefficients predicts the Fricke sign of a Maass form with 94–96% accuracy, and that predictions for forms with unknown signs agree with a heuristic algorithm…

desk verdict A useful empirical paper whose core claim is probably right, but whose headline numbers are sloppy and whose transfer to unknown-sign forms is only heuristically supported. read the letter →

arxiv 2501.02105 v2 pith:KOZCKVTE submitted 2025-01-03 math.NT cs.LGhep-thstat.ML

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

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Tamara Veenstra

This is my paper · ORCID

classification math.NTcs.LGhep-thstat.ML MSC 11F6611Y3568T05

keywords MaassformsFrickesignmurmurationslineardiscriminantanalysisFouriercoefficientsmachinelearningL-functionsneuralnetworks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the Fricke sign of a Maass newform—the eigenvalue under the Fricke involution that fixes the sign of the functional equation—can be recovered statistically from finitely many Fourier coefficients, without computing the form to the precision normally required. Averaging coefficients by sign reveals murmuration-like oscillations, and a simple linear classifier trained on these coefficients predicts the sign with 94–96% accuracy on forms where the sign is rigorously known. Applied to the 15,423 forms whose signs are unknown, the model's predictions reproduce the same averaging patterns, and on the 4,595 forms where a heuristic algorithm is confident, the predictions match roughly 95% of the time. The reason this matters is that Fricke signs are often unavailable precisely because coefficients cannot be computed accurately enough; a collective statistical readout would supply sign information cheaply across the whole database.

What carries the argument

The central object is the normalized feature vector $$D = \{((-1)^{\$\sigma$(f)} a_n)_{n=1}^{1000} : f \in \mathcal{L}\},$$ in which each coefficient is multiplied by the parity sign so that even and odd forms are aligned by root number, together with the factorization of the Fricke sign into local factors $w_N = \prod_{p\mid N} w_p$. The machinery that carries the argument is Linear Discriminant Analysis (LDA), which fits a linear decision boundary under the assumption that the two sign classes share a covariance structure; the paper checks this assumption with a standard equal-covariance test. A second piece of machinery is the averaging operation that produces murmuration plots, which both motivates LDA and validates predictions by comparing average coefficients of predicted-sign forms with known-sign forms. The paper also uses a neural network with the spectral parameter $R$ appended, and a heuristic algorithm that guesses Fricke signs by solving approximate overdetermined linear systems; agreement with that heuristic provides the external check on the unknown-sign predictions.

What would settle it

Take a random sample of the 15,423 forms with unknown Fricke sign, compute their signs rigorously at higher precision, and compare with the LDA predictions; if the error rate on that sample is near 50% rather than near 5%, the transfer claim is false.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Fricke sign of a Maass form is learnable from the first 1,000 Fourier coefficients, with accuracy far above the 86% obtainable from prime-indexed coefficients alone, and robust to masking the coefficients whose indices share a factor with the level—the obvious place where the sign is encoded. The authors argue the classifier is not merely reading off $a_p$ for primes dividing the level, because setting those coefficients to zero leaves accuracy nearly unchanged for the best feature set and because level-1 forms, where no coefficient directly encodes the sign, are classified successfully. Instead, the full coefficient vector carries extra predictive signal: indices with one or two prime factors give 95.3% accuracy, so the multiplicative structure itself appears informative. This connects the predictive task to murmurations: the same sign-conditioned averages that oscillate in the plots are what the linear classifier exploits.

Load-bearing premise

The classifier's transfer to the 15,423 unknown-sign forms assumes that forms whose signs are unknown because of computational difficulty are statistically similar, on the coefficient features used, to forms whose signs are known.

Editorial extensions

If this is right

If the central claim holds, Fricke signs for the 15,423 unknown forms can be assigned probable values using only stored coefficients, matching independent heuristics on the subset where the heuristic is confident.
The accuracy advantage of full coefficient vectors over prime-indexed vectors (96.1% versus 86.2%) implies the predictive information is not contained only in $a_p$ for primes dividing the level, nor in primes generally; composite-index coefficients add real signal.
Because LDA works without hyperparameter tuning, the sign is nearly linearly separable in coefficient space after parity normalization, suggesting a simple statistical description rather than a deep structural one.
Because the model transfers across levels without being given the level, the learned boundary is a function of coefficient patterns rather than of the level or analytic conductor alone.
The connection to murmurations suggests sign-conditioned coefficient averages are a stable phenomenon, not an artifact of a particular classifier.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the root number is the product of parity and Fricke sign, a classifier with this accuracy also yields a root-number estimator; this could be used to prioritize candidates for rigorous certification.
The finding that full coefficient vectors beat prime-indexed ones suggests a testable hypothesis: the signal lives in the multiplicative semigroup structure of indices, so features derived from divisor counts should retain most of the accuracy.
Since unknown signs become more frequent at higher level (as the paper's own level-by-level plot shows), a level-stratified retraining experiment would be the cleanest way to test whether the transfer assumption holds.
The same averaging-plus-classifier pipeline could be tried on other expensive invariants of automorphic forms, such as symmetry type or eigenvalue location, wherever a labeled subset exists.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main empirical finding is plausible and likely correct: LDA and a small neural net can predict the Fricke sign of a Maass form from its first 1000 coefficients at around 95% accuracy, even after zeroing the coefficients that directly encode the sign. The parity-corrected murmuration plots are a genuine improvement over [BLLD+24], and the zeroed-feature experiment in Section 2.4 is the right check. I think the paper is worth engaging with, but it needs cleanup.

What is actually new: applying standard classifiers to Fricke signs, and the observation that including composite-index coefficients helps beyond primes alone. The latter is curious and not fully explained, but the experiments in Section 2.4 and Figure 2.7 are honest attempts to probe it.

The soft spots, in order of severity. First, the abstract swaps the even/odd accuracy figures: it says 96% even, 94% odd, while Table 2.2 gives 0.9488 even and 0.9633 odd. That is a concrete error that will embarrass the authors. Second, no code is shipped, and there are no confidence intervals or repeated-seed variance for the neural net, so the reader cannot tell how stable the 95% figures are. Third, and most substantive, the transfer from labeled to unknown-sign forms is not established. The unknown fraction grows with level (Figure 2.9), and the 4,595 Hejhal-confirmed forms are a selective subset; the paper does not show their level distribution or that Hejhal convergence is independent of level. Without that, the 95% agreement with Hejhal is suggestive but not proof that predictions on the full unknown set are reliable.

The direct-encoding issue is real but partially handled. For squarefree N up to 105, a_p for p|N gives w_p exactly, so the unzeroed a_n features contain a tautological readout. The a'_n experiment removes that and still gets 94-96%, which is the actual result. The paper says this clearly enough, but the abstract's emphasis on the unzeroed accuracy overstates the novelty.

The math looks solid, definitions are standard, and the citation pattern is fine. This is a computational paper, not a theorem paper, and the claims are framed as empirical. Who is it for? People working on murmurations, machine learning for L-functions, and the LMFDB Maass form database. It deserves a serious referee; I would send it to review after asking for the abstract fix, error bars, and a level-distribution check for the Hejhal subset.

Referee Report

4 major / 6 minor

Summary. The paper applies supervised machine learning, specifically Linear Discriminant Analysis (LDA) and feed-forward neural networks, to the first 1000 Fourier coefficients of Maass newforms from the LMFDB in order to predict the Fricke sign. On the 19,993 forms with rigorously known sign, LDA is reported to reach about 96% validation accuracy. The trained model is then applied to the 15,423 forms with unknown Fricke sign, and the resulting predictions are checked in two ways: by comparing averaged coefficient patterns ('murmurations') for predicted signs with those for known signs, and by comparing with heuristic Hejhal guesses on a subset of 4,595 forms, where agreement is about 95%. The paper also addresses the concern that the Fricke sign is directly encoded in coefficients at primes dividing the level by repeating experiments with a modified feature in which such coefficients are zeroed.

Significance. If the transfer claim holds, the paper provides a useful empirical data-scientific tool for guessing missing Fricke signs in the LMFDB and evidence that coefficient vectors contain recoverable information beyond the direct local-factor readout. The paper is commendably explicit about the direct-encoding confound and tests a zeroed feature variant, and it compares its predictions with an independent heuristic algorithm. The main weaknesses are the lack of uncertainty quantification and, more importantly, the absence of evidence that the 4,595-form Hejhal-confirmed subset is representative of the full unknown-sign set, which is load-bearing for the claim that predictions on all unknown-sign forms are reasonable.

major comments (4)

[Abstract and Section 2.3/Table 2.2] The abstract states 96% (resp. 94%) accuracy for even (resp. odd) parity, but Section 2.3 and Table 2.2 report 94.9% for even forms and 96.3% for odd forms. Since these are the headline accuracy figures, the abstract should be corrected to match the table.
[Sections 2.3, 2.7, Table 2.4, Figure 2.9] The application to the 15,423 unknown-sign forms is a central output of the paper, but the only direct validation is on the 4,595 forms for which the heuristic Hejhal algorithm converged. Figure 2.9 shows that the unknown fraction increases with level, and the convergence of a heuristic root-finding algorithm is likely easier at lower levels and for larger eigenvalue gaps. The paper does not report the level or spectral-parameter distribution of the 4,595 Hejhal-confirmed forms, nor accuracy stratified by level. Without such evidence, the claim that predictions on the full unknown-sign set are 'reasonable' is not established.
[Section 2.4, Eq. (2.4)] The sentence 'Similarly, when gcd(n,N)>1, we have a_n = 0' is inconsistent with Eq. (2.4), which gives the nonzero value a_p = -w_p/sqrt(p) for each prime p dividing N. The intended statement is presumably that entries not rigorously computed are set to zero in the database. This matters because the a'_n experiment is the main evidence that the classifier learns beyond the direct encoding, so the exact zeroing rule (which indices are zeroed, and whether coefficients with mixed prime factors are handled correctly) must be stated precisely.
[Tables 2.2-2.6 and Section 2.3] All accuracy figures are single-split point estimates with no confidence intervals, standard errors, or repeated-seed variation. The reported differences between feature sets, such as 0.9612 for a_n versus 0.9456 for a'_n, could be within sampling noise. Please provide bootstrap intervals or results over several random splits, especially for the comparison that supports the 'learning something more' claim.

minor comments (6)

[Figures 2.1 and 2.2] The text says these figures provide 'clear evidence' of separation, but no quantitative measure is given; consider adding a simple statistic such as the L2 distance between the averaged coefficient sequences or the area between the curves.
[Section 2.3] The training sizes 7772 and 5023 for even and odd forms are not tied to the described 80-20 splits; please clarify how these counts are obtained from Table 2.1.
[Section 2.3] The use of Box's M test to 'satisfy' equal covariance is not rigorous: rejecting equality for only 33 of 1000 features is not the same as establishing equality, and the test is sensitive to sample size. A direct comparison of covariance matrices would be more appropriate.
[Sections 2.3 and 2.6] The phrase 'without any hyperparameter tuning' appears in Section 2.3, but the neural network experiments in Section 2.6 use Adam with a learning rate of 1e-3 and 4e4 iterations; the claim should be restricted to the LDA experiments.
[Section 2.7 and Table 2.5] There is a typo in the caption of Table 2.5: 'Maaass forms' should be 'Maass forms', and the space in 'F ricke sign' in the Section 2.3 heading should be removed.
[General] No code, data, or reproducibility statement is included. Since the experiments are computational, please provide a link to a repository with the exact data-processing steps, random seeds, and model configurations, or at least specify the seed and split procedure.

Circularity Check

1 steps flagged · score 2.0 of 10

Acknowledged partial tautology in raw-coefficient LDA accuracy; central prediction claim remains independently supported.

self definitional [Section 2.1, Eq. (2.4); Section 2.3 validation accuracy; Section 2.4 a'_n experiment]
"Given complete information about the coefficients, the Fricke sign is easily computable; on Γ0(N ) with N squarefree, the coefficient aN encodes the Fricke sign. ... If p divides the level N then we have (2.4) a_p = −w_p /√p. ... we trained the LDA using all available training data (12795 observations) and recorded 96 .1% accuracy on the validation data."

Under Eq. (2.4), the Fricke sign of a labeled form is directly recoverable from the feature vector a_n, because a_p = −w_p/√p for every p|N and w_N is the product of the local signs w_p. An LDA trained on the raw (a_n) vector can therefore reach high validation accuracy by reading the label out of the input features, so the 96.1% figure is partly true by construction. The paper explicitly anticipates this objection, defines a'_n = 0 when gcd(n,N)>1 (matching the unknown-sign data), and reports 94.56% accuracy, so the transfer-to-unknown-sign claim is not carried by the tautological features. This is a partial, acknowledged circularity in one headline number, not a load-bearing derivation step.

full rationale

The only genuine circularity in the paper is the raw-coefficient LDA accuracy: for p|N, a_p = −w_p/√p, so the sign is literally embedded in the input for those indices. The paper identifies this in Section 2.4, removes the direct encoding via a'_n, and obtains essentially unchanged accuracy (94.56% vs. 96.12%), which is real evidence that the classifier is learning more than the deterministic readout. The application to the 15,423 unknown-sign forms is supported by this a'_n experiment and by a separate comparison with Hejhal's heuristic on 4,595 forms, an external though heuristic benchmark. The murmuration checks on predicted labels are weaker, since a classifier trained on the same features will tend to reproduce separation in those features, but the paper presents them only as reasonableness checks, not as the load-bearing derivation. Prior-work citations are contextual and not used to prove the ML result. Therefore the central claim retains independent content, and the partial tautology is explicitly acknowledged; a score of 2 reflects that minor, non-load-bearing circularity rather than a serious defect.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

This is an empirical machine learning study, not a derivation. The central claims rest on the reliability of the LMFDB dataset, the correctness of the labeled Fricke signs, standard facts about Hecke and Atkin-Lehner operators, and the trustworthiness of the heuristic Hejhal subset used for validation. The classifier parameters themselves are fitted objects. No new particles, forces, dimensions, or other invented entities are introduced.

free parameters (3)

LDA discriminant weights and threshold = 1000-dimensional coefficient vector plus threshold, values not reported
The LDA classifier is trained on labeled Maass form coefficient vectors; its discriminant coefficients and bias determine the predictions and are fitted to the training data.
Neural network weights and biases = Architecture weights and biases, values not reported
The neural network in Figure 2.11 is trained by Adam over 4e4 iterations; the fitted weights drive the sign predictions and depend on the chosen architecture and hyperparameters.
Per-feature normalization statistics for NN inputs = Per-feature means and variances from the dataset
The input features to the neural network are normalized to unit variance over the dataset; these scaling statistics are estimated from data and affect the trained model.

assumptions (5)

domain assumption Selberg eigenvalue conjecture for the dataset: spectral parameter R satisfies lambda = 1/4 + R^2 with R real and nonnegative
Invoked in Section 2.1 to write eigenvalues in the form used throughout the analysis; all dataset forms are treated as satisfying this conjecture.
standard math Hecke multiplicativity and Atkin-Lehner local sign relations, including a_p = -w_p / sqrt(p) for p dividing the level
Used in Eq (2.3) and (2.4) to decompose the global Fricke sign and to explain direct encoding of the sign in Fourier coefficients.
domain assumption The LMFDB rigorous computations and labeled Fricke signs for the 19,993 known-sign forms are correct
All training and validation labels come from LMFDB rigorous computations; if any labeled signs were wrong, the reported accuracy would be affected.
ad hoc to paper The 4,595 heuristic Hejhal outputs labeled 'probably correct' are accurate enough to serve as comparison ground truth
Section 2.7 treats these heuristic expansions as a validation set for LDA and neural network predictions; the heuristic has no rigorous proof of correctness.
domain assumption LDA's equal-covariance assumption holds for the feature distributions
The paper checks Box's M test for individual features and finds approximate equality, but LDA's optimality and the validity of its probability model still rely on this assumption.

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Cite this review

Pith. "Pith review of Learning Fricke signs from Maass form Coefficients." pith.science (2026). https://pith.science/paper/KOZCKVTE

@misc{pith2026250102105,
  author       = {Pith},
  title        = {Pith review of: Learning Fricke signs from Maass form Coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOZCKVTE}},
  note         = {Machine review of arXiv:2501.02105}
}

read the original 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.

Figures

Figures reproduced from arXiv: 2501.02105 by the authors.

**Figure 2.1.** Average value of ap over Maass forms with given Fricke sign, with and without separating by symmetry 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗

**Figure 2.2.** Average value of (−1)σ(f) ap over Maass Forms separated by Fricke sign. 2.3. Predicting the Fricke sign with LDA. The clear separation in Fricke sign provided by averaging (−1)σ(f)ap indicates that it may be possible to predict the Fricke sign based on these features, perhaps even using a fairly simple technique. We choose to undertake Linear Discriminant Analysis (LDA) to learn a linear decision boundary between cl… view at source ↗

**Figure 2.3.** Comparing the distribution of a7 for Maass forms in L1 and 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 = 1 2π √ 4 − x2, given by the (vertical) Sato–Tate distribution [Sar87]. We ran supervised learning experiments on the labeled dataset D1 ` D−1. In e… view at source ↗

Figures from the paper (11 more)

**Figure 2.4.** Figure 2.4: Average value of ap(−1)σ(f) for Maass forms with odd parity, separated by Fricke sign [PITH_FULL_IMAGE:figures/full_fig_p007_2_4.png]

**Figure 2.5.** Figure 2.5: Average value of ap(−1)σ(f) for Maass forms with even parity, separated by Fricke sign. We conclude from equation (2.4) that the Fricke sign is determined by the sequence (sgn(ap)) of signs and the number of prime factors of N. If the Fricke sign is unknown, we do no…

**Figure 2.6.** Figure 2.6: Comparing average values of ap versus a ′ p by Fricke sign In [PITH_FULL_IMAGE:figures/full_fig_p008_2_6.png]

**Figure 2.7.** Figure 2.7: Accuracy on validation set as number of an increases As mentioned earlier, it is surprising that LDA performs better when using (an) 1000 n=1 than when using just (ap)p<1000. Indeed, the Fourier coefficients are multiplicative, and so there isn’t more information in …

**Figure 2.8.** Figure 2.8: Comparing the distribution of Fricke signs by level. Unknown Fricke signs are denoted by 0. separation between consecutive eigenvalues, forcing Heisenberg-Uncertainty tradeoffs and less wellbehaved test functions in the trace formula used to compute the Maass forms)…

**Figure 2.9.** Figure 2.9: Comparing the percentages of known and unknown Fricke sign by level. Building on this theme, we experimented with many different subsets of Maass forms of various levels and did not find any that were especially easy or hard to predict. The only exception was that if…

**Figure 2.10.** Figure 2.10: Exploring accuracy of LDA on subsets of Maass forms for given levels. The size of the dot corresponds to the size of the training set. a2 a3 ... apd R n (1) 1 n (1) 2 n (1) 3 n (1) 4 n (1) 5 n (2) 1 Prob(wN = 1) ReLu σ [PITH_FULL_IMAGE:figures/full_fig_p012_2_10.png]

**Figure 2.11.** Figure 2.11: Neural network architecture used for predicting the Fricke sign from feature vectors of the form (a2, a3, . . . , apd , R) where pi denotes i th prime number, and R denotes the spectral parameter. Here n (j) i denotes i th node in j th layer. The spectral parameter …

**Figure 2.12.** Figure 2.12: Saliency map of a neural network trained for predicting Fricke sign. 2.7. Comparison of Hejhal’s algorithm, LDA, and neural networks. The rigorous computations of Maass forms in the LMFDB were computationally expensive. It is easier to make heuristic guesses at the…

**Figure 2.13.** Figure 2.13: Variances of the coefficients ap for Maass forms from [LMF24]. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_2_13.png]

**Figure 2.14.** Figure 2.14: Dependence of the accuracy on the number of the coefficients ap (for p prime) used as the input to the neural network trained for predicting Fricke sign. Here R denotes the spectral parameter. homogenous linear system in the coefficients. The non-linearity of the gr…

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A. R. Booker, M. Lee, D. Lowry-Duda, A. Seymour-Howell, N. Zubrilina Murmurations of Maass forms , arXiv:2409.00765

arXiv

[2] [2]

A. R. Booker, A. Str\"ombergsson, A. Venkatesh. Effective computation of Maass cusp forms , International Mathematics Research Notices, 2006

work page 2006

[3] [3]

Certification of Maass cusp forms of arbitrary level and character

K. Child Certification of Maass cusp forms of arbitrary level and character , arXiv:2204.11761

work page Pith review arXiv

[4] [4]

Friedlander, and Henryk Iwaniec

Duke, William, John B. Friedlander, and Henryk Iwaniec. The subconvexity problem for Artin L –functions, Inventiones Mathematicae 149 (2002): 489-577

work page 2002

[5] [5]

Farmer and S

D. Farmer and S. Lemurell Maass forms and their L -functions , arXiv:0506102

work page

[6] [6]

Kazalicki, D

M. Kazalicki, D. Vlah Ranks of elliptic curves and deep neural networks , Research in Number Theory, 9(3), 2023

work page 2023

[7] [7]

Hastie, R

T. Hastie, R. Tibshirani, and J. Friedman, The elements of statistical learning: data mining, inference, and prediction , NY Springer, 2001

work page 2001

[8] [8]

He, K.-H

Y.-H. He, K.-H. Lee, and T. Oliver, Machine-learning the Sato--Tate conjecture , J. Symb. Comput. 111 (2022), 61--72

2022