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arxiv: 2606.27190 · v1 · pith:VML3EIQ6new · submitted 2026-06-25 · 🧮 math.NT

Determining Newforms via various relations among Fourier Coefficients

Arvind Kumar , Moni Kumari , Prabhat Kumar Mishra This is my paper

Pith reviewed 2026-06-26 02:45 UTC · model grok-4.3

classification 🧮 math.NT
keywords newformsFourier coefficientsmultiplicity onedensity of ratiostwist inequivalentnon-CM formsautomorphic forms
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The pith

For twist-inequivalent non-CM newforms the ratios of normalized Fourier coefficients at prime powers are quantitatively dense in the real line.

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

The paper establishes that arithmetic relations among Fourier coefficients of two newforms can be used to identify which forms they come from. For pairs of twist-inequivalent non-CM newforms it proves quantitative distribution results for differences, products, and ratios of the normalized coefficients at prime powers. These distribution statements are then applied to give quantitative versions of multiplicity-one theorems and density-one analogues of a theorem of Gafni–Thorner–Wong, together with an explicit criterion that distinguishes the forms by the density of their coefficient ratios.

Core claim

For any pair of twist-inequivalent non-CM newforms the ratios of their normalized Fourier coefficients at prime powers are quantitatively dense in the real line; this density supplies a new criterion for determining the underlying newforms from the distribution of their coefficient ratios.

What carries the argument

Quantitative density of the ratios of normalized Fourier coefficients at prime powers between twist-inequivalent non-CM newforms.

If this is right

  • Quantitative refinements of the multiplicity-one theorem become available.
  • Density-one analogues of the Gafni–Thorner–Wong theorem hold for the coefficient relations.
  • Newforms can be distinguished solely by the distribution of ratios of their Fourier coefficients.
  • The same quantitative distribution statements apply to differences and products of the coefficients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The density criterion may allow reconstruction of a newform from sufficiently many coefficient ratios without knowing the level or weight in advance.
  • Similar ratio-density statements could be tested for other families of automorphic forms where Fourier coefficients are known to satisfy Sato–Tate type laws.
  • Effective versions of the density results might yield explicit bounds usable in computational identification of newforms from tables of coefficients.

Load-bearing premise

The newforms under comparison must be twist-inequivalent and non-CM.

What would settle it

Exhibit two twist-inequivalent non-CM newforms such that the set of ratios of their normalized Fourier coefficients at prime powers fails to be dense in any interval of the real line.

read the original abstract

In this article, we investigate how arithmetic relations among the Fourier coefficients of two newforms can be used to determine the underlying forms. For pairs of twist-inequivalent non-CM newforms, we obtain quantitative results on the distribution of differences, products, and ratios of normalized Fourier coefficients at prime powers. As applications, we derive quantitative refinements of multiplicity one and establish density one analogues of a theorem of Gafni--Thorner--Wong. We further show that, for twist-inequivalent newforms, the ratios of Fourier coefficients are quantitatively dense in the real line, which provides a new criterion for determining newforms through the distribution of their coefficient ratios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper investigates arithmetic relations among the Fourier coefficients of newforms to determine the underlying forms. For pairs of twist-inequivalent non-CM newforms, quantitative results are obtained on the distribution of differences, products, and ratios of normalized Fourier coefficients at prime powers. These yield quantitative refinements of multiplicity one theorems and density-one analogues of a theorem of Gafni--Thorner--Wong. The paper further proves that the ratios of Fourier coefficients for twist-inequivalent newforms are quantitatively dense in the real line, providing a new criterion for determining newforms via the distribution of coefficient ratios.

Significance. If the quantitative distribution results hold with effective error terms, the work supplies new, explicit tools for distinguishing newforms from coefficient data alone. The density criterion for ratios offers a novel, falsifiable test that could be implemented computationally and may interact usefully with existing multiplicity-one results in the literature on Hecke eigenvalues.

minor comments (3)
  1. [§2.3] §2.3: the normalization of the Fourier coefficients a_p(f)/sqrt(p) is introduced without an explicit reminder that the Sato-Tate measure is with respect to this normalization; a one-sentence clarification would prevent reader confusion when the distribution statements are stated later.
  2. [Theorem 4.2] Theorem 4.2: the density-one statement is stated with an implicit dependence on the levels of the two forms; making the dependence on N1 N2 explicit in the statement would strengthen the claim.
  3. [§5] The proof of the ratio-density result invokes an effective version of a theorem from the literature; the precise reference and the dependence of the implied constant on the weight and level should be recorded in the statement of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard newform theory

full rationale

The abstract and stated claims rely on established results from the theory of newforms, Hecke eigenvalues, and distribution theorems for twist-inequivalent non-CM forms. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the paper's own inputs. The density criterion and multiplicity refinements are presented as consequences of coefficient distributions at prime powers, with the non-CM and twist-inequivalent restrictions explicitly required rather than smuggled in. This matches the most common honest finding for papers that apply known tools without re-deriving them circularly.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.1-grok · 5636 in / 1033 out tokens · 50417 ms · 2026-06-26T02:45:21.495758+00:00 · methodology

discussion (0)

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Reference graph

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