Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

Published 19 Apr 2026 in math.NT | (2604.17532v1)

Abstract: We prove an unconditional, effective joint Sato-Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f'$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to a wide range of measurable subsets of $[-2,2]^2$. Indeed, our theorem applies to any measurable region whose boundary consists of a finite number of continuous curves of finite length. As a consequence, we develop a unified framework to study various arithmetic properties of Fourier coefficients of symmetric power $L$-functions attached to $f$ and $f'$. In particular, for these coefficients (and their polynomial expressions), we obtain effective distribution results, quantitative statements on simultaneous sign behaviour, and bounds for the first sign change.

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Summary

The paper establishes effective joint Sato-Tate distribution theorems with error bounds explicitly tied to the geometric properties of measurable regions.
It demonstrates that symmetric power coefficients exhibit predictable sign patterns, providing explicit density formulas and effective asymptotic results.
The methodology partitions the coefficient space into grid boxes, refining quantitative analysis of modular forms and enabling precise prime counting functions.

Effective Joint Sato-Tate Distribution and Sign Behavior of Symmetric Power Coefficients

Introduction and Context

The Sato-Tate conjecture, now established in significant generality for modular forms, governs the statistical equidistribution of normalized Fourier coefficients $a(p)$ of non-CM newforms on the interval $[-2,2]$ relative to the Sato-Tate measure. Recent quantitative versions, notably by Thorner and subsequent refinements, extend this equidistribution to error-controlled, effective counts within intervals or, for joint distributions, rectangles in $[-2,2]^2$ . The work under consideration provides a substantial generalization to more elaborate measurable subsets, deriving effective joint Sato-Tate theorems whose error estimates explicitly depend on geometric attributes of the underlying regions. This enables the systematic study of analytic and arithmetic properties—particularly sign patterns—of symmetric power coefficients and their polynomial combinations.

Main Theoretical Results

The paper proves two principal effective joint Sato-Tate distribution theorems for the Fourier coefficients $(a(p),a'(p))$ of two non-CM, twist-inequivalent newforms $f$ and $f'$ :

For any Borel-measurable $E\subset [-2,2]^2$ with boundary a finite union of continuous curves of finite length, the counting function

$\pi_{f,f',E}(x) = \#\{p \le x : (a(p), a'(p)) \in E\}$

satisfies, for sufficiently large $x$ ,

$\pi_{f,f',E}(x) = \mu_{\rm JST}(E)\pi(x) + O\left( L\alpha\,\frac{\pi(x)}{{\mathcal{M}}(x)^{1/3}}\right),$

where $[-2,2]$ 0 is the total length of the boundary and $[-2,2]$ 1 the number of boundary components. The function $[-2,2]$ 2 scales as $[-2,2]$ 3.

If the boundary satisfies a mild geometric intersection condition (Hypothesis 1), the error term improves to

$[-2,2]$ 4

with $[-2,2]$ 5 governing intersection multiplicity with vertical lines.

This framework encompasses regions bounded by algebraic or piecewise-smooth curves, thus subsuming most arithmetic sets of interest.

Figure 1: Density of primes $[-2,2]$ 6 for which $[-2,2]$ 7, $[-2,2]$ 8, and $[-2,2]$ 9 are simultaneously positive.

These theorems quantitatively realize the limiting joint Sato-Tate measure distribution with explicit dependence on the boundary data, extending previous results for rectangles and hyperbolic regions.

Applications to Symmetric Power Coefficients

The effective joint distribution results admit a broad range of additional consequences for symmetric power $[-2,2]^2$ 0-functions. For integer $[-2,2]^2$ 1 and polynomials $[-2,2]^2$ 2, polynomial expressions in symmetric power coefficients such as $[-2,2]^2$ 3 are shown to exhibit effective quantitative equidistribution relative to the Sato-Tate pushforward measure. This underpins precise asymptotics for various sign and comparison statistics.

Simultaneous Sign Distributions

A particularly notable application is the density of primes where $[-2,2]^2$ 4 (or more general polynomial combinations) displays specified sign patterns. The paper obtains—for all $[-2,2]^2$ 5—an explicit and effective formula for the density of primes for which $[-2,2]^2$ 6 is positive (density $[-2,2]^2$ 7) or negative ( $[-2,2]^2$ 8), with error decaying as a negative power of $[-2,2]^2$ 9.

If either $(a(p),a'(p))$ 0 or $(a(p),a'(p))$ 1 is odd, $(a(p),a'(p))$ 2; if both are even, a sign bias in favor of positivity arises, given by

$(a(p),a'(p))$ 3

This result subsumes and generalizes classical sign distribution theorems for modular forms and their symmetric powers, yielding effective error terms and applicable to polynomial conditions of considerable generality.

Comparison of Coefficient Sizes

The same machinery applies to inequalities between coefficients, for instance, the asymptotic frequency of primes with $(a(p),a'(p))$ 4. If $(a(p),a'(p))$ 5, or both $(a(p),a'(p))$ 6 and $(a(p),a'(p))$ 7 are odd, this occurs with density $(a(p),a'(p))$ 8 up to a power-saving error.

Figure 2: Density of primes $(a(p),a'(p))$ 9 for which $f$ 0, $f$ 1, and $f$ 2.

The results not only recover and make effective several previous asymptotic and sign equidistribution statements in the literature but also quantifiably refine them.

Bounds for First Sign Change

An effective upper bound is provided for the least prime $f$ 3 for which $f$ 4, with the current (unconditional) bound

$f$ 5

for some constant $f$ 6. Under GRH, this can be sharpened to a polylogarithmic bound in the level.

Such bounds provide the first simultaneous sign-change results at the prime level for general symmetric power coefficients, previously unavailable even in non-effective form.

Methodological Contributions

The proof strategy entails partitioning the region $f$ 7 into grid boxes and precisely indexing those intersecting the desired sets, with the error terms sharply quantified via geometric attributes of the boundary. The approach enables the error to scale independently of the region's complexity, provided it has finite geometric description, and couples seamlessly with previous Sato-Tate error term analysis.

Crucially, this paradigm allows polynomial images of the joint coefficient tuples $f$ 8 to be analyzed via the measure-preserving properties of the Sato-Tate law, both for sign and for comparison statistics.

Theoretical and Practical Implications

These results tightly connect geometric features of regions in the joint coefficient space with analytic behavior of counting functions for primes—demonstrating an intimate link between equidistribution geometry and arithmetic. The explicit effectiveness enables unconditional quantitative statements for simultaneous sign and size properties of symmetric power $f$ 9-function coefficients, previously unavailable in such generality.

On the theoretical side, this facilitates further study of biases and statistical independence phenomena in the space of automorphic forms, especially regarding the interplay between higher symmetric powers and modular forms. Practically, the explicit error terms allow sharpened statistical investigations into sign behaviors and dominance patterns in computational and theoretical experiments.

The error improvement under GRH suggests future directions for pushing conditional bounds to the limit of available analytic techniques on Rankin-Selberg convolutions.

Conclusion

This work offers a unified, highly effective framework for joint Sato-Tate equidistribution for pairs of newforms, with direct arithmetic applications to sign patterns and comparison of symmetric power coefficients. The explicit, geometric dependence of the error terms, together with applications to longstanding problems about sign change and domination, substantially advances quantitative analytic number theory relating to modular forms and automorphic $f'$ 0-functions.

Further refinements along these lines, particularly concerning minimal sign change and joint equidistribution for other non-abelian settings, remain promising directions for subsequent research, as do potential conditional reductions of the main error terms based on analytic properties of underlying $f'$ 1-functions.