arXiv subjects

Moni Kumari

Publications and source records attributed to Moni Kumari.

13 recordsLinked to original sources

arXiv2026

Determining newforms via arithmetic relations among Fourier coefficients

We investigate the distribution of primes satisfying arithmetic inequalities involving the Fourier coefficients of two non-CM newforms at prime powers. More precisely, we establish asymptotic formulas for the number of primes for which the differences, products, and ratios of the Fourier coefficients satisfy prescribed inequalities, together with explicit estimates for the corresponding densities. The proofs combine an effective joint Sato--Tate theorem with a geometric analysis of the associated semi-algebraic regions. As applications, we obtain new multiplicity one criteria, improve a theorem of Matom\"aki on small differences between Fourier coefficients, establish density-one analogues in the spirit of the Atkin--Serre conjecture, and derive a new characterization of twist-equivalence through the distribution of ratios of Fourier coefficients.

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arXiv2026

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

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 any measurable region of $[-2,2]^2$ whose boundary consists of finitely many 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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arXiv2026

On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms

In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-\epsilon}$ for almost all primes $p$ and for any $\epsilon > 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character.

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arXiv2022

Ramanujan-style congruences for prime level

We establish Ramanujan-style congruences modulo certain primes $\ell$ between an Eisenstein series of weight $k$, prime level $p$ and a cuspidal newform in the $\varepsilon$-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight $k$ for $Γ_0(p)$. Under a mild assumption, this refines a result of Gaba-Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler-Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by $\ell.$ The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.

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arXiv2022

The quantitative distribution of Hecke eigenvalues of Maass forms

Let $f$ be a normalized Hecke-Maass cusp form of weight zero for the group $SL_2(\mathbb Z)$. This article presents several quantitative results about the distribution of Hecke eigenvalues of $f$. Applications to the $Ω_{\pm}$-results for the Hecke eigenvalues of $f$ and its symmetric square sym$^2(f)$ are also given.

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arXiv2022

Coprimality of Fourier coefficients of eigenforms

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for which $(a_1(p), a_2(p))=1$. We also study the average order of the number of prime divisors of $(a_1(p), a_2(p))$.

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arXiv2022

On the Lang--Trotter conjecture for Siegel modular forms

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.

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arXiv2021

Divisors of Fourier coefficients of two newforms

For a pair of distinct non-CM newforms of weights at least 2, having rational integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain an estimate for the set of primes $p$ such that $$ ω(a_1(p)-a_2(p)) \le [ 7k+{1}/{2}+k^{1/5}],$$ where $ω(n)$ denotes the number of distinct prime divisors of an integer $n$ and $k$ is the maximum of their weights. As an application, under GRH, we show that the number of primes giving congruences between two such newforms is bounded by $[7k+{1}/{2}+k^{1/5} ]$. We also obtain a multiplicity one result for newforms via congruences.

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arXiv2020

The first simultaneous sign change for Fourier coefficients of Hecke-Maass forms

Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $λ_f(n)$ and $λ_g(n)$, and we may normalize $f$ and $g$ so that $λ_f(1)=1=λ_g(1)$. In this article, we first prove that the sequence $\{λ_f(n)λ_g(n)\}_{n \in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.

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arXiv2020

Construction of Poincaré-type series by generating kernels

Let $Γ\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetildeΓ$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in{\mathbb R}$ under the action of $\widetildeΓ$. The space of such functions admits the action of the hyperbolic Laplacian $Δ_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlović (where $k=0$), we use the spectral expansion associated to $Δ_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of $Δ_k$.

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arXiv2018

Non-vanishing of Hilbert Poincaré series

We prove some non-vanishing results of Hilbert Poincaré series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincaré series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et. al.

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arXiv2018

Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms

In this article, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms $f$ and $g$ with Fourier coefficients $a_n$ and $b_n$ respectively, we consider the following questions: existence of infinitely many primes $p$ such that $a_p b_p\neq 0$; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

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arXiv2018

On the parity of the Fourier coefficients of hauptmoduln $j_{N}(z)$ and $j_{N}^{+}(z)$

We obtain some interesting results about the parity of the Fourier coefficients of hauptmoduln $j_{N}(z)$ and $j_{N}^{+}(z),$ for some positive integers $N$. We use elementary methods and the techniques of O. Kolberg's proof for the parity of the partition function.

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