Related papers: Comparing Hecke eigenvalues of newforms

Let f be a non-CM newform of weight k > 1. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is…

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and…

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier…

In this article, we estimate the density of the set of primes $p$ such that the $p$-th Hecke eigenvalue of an Ikeda lift is divisible by a fixed positive integer. One of the main ingredients involves the study of abelian subfields of fixed…

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…

Let F in S_k(Sp(2g, Z)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues mu_F(n). Suppose that the associated automorphic representation pi_F is locally tempered everywhere. For each c>0 we consider the set of…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

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 $$ \omega(a_1(p)-a_2(p)) \le […

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in…

Let $\pi$ be a cuspidal automorphic representation for GL(2)/$\mathbb{Q}$ that is self-dual. In this Note we show that there exists a positive upper Dirichlet density of primes at which the associated Hecke eigenvalues of $\pi$ are larger…

In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.

In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from the signs of their Hecke eigenvalues. To establish our theorem, we obtain a result towards simultaneous sign changes of eigenvalues…

Let $n_0(N,k)$ be the number of initial Fourier coefficients necessary to distinguish newforms of level $N$ and even weight $k$. We produce extensive data to support our conjecture that if $N$ is a fixed squarefree positive integer and $k$…

Given a congruence of Hecke eigenvalues between newforms of weight $2$, we prove, under certain conditions, a congruence between corresponding weight-$3/2$ forms.

For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $ N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\neq 0$,…

We revisit a theorem of Ram Murty about the number of initial Fourier coefficients that two cuspidal eigenforms of different weights can have in common. We prove an explicit upper bound on this number, and give better conditional and…

We show that the first sign change of Hecke eigenvalues of classical newforms has a finite mean, which we also compute. We distinguish between the first negative prime Hecke eigenvalue, and the first negative Hecke eigenvalue. This problem…

Let $f$ and $g$ be two distinct newforms which are normalized Hecke eigenforms of weights $k_1, k_2 \ge 2$ and levels $N_1, N_2 \ge 1$ respectively. Also let $a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients of $f$ and $g$…

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and…

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular…