Related papers: Sign changes of coefficients of half integral weig…

We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form $F$ of even integral weight on the Siegel modular group of arbitrary genus $ g\geq 2 $.

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that…

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no…

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $\Gamma_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the…

We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $\Gamma_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to…

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum;…

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author…

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively.

We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and…

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we…

Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems…

We prove that, for fixed tame level (N,p) = 1, there are only finitely many Hecke eigenforms f of level Gamma_1(N) and even weight with a_p(f) = 0 which are not CM.

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for…

Let $k$ be an even integer and $S_k$ be the space of cusp forms of weight $k$ on $\SL_2(\ZZ)$. Let $S = \oplus_{k\in 2\ZZ} S_k$. For $f, g\in S$, we let $R(f, g) = \{ (a_f(p), a_g(p)) \in \mathbb{P}^1(\CC)\ |\ \text{$p$ is a prime} \}$ be…

Let $f$ be a newform of weight $2$ on $\Gamma_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $\Gamma_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a…

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…

We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$…

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…

We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $\Gamma_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$…