Related papers: Fourier Coefficients Of Some Cusp Forms

In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant…

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 $k$…

We study the zeros of cusp forms in the Miller basis whose vanishing order at infinity is a fixed number $m$. We show that for sufficiently large weights, the finite zeros of such forms in the fundamental domain, all lie on the circular…

Let $f$ be a primitive Maass cusp form for a congruence subgroup $\Gamma_0(D) \subset $ SL($2,\mathbb{Z}$) and $\lambda_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $\lambda_f(n)$…

In 1947, Lehmer conjectured that the Ramanujan \tau-function \tau(m) is non-vanishing for all positive integers m, where \tau(m) are the Fourier coefficients of the cusp form \Delta of weight 12. It is known that Lehmer's conjecture can be…

Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper we give a representation theoretical explanation for this…

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log…

In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying…

In this article, we investigate the $p$-adic valuation $\nu_p$ of quantities such as the factorial $n!$, the hyperfactorial $H(n)$ or the superfactorial $\mathrm{sf}(n)$. In particular, we obtain simple bounds (both upper and lower) for…

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N…

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert…

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the…

A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the…

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} +…

In this article, we investigate a non-Archimedean analogue of a question of Atkin and Serre. More precisely, we derive lower bounds for the largest prime factor of non-zero Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms…

Kaneko's formula expresses the Fourier coefficients of the elliptic modular $j$-function as finite sums of singular moduli. First published as a short article in 1996, it was presented as a consequence of Zagier's work inspired by Borcherds…

We study modular forms for $\textrm{SL}_2(\mathbb{Z})$ with no negative Fourier coefficients. Let $A(k)$ be the positive integer where if the first $A(k)$ Fourier coefficients of a modular form of weight $k$ for $\textrm{SL}_2(\mathbb{Z})$…

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small…

In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half…