Related papers: Fourier coefficients of noncongruence cuspforms

Serre proved that any holomorphic cusp form of weight one for $\Gamma_1(N)$ is lacunary while a holomorphic modular form for $\Gamma_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of…

Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,\chi_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $\chi_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen…

Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup norm of a half integral weight cusp form is bounded in terms of the level.

We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of…

In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero…

Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic…

In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a non-trivial nebentypus as well as Fourier…

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ of conductor $N$ with complex multiplication (CM) by $\mathbb{Q}(i)$, and $f_E$ is the corresponding cuspidal Hecke eigenform in $S^{\mathrm{new}}_2(\Gamma_0(N))$. Then $n$-th Fourier…

We prove that if $f$ is a non zero cusp form of weight $k$ on $\Gamma_0(N)$ with character $\chi$ such that $N/(\text{conductor }\chi)$ square-free, then there exists a square-free $n\ll_{\epsilon} k^{3+\epsilon}N^{7/2+\epsilon}$ such that…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

Motivated by a variant of Atiyah-Floer conjecture proposed in \cite{L2} and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional…

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers.…

From a result of Waldspurger, it is known that the normalized Fourier coefficients $a(m)$ of a half-integral weight holomorphic cusp eigenform $\f$ are, up to a finite set of factors, one of $\pm \sqrt{L(1/2, f, \chi_m)}$ when $m$ is…

Using the theory of Stienstra and Beukers, we prove various elementary congruences for the numbers \sum \binom{2i_1}{i_1}^2\binom{2i_2}{i_2}^2...\binom{2i_k}{i_k}^2, where k,n \in N, and the summation is over the integers i_1, i_2, ...i_k…

A new upper bound is given for the dimension of the space of holomorphic cusp forms of weight one and prime level $q$: $$ \hbox{dim}\, S_1(q) << q^{11/12} \log^4{q} $$ with an absolute implied constant.

In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier…

In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$…

We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier…

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight…

We study two analogs, for modular forms over $\mathbb{F}_{q}(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the $\mathbb{C}_{\infty}$-pairing is provided…