Sign Changes Of Fourier Coefficients Research Articles

The first negative Fourier coefficient of an Eisenstein series newform

AbstractThere have been a number of papers on statistical questions concerning the sign changes of Fourier coefficients of newforms. In one such paper, Linowitz and Thompson gave a conjecture describing when, on average, the first negative sign of the Fourier coefficients of an Eisenstein series newform occurs. In this paper, we correct their conjecture and prove the corrected version.

On signs of Fourier coefficients of Hecke-Maass cusp forms on 𝐺𝐿₃

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group S L 3 ( Z ) \mathrm {SL}_3(\mathbb Z) . When the underlying form is self-dual, we show that there are ≫ ε X 5 / 6 − ε \gg _\varepsilon X^{5/6-\varepsilon } sign changes among the coefficients { A ( m , 1 ) } m ≤ X \{A(m,1)\}_{m\leq X} and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients A ( m , m ) A(m,m) for generic G L 3 \mathrm {GL}_3 cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of G L 3 \mathrm {GL}_3 cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

AbstractLet f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$ . More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$ , where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$ .For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$ .

On sign changes of Fourier coefficients of Hermitian cusp forms of degree two

We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.

On the first sign change of Fourier coefficients of cusp forms at sum of two squares

On the first sign change of Fourier coefficients of cusp forms at sum of two squares

Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.

On sign changes of Jacobi forms

In this paper, using two kinds of expansion of a Jacobi cusp form, we prove that there are infinitely many sign changes of Fourier coefficients of a Jacobi cusp form of real weight. We also give an explicit upper bound for the first sign change when the weight is an even integer, the index is a positive odd square-free integer and the group is the congruence subgroup of square-free level.

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

In this paper, 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 [Formula: see text] and [Formula: see text] with Fourier coefficients [Formula: see text] and [Formula: see text] respectively, we consider the following questions: existence of infinitely many primes [Formula: see text] such that [Formula: see text]; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

A note on signs of Fourier coefficients of two cusp forms

Kohnen and Sengupta (Proc. Am. Math. Soc. 137(11) (2009) 3563–3567) showed that if two Hecke eigencusp forms of weight $$k_1$$ and $$k_2$$ respectively, with $$1<k_1<k_2$$ over $$\Gamma _0({N})$$ , have totally real algebraic Fourier coefficients $$\lbrace a(n) \rbrace $$ and $$\lbrace b(n) \rbrace $$ respectively for $$n \ge 1$$ with $$a(1)=1=b(1)$$ , then there exists an element $$\sigma $$ of the absolute Galois group $$\mathrm{Gal}({\bar{\mathbb {Q}}}/{\mathbb {Q}})$$ such that $$a(n)^{\sigma } b(n)^{\sigma } < 0$$ for infinitely many n. Later Gun et al. (Arch. Math. (Basel) 105(5) (2015) 413–424) extended their result by showing that if two Hecke eigen cusp forms, with $$1<k_1<k_2$$ , have real Fourier-coefficients $$\lbrace a(n)\rbrace $$ and $$\lbrace b(n)\rbrace $$ for $$n \ge 1$$ and $$a(1)b(1) \ne 0$$ , then there exists infinitely many n such that $$a(n)b(n) > 0$$ and infinitely many n such that $$a(n)b(n) < 0$$ . When $$k_1=k_2$$ , the simultaneous sign changes of Fourier coefficients of two normalized Hecke eigen cusp forms follow from an earlier work of Ram Murty (Math. Ann. 262 (1983) 431–446). In this note, we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $$\Gamma _0({N})$$ where the coefficients lie in an arithmetic progression. Next, we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigencusp forms for the full modular group.

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.

On the first sign change of Fourier coefficients of cusp forms

Let f be a nonzero cusp form of even integral weight k⩾2 on the Hecke congruence subgroup Γ0(N) with N squarefree. Suppose that the normalized Fourier coefficients λf(n) of f are real. We prove that the first sign change of λf(n) occurs in the range n≪(kN)2+ε. This improves upon the earlier result of Choie and Kohnen [2].

On sign changes of Fourier coefficients of modular forms

It is known that if a nonzero cusp form has real Fourier coefficients, then its Fourier coefficients change signs infinitely often. In this paper, we prove that there is a codimension one subspace in the space of holomorphic modular forms of square-free level such that all of its non-zero forms have similar sign change property.

Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

Sign changes of Fourier coefficients of Hilbert modular forms

Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed.

Sign changes of Fourier coefficients of half-integral weight cusp forms

We prove a quantitative result for the number of sign changes of the Fourier coefficients of half-integral weight cusp forms in the Kohnen plus space, provided the Fourier coefficients are real numbers.

The sign changes of Fourier coefficients of Eisenstein series

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 statistical nature. Many of these results are Eisenstein series analogs of well-known theorems for cusp forms. For instance, we determine how often the pth Fourier coefficients of an Eisenstein newform begin with a fixed sequence of signs $$\varepsilon _p = \{\pm 1, 0\}$$ . Moreover, we prove the following variant of the strong multiplicity-one theorem: an Eisenstein newform is uniquely determined by the signs of its Hecke eigenvalues with respect to any set of primes with density greater than $$1/2$$ .

A short remark on sign changes of Fourier coefficients of half-integral weight modular forms

We give an upper bound on the first sign change of the Fourier coefficients of a cusp form of half-integral weight.

A SHORT NOTE ON FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS

We give an unconditional proof of a result on sign changes of Fourier coefficients of cusp forms of half-integral weight that before was proved only under the hypothesis of Chowla's conjecture.

Sign changes of Fourier coefficients of Maass eigenforms

Let f be a Maass eigenform that is a new form of level N on Γ0(N), with Laplace eigenvalue 1/4 + νf2. Then, all its Fourier coefficients {λf(n)}n=1∞ are real, and we may normalize so that λf(1) = 1. It is proved, in this paper, that the first sign change in the sequence {λf(n)}n=1∞ occurs at some n satisfying n ≪ {(3+|νf|2)N}1/2−δ and (n, N) = 1. This generalizes the previous results for holomorphic Hecke eigenforms.