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Let f and g be two distinct normalized primitive Hecke–Maass cusp forms of weight zero with Laplacian eigenvalues 1 4 + u 2 and 1 4 + v 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the non-trivial upper bounds for the sum ∑ n ∈ S | λ f (n) λ g (n) | , where S is a suitable subset of Z + ∩ [ 1 , x ] with certain properties. [ABSTRACT FROM AUTHOR]

Let f be a cusp form for SL (2 , Z) , and λ f (n) , the n-th Fourier coefficient of the L-function attached to f. Here a new bound is established for the exponential sum ∑ n ≤ x Λ (n) λ f (n) e (n θ α) , where 0 < θ < 1 and α ≠ 0 is a fixed parameter. This improves previous results when 1 / 2 ≤ θ < 1 . [ABSTRACT FROM AUTHOR]

Let π be a cuspidal automorphic representation of GL 2 (A Q) associated to holomorphic forms with Fourier coefficients a π (n) . Consider an automorphic representation Π which is equivalent to sym m π or π × sym m π . We establish uniform upper bounds for ∑ n ⩽ X | a Π (| f (n) |) | , where f (x) ∈ Z [ x ] is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results. [ABSTRACT FROM AUTHOR]

Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λSymmf (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Symm f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λSymmf (n) and establish its asymptotic formula. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. And set Q (x) a primitive integral positive-definite binary quadratic form of fixed discriminant D < 0 with the class number h (D) = 1 . In this paper, we establish a lower bound for the analytic density of the set { p : p = Q (x) for some x ∈ Z 2 , λ f (p i) λ f (p j) < λ g (p i) λ g (p j) } , where j ⩾ 1 , 0 ⩽ i ⩽ j are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of λ f (p j) and λ g (p j) in a given interval supported on the binary quadratic form Q (x) . Similar problems for triple product L-functions are also studied. [ABSTRACT FROM AUTHOR]

There 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. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

Let 2 ≤ j ≤ 8 be any fixed positive integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . Denote by λ sym 2 f (n) the nth normalized coefficient of the Dirichlet expansion of the symmetric square L-function L (sym 2 f , s) attached to f. In this paper, we are interested in the average behaviour of the following summatory function ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym 2 f j (a 2 + b 2 + c 2 + d 2) for x ≥ x 0 (sufficiently large), which improves and generalizes the recent works of Sharma and Sankaranarayanan (Res Number Theory 8:19, 2022, Rend Circ Mat Palermo II Ser 72:1399–1416, 2023). [ABSTRACT FROM AUTHOR]

Let H k ∗ be the set of all normalized primitive holomorphic cusp forms of even weight k ≥ 2 for S L 2 (Z) . For f ∈ H k ∗ , let λ sym j f (n) be the nth normalized Fourier coefficient of the jth symmetric power L-function attached to f. In this paper, we establish asymptotic formulas for the power sums of λ sym j f (n) and improve previous results. [ABSTRACT FROM AUTHOR]

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 σ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of f | | k σ . [ABSTRACT FROM AUTHOR]

Suppose that H ∗ is the set of all primitive cusp forms f of even integral weight k ≥ 2 for the full modular group S L 2 (Z) . In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product L-function L (s , f ⊗ f ⊗ f) and the related L-function L (s , sym 2 f ⊗ f) attached to f on average, which improves previous results. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct normalized primitive holomorphic cusp forms of weights k 1 , k 2 and levels N 1 , N 2 , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the first change of sequences { λ f (n i) } n ∈ N and { λ f (n j) λ g (n l) } n ∈ N with positive integers i ≥ 2 and j , l ≥ 1 . And we also derive the lower bounds for the sequences { λ f (n i) } n ∈ N , { λ f (n j) λ f (n l) } n ∈ N of the same signs in the interval [1, x], which remains the best possible results in terms of order of magnitude, where i ≥ 2 , j , l ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the n-th normalized Fourier coefficient of primitive holomorphic cusp form of even integral weight k ≥ 2 for the full modular group S L (2 , Z) . In this paper, we first refine related previous results on the l-th power sums of the Fourier coefficient λ f (n) . Then we establish asymptotic formulas of the variance of λ f l (n) . [ABSTRACT FROM AUTHOR]

Let f and g be distinct primitive holomorphic cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function λ f (n i) λ f (n j) and λ f (n i) λ g (n j) for any integers i , j ≥ 1 . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 2 + ε for arbitrarily small ε > 0 . In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients λ f × f × f (n) and λ f × f × g (n) of triple product L-functions L (f × f × f , s) and L (f × f × g , s) , respectively. [ABSTRACT FROM AUTHOR]

Let a E (n) be the normalized Frobenius traces. For any multiplicative function f satisfying some mild conditions, we investigate the sum ∑ n = 1 N f (n) a E (n) and obtain a nontrivial bound. We also prove cancelation in the sum of Frobenius traces twisted by the coefficients of automorphic L-function on GLm. [ABSTRACT FROM AUTHOR]

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an even integer. Let f be a primitive holomorphic cusp form of weight k, with λ f (n) being its n-th Fourier coefficient. We explicitly determine the dependence on the weight aspect by proving ∑ n ≤ X λ f (n) e n 2 α + n β ≪ (X k) ε X 21 22 k 31 22 uniformly for k, and any α , β ∈ R , where the implied constant depends only on ε . In addition we obtain an analogue of the Prime Number Theorem associated to the Fourier coefficients of cusp forms ∑ n ≤ X Λ (n) λ f (n) e n 2 α + n β ≪ X exp - c log X log X + log k uniformly for k < X 1 / 31 - ε and any α , β ∈ R , where the implied constant is absolute. [ABSTRACT FROM AUTHOR]

Let f be a Hecke cusp form of even integral weight k for the full modular group S L (2 , Z) . Denote by λ sym j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function associated with f. In this paper, we derive a upper bound for the first negative coefficient for sequence { λ sym j f (n) } n ≥ 1 in terms of the weight of f. Furthermore, we also investigate the length of the maximal string of terms with the same sign in an interval [1, x]. [ABSTRACT FROM AUTHOR]

Let j ≥ 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . Denote by λ sym j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L (sym j f , s) attached to f. In this paper, we are interested in the average behaviour of the following sum ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym j f 2 (a 2 + b 2 + c 2 + d 2) , where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence. [ABSTRACT FROM AUTHOR]

Let f(z) be a holomorphic Hecke eigenform of even weight k for SL(2, ℤ), and denote by L(s, symmf) the corresponding symmetric power L-function associated with f. Denote by ⋋symmf (n) the nth normalized coefficient of L(s, symmf). In this paper, we investigate the sum Σn≤x⋋symmf (n) for m ≥ 2 and get the uniform upper bound extending the previous results. [ABSTRACT FROM AUTHOR]

Let Hk denote the space of holomorphic Hecke cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Let λf (n), λg(n), and λh(n) be the nth Hecke-normalized Fourier coefficients of three distinct holomorphic Hecke cusp forms f(z), g(z), h(z) ∈ H k i ′ , i ′ = 1 , 2 , 3 , respectively. In this paper, we establish an asymptotic formula for the summatory function (under suitable conditions)Σn≤x λ f 2 (ni) λ g 2 (nj), which generalizes the results of Lü [17]. As an application, we also consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms of the type Σn≤x |λf (n)λg(ni)| and Σn≤x|λf (n)λg(ni)λh(nj)|, which also generalizes the work of Lü [17]. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 and N be positive integers and let χ be a Dirichlet character modulo N. Let f(z) be a modular form in M k (Γ 0 (N) , χ) . Then we have a unique decomposition f (z) = E f (z) + S f (z) , where E f (z) ∈ E k (Γ 0 (N) , χ) and S f (z) ∈ S k (Γ 0 (N) , χ) . In this paper, we give an explicit formula for E f (z) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities. [ABSTRACT FROM AUTHOR]

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

Let λ π (1 , ... , 1 , n) be the normalized Fourier coefficients of an even Hecke–Maass form π for S L (m , Z) with m ≥ 3 , and r 3 (n) = # { (n 1 , n 2 , n 3) ∈ Z 3 : n = n 1 2 + n 2 2 + n 3 2 } . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients λ π (1 , ... , 1 , n) and r 3 (n) , which improves previous results (even under the generalized Ramanujan conjecture). [ABSTRACT FROM AUTHOR]

Let ϕ (z) be a primitive Hecke–Maass cusp forms with Laplace eigenvalue 1 4 + t 2 . Denote by L (s , sym m ϕ) the m-th symmetric power L-function associated to ϕ and by λ sym m ϕ (n) the n-th coefficient of the Dirichlet expansion of L (s , sym m ϕ) . For any nonzero integer ℓ we prove ∑ n ⩽ x λ ϕ (n) λ ϕ (n + ℓ) ≪ ϕ , ℓ x (log x) 0.187 (x ⩾ 3). This improves Holowinsky's corresponding result, which requires 1 6 in place of 0.187. for all x ⩾ 3 . Further assuming that L (s , sym 10 ϕ) and L (s , sym 12 ϕ) are automorphic cuspidal, we obtain a conditional generalization to the symmetric square case: ∑ n ⩽ x λ sym 2 ϕ (n) λ sym 2 ϕ (n + ℓ) ≪ ϕ , ℓ x (log x) 0.196 (x ⩾ 3). [ABSTRACT FROM AUTHOR]

We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of Γ 0 (N) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class t Z 2 , we investigate the sign changes in the t p 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. [ABSTRACT FROM AUTHOR]

We prove that given any ϵ > 0 and a primitive adelic Hilbert cusp form f of weight k = (k 1 , k 2 , ... , k n) ∈ (2 Z) n and full level, there exists an integral ideal m with N (m) ≪ ϵ Q f 9 / 20 + ϵ such that the m -th Fourier coefficient of C f (m) of f is negative. Here n is the degree of the associated number field, N (m) is the norm of integral ideal m and Q f is the analytic conductor of f. In the case of arbitrary weights, we show that there is an integral ideal m with N (m) ≪ ϵ Q f 1 / 2 + ϵ such that C f (m) < 0 . We also prove that when k = (k 1 , k 2 , ... , k n) ∈ (2 Z) n , asymptotically half of the Fourier coefficients are positive while half are negative. [ABSTRACT FROM AUTHOR]

In this article, we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of GL 2 + (Q) and Aut (C) on modular forms. [ABSTRACT FROM AUTHOR]

In this article, we investigate the sign changes of the sequence of coefficients at sum of two squares where the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group. We provide the quantitative result for the number of sign changes of the sequence in a small interval. [ABSTRACT FROM AUTHOR]

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane R e s > 1 2 . This exhibits a high fluctuation of the coefficients at squarefree integers and improves a sign-change result in Lau et al. (Mathematika 62:866–883, 2016). [ABSTRACT FROM AUTHOR]

In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous non-vanishing of Fourier coefficients of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights at powers of prime ideals. [ABSTRACT FROM AUTHOR]

Let Sk(N) be the space of all holomorphic cusp forms of even integral weight k for the congruence group Γ0(N). For any f∈Sk(N) with ‖f‖2=1, we study the higher-power moments of ∑n≤xaf(n), where af(n) is the nth normalized Fourier coefficient of f. Furthermore, as an application, we investigate the higher-power moments of Fourier coefficients in arithmetic progressions. [ABSTRACT FROM AUTHOR]

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight. [ABSTRACT FROM AUTHOR]

Let π1,π2,…,πk be cuspidal automorphic representations of GL(2) or GL(3) with trivial conductor and trivial central character, and π:=π1⊞π2⊞⋯⊞πk be the isobaric representation associated to them. We establish that there exists a positive constant ϑk<1 such that for any α∈R, any x≥1 and any positive ε, one has ∑n≤xλπ1⊞π2⊞⋯⊞πk(n)e(αn)≪πj,εxϑk+ε,where the implied constant does not depend on α. We also consider some isobaric representations containing the trivial representation. As an application, we consider averages of shifted convolution sums of the type ∑h=1H∑n=1Xa(n)λπ1⊞π2⊞⋯⊞πk(n+h),where a(n) is any arithmetic function. [ABSTRACT FROM AUTHOR]

We use the theory of modular forms to give formulas for the number of representations of n by all sextenary quadratic forms with coefficients 1 , 2 , 3 , 6 . We also apply our results to write newforms in S 3 ( Γ 0 ( 24 ) , χ ) in terms of eta quotients. [ABSTRACT FROM AUTHOR]

Recently, combining a product-to-sum formula and conditions for the non-representability of integers by certain ternary quadratic forms, Williams gave ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. In this paper, we generalize Williams’ results by utilizing theta function identities. [ABSTRACT FROM AUTHOR]

Assume that α , β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation ∑ n ∈ ( X , 2 X ] α ( n ) β ( n − h ) γ ( n + h ) averaging over a short variable h . More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh. [ABSTRACT FROM AUTHOR]

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a ( n) be its n-th Fourier coefficient. We consider the sum $${S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}$$ and prove that S has an asymptotic formula when β = 1/2 and α is close to $$\pm 2\sqrt {q/D}$$ for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S . We also consider the sum $${S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}$$ with ø( x) ∈ C (0,+∞) and prove that S has better upper bounds than S at some special α and β. [ABSTRACT FROM AUTHOR]

Let a( n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of $$SL_2({\mathbb R})$$ . This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of $$a(n)a(n+r)$$ for a fixed positive integer r. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, () for a fixed positive integer r, concerning both non-vanishing and non-negativity. [ABSTRACT FROM AUTHOR]

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles. [ABSTRACT FROM AUTHOR]

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms. [ABSTRACT FROM AUTHOR]

We give an exact expression (Theorem 3.2) for the Fourier coefficients of logarithmic vector-valued Poincaré series associated to representations where $$\rho (T)$$ is a single Jordan block. [ABSTRACT FROM AUTHOR]

In this note, we give an explicit basis for the harmonic weak forms of weight two. We also show that their holomorphic coefficients can be given in terms of regularized inner products of weight zero weakly holomorphic forms. [ABSTRACT FROM AUTHOR]