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In this article, we consider the problem of determining formulas for the number of representations of a natural number n by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain conditions on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain the modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain the modularity property of the generating function of mixed forms involving figurate numbers (including the squares and triangular numbers) with coefficients and forms of the type m 2 + m n + n 2 with coefficients. In particular, we show the modularity of the generating function of odd number of squares and odd number of triangular numbers (with coefficients). As a consequence, explicit formulas for the number of representations of these mixed forms are obtained using a basis of the corresponding space of modular forms of integral weight. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in Ono et al. (Aequat Math 50:73–94, 1995). In 2016, Xia et al. (Int J Number Theory 12:945–954) considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the (p, k) parametrization method. We also derive these 21 formulas using our method and further obtain as a consequence, the (p, k) parametrization of the Eisenstein series E 4 (τ) and its duplications. It is to be noted that the (p, k) parametrization of E 4 and its duplications were derived by a different method in [3, 8]. We illustrate our method with several examples. [ABSTRACT FROM AUTHOR]

We give a general conjecture concerning the existence of Eisenstein congruences between weight k ≥ 3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ of conductor N. The special cases M = 1 and M = p prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof. [ABSTRACT FROM AUTHOR]

Lehmer's conjecture that τ (n) never vanishes remains open. In the spirit of this conjecture, it is natrual to consider for a given integer α , whether there exists n ∈ Z + such that τ (n) = α . Several recent papers exclude many odd α involving one prime factor or small values of the τ -function using the theory of congruences of τ (n) , Lucas sequences, integer points of elliptic curves, and the Thue equation. In this paper, we generalize the results and methods to infinitely many odd α involving two prime factors, showing that τ (n) ≠ ± p 1 p 2 b for arbitrary b ∈ Z , where 3 ≤ p 1 , p 2 ≤ 31 are primes such that p 1 p 2 ≤ 247 or p 1 = p 2 . Moreover, together with the previous results, we get τ (n) ≠ 3 b 1 5 b 2 7 b 3 11 b 4 13 b for b ∈ Z , and b i ∈ { 0 , 1 } , i = 1 , 2 , 3 , 4 . Besides, we show that | τ (n) | > | τ (3) | for all odd τ (n) , n ≠ 1 . [ABSTRACT FROM AUTHOR]

Let f and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group S L (2 , Z) . We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑ n ≤ x λ sym i f × sym j g (n) , ∑ n ≤ x λ f (n i) λ g (n j) , ∑ n ≤ x | λ sym i f × sym j g (n) | , ∑ n ≤ x | λ f (n i) λ g (n j) | , and ∑ n ≤ x max { | λ sym i f × sym j g (n) | 2 φ , | λ sym i f × sym j g (n + h) | 2 φ } , where φ > 0 and h is a fixed positive integer. [ABSTRACT FROM AUTHOR]

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that lim x → ∞ # { 1 ≤ n ≤ x ∣ τ (n) ≠ 0 } x > 1 - 1.15 × 10 - 12 , where τ (n) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ (n) ≠ 0 . [ABSTRACT FROM AUTHOR]

We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form. [ABSTRACT FROM AUTHOR]

We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central L -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo ℓ analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo ℓ on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work. [ABSTRACT FROM AUTHOR]

We investigate the linear dependence among odd periods of modular forms. We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three odd periods are linearly independent. We also prove an asymptotic result for an arbitrary number of odd periods. The results are achieved by studying cuspidal projections of products of two Eisenstein series. [ABSTRACT FROM AUTHOR]

This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues. [ABSTRACT FROM AUTHOR]

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 , there exist infinitely many natural numbers 푛 such that τ ⁢ (p n) has at least 2 (1 - ϵ) ⁢ log ⁡ n log ⁡ log ⁡ n distinct prime factors for almost all primes 푝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above. [ABSTRACT FROM AUTHOR]

In this article, we obtain finite field analogues of classical summation identities connecting F 3 -Appell series and 4 F 3 -classical hypergeometric series. As an application, we establish a new summation formula satisfied by the 4 F 3 -Gaussian hypergeometric series. We further express certain special values of 4 F 3 -Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of 4 F 3 -Gaussian hypergeometric series. [ABSTRACT FROM AUTHOR]

For fixed positive integers k and N, there are only finitely many holomorphic eta quotients of weight k for the congruence subgroup \(\Gamma_{0}(N)\). In this article, we obtain a similar finiteness result for holomorphic generalized eta quotients of weight k on \(\Gamma_1(p)\) where k is a positive integer and p is a prime number. We also obtain a criterion for two holomorphic generalised eta quotients which represent the same one. [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]

In this paper, we consider twisted Dirichlet series attached to quasimodular forms, study their analytic properties, and prove an analogue of Weil's converse theorem for quasimodular forms over congruence subgroups. We also give some applications of our results to a certain q-series and sign changes of the Fourier coefficients of quasimodular forms. [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 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]

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]

Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to $2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$) and odd parts at most half the peak. We prove that two-variable generating functions for $\unicode[STIX]{x1D707}(m,n)$ and $\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single $C^{\infty }$ function in $\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables $w$ and $q$ , how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak. [ABSTRACT FROM AUTHOR]

In this work, we establish Ramanujan–Mordell type formulas associated to certain quadratic forms of discriminants 20k or 32k. In the end, a remark is given to illustrate potential extensions to other related cases. [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]

Let L ∈ { 6 , 9 , 12 } . We determine the generating functions of certain combinations of three ranks and three cranks modulo L in terms of eta quotients. Then, using the periodicity of signs of these eta quotients, we compare their values with the values of p (n) L / 3 . [ABSTRACT FROM AUTHOR]

We study the angle changes of Fourier coefficients of cusp forms and q-exponents of generalized modular functions at primes. More precisely, we prove that both these subsequences, under certain conditions, fall infinitely often outside any given wedge W (θ 1 , θ 2) : = { r e i θ : r > 0 , θ ∈ [ θ 1 , θ 2 ] } with 0 ≤ θ 2 - θ 1 < π . [ABSTRACT FROM AUTHOR]

We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums ∑ a + p b = n σ (a) σ (b) , ∑ p 1 a + p 2 b = n σ (a) σ (b) and ∑ a + p 1 p 2 b = n σ (a) σ (b) where p , p 1 , p 2 are primes. [ABSTRACT FROM AUTHOR]

We extend the author's earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3 / 2 and weight 2 for the dual Weil representation for an even lattice. These formulas can be used to compute Borcherds products for orthogonal groups of type O(2, 1) and O(2, 2). [ABSTRACT FROM AUTHOR]

We calculate the Jacobi Eisenstein series of weight k≥3 for a certain representation of the Jacobi group, and evaluate these at z=0 to give coefficient formulas for a family of modular forms Qk,m,β of weight k≥5/2 for the (dual) Weil representation on an even lattice. The forms we construct have rational coefficients and contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for Qk,m,β and the Eisenstein series of Bruinier and Kuss p-adically to get an efficient algorithm. The main application is in constructing automorphic products. [ABSTRACT FROM AUTHOR]

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct χ5, the cusp form of lowest weight for the group Sp(2,Z). In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice A1 and Igusa's form χ5 appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates. [ABSTRACT FROM AUTHOR]

We study the values of | L ( 1 , F ) | for Hecke–Maass cusp forms F on S L ( n , Z ) ( n ≥ 3 ) of large Langlands parameters. New unconditional results on the extreme values and conditional results on the size range are derived, which determine precisely the order of magnitude of L ( 1 , F ) . In addition, we enhance the new average estimate toward the Ramanujan Conjecture due to Matz and Templier. An application of the Hecke multiplicativity to the Littlewood–Richardson rule for a product of two Schur polynomials is cultivated. [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]

Suppose k is a positive integer. In this work, we establish formulas for the number of representations of integers by the quadratic forms x 1 2 + ⋯ + x k 2 + m ( x k + 1 2 + ⋯ + x 2 k 2 ) for m ∈ { 2 , 4 } . [ABSTRACT FROM AUTHOR]

Under a slightly better zero-free region of the corresponding modular L -function, we get a very small bound for the size of first sign change of Hecke eigenvalues for the classical modular forms. When it comes to the prime argument, we derive, for almost all primitive forms, a small bound for the first sign change on prime numbers. [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 study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $$\mathbb {Z}+\frac{1}{2}$$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $$\Gamma _0(4)$$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincaré series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expression for radial limits associated to the partition rank and crank functions previously studied by Ono, Rhoades, and Folsom. [ABSTRACT FROM AUTHOR]

In recent work of Bringmann, Guerzhoy, and the first author, p -adic modular forms were constructed from mock modular forms. We look at a specific case, starting with the weight −10 mock modular form, and prove explicit congruences. [ABSTRACT FROM AUTHOR]

In this note, we first see that the isomorphism in [26] between vector-valued modular forms and scalar-valued modular forms holds for more general discriminant forms. With this established, we shall prove the Zagier duality for canonical bases for spaces of weakly holomorphic modular forms satisfying some ϵ -condition, thus obtain the complete grids for the duality. Finally we raise a question on certain integrality of the Fourier coefficients of reduced modular forms and provide a partial solution. [ABSTRACT FROM AUTHOR]

In this paper we obtain asymptotic formulas for the Fourier coefficients of an infinite family of crank-generating functions. Moreover, we use this result to show that the crank obeys certain inequalities. This implies that the crank cannot explain any partition congruences in the usual way beside the three deduced by Ramanujan. [ABSTRACT FROM AUTHOR]

Given a fixed modular form of level 1 we define a family of linear operators between spaces of cusp forms by use of the Rankin-Cohen brackets and we compute the adjoint maps of such family with respect to the usual Petersson inner product. This is done in terms of the effect on the Fourier development of cusp forms. This is a generalization of a result due to W. Kohnen. As an application we prove certain relations among Fourier coefficients of cusp forms. [ABSTRACT FROM AUTHOR]

We prove that the orbit of a non-periodic point at prime values of the horocycle flow in the modular surface is dense in a set of positive measure. For some special orbits we also prove that they are dense in the whole space—assuming the Ramanujan/Selberg Conjectures for GL 2 / Q . In the process, we derive an effective version of Dani's Theorem for the (discrete) horocycle flow. [ABSTRACT FROM AUTHOR]

We characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group . [ABSTRACT FROM AUTHOR]

Let Using the theory of Stienstra and Beukers (Math. Ann., 271:269-304, ), we prove that the numbers A( n) and A( n−1)−16 A( n−2) satisfy three term congruence relations similar to those satisfied by Apery numbers. Moreover, for k≥3 and p prime, we prove divisibility by p of some simple linear combinations of the numbers A( n), for $n\in \mathbb{N}$. To obtain this, we study the arithmetic properties of the Fourier coefficients of certain holomorphic and weakly holomorphic modular forms. [ABSTRACT FROM AUTHOR]

In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula. [ABSTRACT FROM PUBLISHER]

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level Γ 0 (N) and integral weight k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to Γ 0 (N) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono's construction of harmonic lifts of Poincaré series. [ABSTRACT FROM AUTHOR]