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We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular L -values for short intervals. [ABSTRACT FROM AUTHOR]

In this work, Ramanujan type congruences modulo powers of primes p ≥ 5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t -core partitions known to satisfy Ramanujan type congruences for p = 5 , 7 , 11. The vectors of exponents corresponding to products that are modular forms for Γ 1 (p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ 1 (p) of weights 1 ≤ k ≤ 5 for primes 5 ≤ p ≤ 19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p > 19. Canonical bases for modular forms of level 5 ≤ p ≤ 19 are constructed by summing weight one Hecke Eisenstein series of levels 5 ≤ p ≤ 19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ 1 (p) and Γ (p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ 1 (p) as a direct sum of subspaces of modular forms for Γ (p) of the form q r / p Z [ [ q ] ]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences. [ABSTRACT FROM AUTHOR]

We construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a P1${\mathbb {P}}^1$‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k$k$‐Hodge bundle determined by Gheorghita–Tarasca and by Korotkin–Sauvaget–Zograf. [ABSTRACT FROM AUTHOR]

Viazovska proved that the E8 lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations. [ABSTRACT FROM AUTHOR]

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

If t is a positive integer, then a partition of a non-negative integer n is a t-core if none of the hook numbers of the associated Ferrers-Young diagram is divisible by t. Let a t (n) denote the number of t-core partitions of n. The purpose of this article is to prove some infinite families of congruences modulo 3 for the 5-core partition function a 5 (n) and parity results for the 7-core partition function a 7 (n) by using elementary generating functions and q-series manipulations. We further verify the congruences for the 5-core partition function a 5 (n) using the theory of modular forms. [ABSTRACT FROM AUTHOR]

We study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma. [ABSTRACT FROM AUTHOR]

Let Γ be a congruence subgroup such that Γ 1 (N) ⊂ Γ ⊂ Γ 0 (N) for some positive integer N. For a positive integer k, let M k, Z (Γ) be the set of modular forms of weight k on Γ with integral Fourier coefficients. Let R k (Γ) be the set of common zeros in the upper half plane H of all the modular forms of weight k on Γ. In this note, we prove that the density of modular forms in M k, Z (Γ) with an algebraic zero z ∉ R k (Γ) is zero. [ABSTRACT FROM AUTHOR]

This article addresses a new odd entire function of order one structured by the Fourier sine integral, which is the solution of the one-dimensional diffusion equation in theory of modular form. [ABSTRACT FROM AUTHOR]

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems. [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]

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method. [ABSTRACT FROM AUTHOR]

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x , y , and z , using creative telescoping, yielding modular forms expressed as pullbacked 2 F 1 hypergeometric functions, can be obtained much more efficiently by calculating the j -invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p = x y z . In other cases where creative telescoping yields pullbacked 2 F 1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2 F 1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve. [ABSTRACT FROM AUTHOR]

In recent years, Rogers and Zudilin developed a method to write L -values attached to elliptic curves as periods. In order to apply this method to a broader collection of L -values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the L -values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at k > 1 explicitly as a period. As a side result, we give simple integral expressions for the generating functions of L (E , k) when even (or odd) k runs over positive integers. [ABSTRACT FROM AUTHOR]

The Birch and Swinnerton-Dyer conjecture – which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems – and its generalizations are a significant focus of number theory research. A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime p for certain rational elliptic curves of rank 1. We generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms f of even weight higher than 2 with Galois representation V containing a Galois-stable lattice T , let W = V / T and let K be an imaginary quadratic field in which the prime p splits. Our main result is that under some conditions, the p -index of the size of the Shafarevich-Tate group of W with respect to the Galois group of K is twice the p -index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna. Significant original adaptations we make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of a localization-modulo-torsion map, and (2) a comparison of different Heegner cycles. [ABSTRACT FROM AUTHOR]

We consider the Diophantine equation 7 x 2 + y 2 n = 4 z 3. We determine all solutions to this equation for n = 2 , 3 , 4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper integer solutions for specific primes p > 7. We computationally verify the criterion for all primes 7 < p < 10 9 , p ≠ 13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x 2 + 7 y 2 n = 4 z 3 , determining all families of solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for n = 4 and primes n ≥ 5. [ABSTRACT FROM AUTHOR]

In a paper of Kedlaya and Medvedovsky [KM19] , the number of distinct dihedral mod 2 modular representations of prime level N was calculated, and a conjecture on the dimension of the space of level N weight 2 modular forms giving rise to each representation was stated. In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K3 hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families. [ABSTRACT FROM AUTHOR]

A folklore conjecture on the partition function asserts that the density of odd values of p (n) is 1 2. In general, for a positive integer t , let p t (n) be the t -multipartition function and δ t be the density of the odd values of p t (n). It is widely believed that δ t exists. Given an odd integer a and an integer b depending on a and t , Judge and Zanello framed an infinite family of conjectural congruence relations on p t (a n + b) (mod 2) which establishes a striking connection between δ a and δ 1. As a special case t = 1 , it implies that δ 1 > 0 if (3 , a) = 1 and δ a > 0. This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for a = ℓ α is a prime power with ℓ ≥ 5 and a = 3. [ABSTRACT FROM AUTHOR]

We compare the homology of a congruence subgroup Γ of GL 2 (Z) with coefficients in the Steinberg modules over Q and over E , where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism ψ Γ , E in the long exact sequence of homology stemming from this comparison has image in H 0 (Γ , St (Q 2 ; R)) generated by classes z β indexed by β ∈ E ∖ Q. We investigate this image. When R = C , H 0 (Γ , St (Q 2 ; C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z β is closely related to periods of modular forms over the geodesic in the upper half plane from β to its conjugate β ′. Assuming GRH we prove that the image of ψ Γ , E equals the entire cuspidal part. When R = Z , we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, H 0 cusp (Γ , St (Q 2 ; Z)). Assuming GRH we prove that for any congruence subgroup, ψ Γ , E always has finite index in H 0 cusp (Γ , St (Q 2 ; Z)) , and if Γ = Γ 1 (N) ± or Γ 1 (N) , then the image is all of H 0 cusp (Γ , St (Q 2 ; Z)). If Γ = Γ 0 (N) ± or Γ 0 (N) , we prove (still assuming GRH) an upper bound for the size of H 0 cusp (Γ , St (Q 2 ; Z)) / Im (ψ Γ , E). We conjecture that the results in this paragraph are true unconditionally. We also report on extensive computations of the image of ψ Γ , E that we made for Γ = Γ 0 (N) ± and Γ = Γ 0 (N). Based on these computations, we believe that the image of ψ Γ , E is not all of H 0 cusp (Γ , St (Q 2 ; Z)) for these groups, for general N. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. We show that each solution of the system of congruence equations x 1 2 + x 2 2 + x 3 2 + x 4 2 ≡ 0 (mod p) and x 1 + x 2 + x 3 + x 4 ≡ 0 (mod p) corresponds to precisely four solutions of the system of Diophantine equations x 1 2 + x 2 2 + x 3 2 + x 4 2 = p 2 and x 1 + x 2 + x 3 + x 4 = p that are pairwise orthogonal over Z , partially answering a conjecture proposed in Wang et al. [10]. The result was obtained by counting the number of solutions of both equations using Gaussian sum and modular forms, and the classical Cayley transformation. [ABSTRACT FROM AUTHOR]

In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.e., # { (x 1 , ... , x 2 k + 1) ∈ Z 2 k + 1 : x 1 2 + ⋯ + x 2 k + 1 2 = p 2 } , proposed by Cooper. [ABSTRACT FROM AUTHOR]

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author. [ABSTRACT FROM AUTHOR]

Given odd, coprime integers a , b (a > 0), we consider the Diophantine equation a x 2 + b 2 l = 4 y n , x , y ∈ Z , l ∈ N , n odd prime, gcd ⁡ (x , y) = 1. We completely solve the above Diophantine equation for a ∈ { 7 , 11 , 19 , 43 , 67 , 163 } , and b a power of an odd prime, under the conditions 2 n − 1 b l ≢ ± 1 (mod a) and gcd ⁡ (n , b) = 1. For other square-free integers a > 3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x , y with (gcd ⁡ (x , y) = 1), l ∈ N and all odd primes n > 3 , satisfying 2 n − 1 b l ≢ ± 1 (mod a) , gcd ⁡ (n , b) = 1 , and gcd ⁡ (n , h (− a)) = 1 , where h (− a) denotes the class number of the imaginary quadratic field Q (− a). For a video summary of this paper, please visit https://youtu.be/Q0peJ2GmqeM. [ABSTRACT FROM AUTHOR]

We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. This occurs, for example, when the Rankin-Cohen bracket covers the entirety of S n. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and when it is determined by the injectivity of the operator itself. This can also be interpreted as using the Rankin-Cohen bracket operator of eigenforms to create evidence for Maeda's Conjecture. [ABSTRACT FROM AUTHOR]

In a previous paper, for any odd integer N ≥ 5 we constructed (N - 1) / 2 modular forms of Γ (N) of rational weight (N - 3) / 2 N and proved that the graded rings of modular forms of weight ℓ (N - 3) / 2 N ( ℓ ∈ Z ≥ 0 ) are generated by our forms for N = 5 , 7, 9. The proof was given by a direct calculation of the structure of the ring. In this paper, we generalize the result to cases when N = 11 and 13 by using Castelnuovo–Mumford criterion on normal generation and Fujita criterion on relations of sections of invertible sheaves. For this purpose, it is needed to handle modular forms of small weight where Riemann Roch theorem has cohomological obstruction. Both rings for N = 11 and 13 are generated by 5 and 6 generators with 15 and 35 concrete fundamental relations, respectively. These relations also give equations of the corresponding modular varieties. We will show that the similar claim does not hold for N = 15 and N = 23 . We also give remarks on relations of some theta constants and further problems. [ABSTRACT FROM AUTHOR]

Let k be an even integer and S k be the space of cusp forms of weight k on SL 2 (Z). Let S = ⊕ k ∈ 2 Z S k. For f , g ∈ S , we let R (f , g) be the set of ratios of the Fourier coefficients of f and g defined by R (f , g) : = { x ∈ P 1 (C) | x = [ a f (p) : a g (p) ] for some prime p } , where a f (n) (resp. a g (n)) denotes the n th Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R (f , g) is finite, then f = c g for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL 2 (Z). We apply it to study the number of representations of a positive integer by a quadratic form. [ABSTRACT FROM AUTHOR]

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic 'arithmetic–geometric mean (AGM)' integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms. [ABSTRACT FROM AUTHOR]

A partition of a positive integer n is called ℓ -regular if none of its parts is divisible by ℓ. Let b 1 1 (n) denote the number of 11-regular partitions of n. In this paper we give a complete description of the behavior of b 1 1 (n) modulo 5 when 5 ∤ n in terms of the arithmetic of the ring ℤ [ − 3 3 ]. This description is obtained by relating the generating function for these values of b 1 1 (n) to a Hecke eigenform, and as a byproduct we find exact criteria for which of these values are divisible by 5 in terms of the prime factorization of 1 2 n + 5. [ABSTRACT FROM AUTHOR]

In this article, we compute the Petersson norm of a family of weight one cusp forms constructed by Hecke and express it in terms of the Rademacher symbol and the regulator of real quadratic fields. [ABSTRACT FROM AUTHOR]

Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions. The theory of singular values is applied to derive expansions for 1 / π of level 20 analogous to those formulated by Ramanujan. [ABSTRACT FROM AUTHOR]

We will construct a p -adic analytic family of D -th Shintani lifting generalized by Kojima and Tokuno for a Coleman family. Consequently, we will have a p -adic L -function which interpolates the central L -values attached to a Coleman family and obtain a congruence between the central L -values. Focusing on the case of p -ordinary, we will obtain two applications. One of them states that a congruence between Hecke eigenforms of different weights sufficiently close, p -adically, derives a congruence between their central L -values. The other one is about the Goldfeld conjecture in analytic number theory. We will show that there exists a primitive form satisfying the conjecture for each even weight sufficiently close to 2, 3-adically, thanks to a result of Vatsal. [ABSTRACT FROM AUTHOR]

We study two general divisor problems related to Hecke eigenvalues of classical Maass cusp forms. We give the relevant estimation and improve previous results. [ABSTRACT FROM AUTHOR]

We give a short proof of Zagier's conjecture/Mersmann's theorem which states that each holomorphic eta quotient of weight 1/2 is an integral rescaling of some eta quotient from Zagier's list of fourteen primitive holomorphic eta quotients. In particular, given any holomorphic eta quotient f of weight 1/2, this result enables us to provide a closed-form expression for the coefficient of q n in the q -series expansion of f , for all n . We also demonstrate another application of the above theorem in extending the levels of the simple (resp. irreducible) holomorphic eta quotients. [ABSTRACT FROM AUTHOR]

Let p k ( n ) denote the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k . We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo 5 for p k ( n ) . Moreover, we will present some new congruences modulo 7 for p 4 ( n ) . [ABSTRACT FROM AUTHOR]

In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a new type of object, which Zagier later called a quantum modular form. Since then, a number of others have studied similar examples. Here we develop the theory in a general context, giving rise to a well-defined class of quantum modular forms. Since elements of this class show up frequently in examples of combinatorial and number theoretical interest, we propose the study of the general properties of this space of quantum modular forms. We conclude by raising fundamental questions concerning this space of objects which merit further study. [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]

One of the properties of the Rogers–Ramanujan continued fraction is its representation as an infinite product given by ( q ) = q 1 / 5 ∏ j = 1 ∞ ( 1 − q j ) ( j 5 ) where ( j p ) is the Legendre symbol. In this work we study the level 13 function R ( q ) = q ∏ j = 1 ∞ ( 1 − q j ) ( j 13 ) and establish many properties analogous to those for the fifth power of the Rogers–Ramanujan continued fraction. Many of the properties extend to other levels ℓ for which ℓ − 1 divides 24, and a brief account of these results is included. [ABSTRACT FROM AUTHOR]

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples. [ABSTRACT FROM AUTHOR]

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q , but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms. [ABSTRACT FROM AUTHOR]

Let p ¯ ( n ) denote the number of overpartitions of n . In this paper, we show that p ¯ ( 5 n ) ≡ ( − 1 ) n p ¯ ( 4 ⋅ 5 n ) ( mod 5 ) for n ≥ 0 and p ¯ ( n ) ≡ ( − 1 ) n p ¯ ( 4 n ) ( mod 8 ) for n ≥ 0 by using the relation of the generating function of p ¯ ( 5 n ) modulo 5 found by Treneer and the 2-adic expansion of the generating function of p ¯ ( n ) due to Mahlburg. As a consequence, we deduce that p ¯ ( 4 k ( 40 n + 35 ) ) ≡ 0 ( mod 40 ) for n , k ≥ 0 . When k = 0 , it was conjectured by Hirschhorn and Sellers, and confirmed by Chen and Xia. Furthermore, applying the Hecke operator on ϕ ( q ) 3 and the fact that ϕ ( q ) 3 is a Hecke eigenform, we obtain an infinite family of congruences p ¯ ( 4 k ⋅ 5 ℓ 2 n ) ≡ 0 ( mod 5 ) , where k ≥ 0 and ℓ is a prime such that ℓ ≡ 3 ( mod 5 ) and ( − n ℓ ) = − 1 . Moreover, we show that p ¯ ( 5 2 n ) ≡ p ¯ ( 5 4 n ) ( mod 5 ) for n ≥ 0 . So we are led to the congruences p ¯ ( 4 k 5 2 i + 3 ( 5 n ± 1 ) ) ≡ 0 ( mod 5 ) for n , k , i ≥ 0 . In this way, we obtain various Ramanujan-type congruences for p ¯ ( n ) modulo 5 such as p ¯ ( 45 ( 3 n + 1 ) ) ≡ 0 ( mod 5 ) and p ¯ ( 125 ( 5 n ± 1 ) ) ≡ 0 ( mod 5 ) for n ≥ 0 . [ABSTRACT FROM AUTHOR]