An eta-quotient of level N is a modular form of the shape f (z) = δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ 0 (N ) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f (z) is a holomorphic modular form that is nonvanishing on the upper half plane and has integer Fourier coefficients at infinity, then f (z) is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of J 0 (2 k )(Q).

Let b 13 (n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 (n) modulo 3 where n ≡ 1 (mod 3). In particular, we identify an infinite family of arithmetic progressions modulo arbitrary powers of 3 such that b 13 (n) ≡ 0 (mod 3).

Let ≥ 5 be prime, let m ≥ 1 be an integer, and let p(n) denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer b (m) of size roughly m 2 such that the module formed from the Z/ m Z-span of generating functions for p b n+1 24 with odd b ≥ b (m) has finite rank. The same result holds with "odd" b replaced by "even" b. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of m; it is +12 24. In this paper, we prove, with a mild condition on , that b (m) ≤ 2m − 1. Our bound is sharp in all computed cases with ≥ 29. To deduce it, we prove structure theorems for the relevant Z/ m Z-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.

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We present an accurate two-dimensional mathematical model for steak cooking based on Flory-Rehner theory. The model treats meat as a poroelastic medium saturated with fluid. Heat from cooking induces protein matrix deformation and moisture loss, leading to shrinkage. Numerical simulations indicate good agreement with experimental data. Moreover, this work presents a new and computationally non-expensive method to account for shrinkage.

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We develop a new algorithm to compute a basis for M k (Γ 0 (N )), the space of weight k holomorphic modular forms on Γ 0 (N ), in the case when the graded algebra of modular forms over Γ 0 (N ) is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.

Abstract. Given a classical modular form f (z), a basic question is whether any of its Fourier coefficients vanish. This question remains open for certain modular forms. For example, let ∆(z) = τ (n)q n ∈ S 12 (Γ 0 (1)). A well-known conjecture of Lehmer asserts that τ (n) = 0 for all n. In recent work, Ono constructed a family of polynomials A n (x) ∈ Q[x] with the property that τ (n) vanishes if and only if A n (0) and A n (1728) do. In this paper, we establish a similar criterion for the vanishing of coefficients of certain newforms on genus zero groups of prime level.