Narrow Results

Let p be a prime and let f be any cusp form of level l ∈ {2,3,5,7,13} whose weight satisfy a certain congruence modulo (p-1). Then we exhibit explicit linear combinations of the coefficients of f that must be divisible by p. For a normalized Hecke eigenform, this translates (under mild restrictions) into the pth coefficient itself being divisible by a prime ideal above p in the ring generated by the coefficients of f. This provides many instances of the so-called non-ordinary primes. We also discuss linear relations satisfied universally on the space of modular forms of these levels. These results extend recent work of Choie, Kohnen and Ono in the level 1 case.

The sum of divisors function σ(m) is defined by

Recently, Williams discovered explicit formulas of the coefficients c(n) in the Fourier series expansions of a class of eta quotients. Motivated by the results obtained by Williams, we find that the coefficients c(2n) in the Fourier series expansions of another class of eta quotients can be represented as a linear combination of σ₃(n), σ₃(n/2), σ₃(n/3) and σ₃(n/6). One example is

Using modular forms we determine the number of representations of a positive integer by certain diagonal octonary quadratic forms with coefficients 1, 3 or 9.

Using modular forms, we determine the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients 1, 3 or 9.

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that

Andrews and Newman introduced the minimal excludant or “$\mathrm{\text{mex}}$” function for an integer partition $\pi$ of a positive integer $n$, $\mathrm{\text{mex}}(\pi )$, as the smallest positive integer that is not a part of $\pi$. They defined $\sigma \mathrm{\text{mex}}(n)$ to be the sum of $\mathrm{\text{mex}}(\pi )$ taken over all partitions $\pi$ of $n$. We prove infinite families of congruence and multiplicative formulas for $\sigma \mathrm{\text{mex}}(n)$. By restricting to the part of $\pi$, Andrews and Newman also introduced $\mathrm{\text{moex}}(\pi )$ to be the smallest odd integer that is not a part of $\pi$ and $\sigma \mathrm{\text{moex}}(n)$ to be the sum of $\mathrm{\text{moex}}(\pi )$ taken over all partitions $\pi$ of $n$. In this paper, we show that for any sufficiently large $X$, the number of all positive integer $n\le X$ such that $\sigma \mathrm{\text{moex}}(n)$ is an even (or odd) number is at least ${𝒪}(loglogX)$.

For $\left|q\right|<1$, define $f_i={\prod }_{n=1}^{\infty }(1-q^{in})$, and let $(A(q),B(q))$ be any of the pairs $(f_1^4,\frac{f_1^8}{f_2^2}),$$(f_1^4,\frac{f_1^{10}}{f_3^2}),$$(f_1^6,\frac{f_2^4}{f_1^2}),$$(f_1^6,\frac{f_1^{14}}{f_2^4}),$$(f_1^{10},\frac{f_2^6}{f_1^2}),$$(f_1^{14},\frac{f_3^5}{f_1}),(f_1^{14},\frac{f_2^8}{f_1^2}).$ For any such pair $(A(q),B(q))$, define the sequences $\{a(n)\}$ and $\{b(n)\}$ to be the coefficients of $q^n$ of $A(q)$ and $B(q)$, respectively. Then for each pair it is shown that $a(n)$ vanishes if and only if $b(n)$ vanishes. In each case, a criterion is given which states precisely when $a(n)=b(n)=0$. Moreover, for the pairs $(f_1^{26},\frac{f_3^9}{f_1}),$$(f_1^{26},\frac{f_2^{16}}{f_1^6})$ it is shown that $a(n)=b(n)=0$ if $12n+13$ satisfies a criteria of Serre for $a(n)=0$.

Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight $15/2$ with varying levels. Proof is based on the Shimura lift.