Timothy HuberAffiliation: University of Texas Rio Grande Valley Title Of Talk: Counting holomorphic modular forms generated by generated by quotients of Klein forms Abstract: Let $a_0$ be an even positive integer. For each fixed integer $N\ge 5$, we give closed formulas for the number of products $$ q^{r} (q^{N};q^{N})_{\infty}^{a_{0}} \prod_{i=1}^{\lfloor N/2 \rfloor} (q^{i},q^{N-i};q^{N})_{\infty}^{a_{i}}, \qquad r:= \frac{N}{24}a_0 + \sum_{i=1}^{\lfloor N/2 \rfloor} \left( \frac{N}{12} - \frac{i}{2} + \frac{i^{2}}{2N} \right) a_i$$ that are holomorphic modular forms for $\Gamma_1(N)$ of weight $a_{0}/2$ by counting integer lattice points satisfying linear congruences in appropriate polytopes. The Fourier coefficients of certain products in these classes satisfy Ramanujan-type congruences for primes and prime powers.
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Last update made Tue Mar 10 21:28:04 CDT 2026.
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