# Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

### YoungJu Choie

Pohang University of Science and Technology, Pohang City, Republic of Korea### Yoon Kyung Park

Gongju National University of Education, Gongju, Republic of Korea### Don B. Zagier

Max Planck Institute for Mathematics, Bonn, Germany

## Abstract

Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$ . We also show that for $N = 2, 3$ and $5$ this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on $\Gamma_0(N)$.

## Cite this article

YoungJu Choie, Yoon Kyung Park, Don B. Zagier, Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1379–1410

DOI 10.4171/JEMS/864