# Journal of the European Mathematical Society

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**Volume 21, Issue 5, 2019, pp. 1379–1410**

**DOI: 10.4171/JEMS/864**

Published online: 2019-02-01

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

YoungJu Choie^{[1]}, Yoon Kyung Park

^{[2]}and Don B. Zagier

^{[3]}(1) Pohang University of Science and Technology, Pohang City, Republic of Korea

(2) Gongju National University of Education, Gongju, Republic of Korea

(3) Max Planck Institute for Mathematics, Bonn, Germany

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 $\Gamma_0(N)$.

*Keywords: *Period, Hecke eigenform, Jacobi theta series, parabolic cohomology, Rankinâ€“Cohen brackets

Choie YoungJu, Park Yoon Kyung, Zagier Don: Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions. *J. Eur. Math. Soc.* 21 (2019), 1379-1410. doi: 10.4171/JEMS/864