Abstract. In this paper we show that both of the Green-Schwarz anomaly factorization formula
for the gauge group E8 × E8 and the Hoˇrava-Witten anomaly factorization formula for the gauge
group E8 can be derived through modular forms of weight 14. This answers a question of J.
H. Schwarz. We also establish generalizations of these factorization formulas and obtain a new
Hoˇrava-Witten type factorization formula.