CONGRUENCES BETWEEN EXTREMAL MODULAR FORMS AND THETA SERIES OF SPECIAL TYPES MODULO POWERS OF 2 AND 3

Heninger, Rains and Sloane proved that the nth root of the theta series of any extremal even unimodular lattice in R^n has integer coefficients if n is of the form 2^i3^j5^k(i ≥ 3). Motivated by their discovery, we find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3. This assertion enables us to prove that the 2nth root and the (3n/2)th root of the extremal modular form of weight n/2 have at least one non-integer coefficient