SUPERLACUNARY CUSP FORMS

Many researchers have studied Euler product identities of weight k = 1/2 and k = 3/2, often related to the Jacobi Triple Product identity and the Quintuple Product identity. These identities correspond to theta series of weight k = 1/2 and k = 3/2, and they exhibit a behavior which is defined as superlacunary. We show there are no eigen-cusp forms of integral weight which are superlacunary. For half-integral weight forms with k greater than or equal to 5/2, we give a mild condition under which there are no superlacunary eigen-cusp forms. These results suggest the nonexistence of similar Euler-Product identities that arise from eigen-cusp forms with weight k not equal 1/2 or 3/2.