The spaces of new forms for harmonic weak Maalss forms and their level raising properties modulo $$\ell $$ ℓ

Abstract

Bringmann et al. (Math. Ann., 355(3):1085–1121, 2013) and Guerzhoy (Proc. Am. Math. Soc., 136:3051–3059, 2008) studied the multiplicity two Hecke theory for weakly holomorphic modular forms on the full modular group. In this note, we consider a generalization of these results to congruence subgroups \(\Gamma _1(N)\) with a new form theory. Using this, we define new forms for harmonic weak Maass forms and study a level raising property modulo \(\ell \) for them.