Rational structures on the space of cusp forms on Γ0(N) through the rationality of periods

Kohnen and Zagier studied two types of cusp forms [Formula: see text] on [Formula: see text] defined by binary quadratic forms. They showed the cusp forms [Formula: see text] have rational even periods, while [Formula: see text] have rational odd periods. In this paper, we examine analogous cusp forms [Formula: see text] on [Formula: see text] for higher levels [Formula: see text]. Using locally harmonic Maass forms on [Formula: see text], we explicitly compute the even (resp. odd) period polynomials of [Formula: see text] (resp. [Formula: see text]). Consequently, we establish that the space of cusp forms of weight [Formula: see text] on [Formula: see text] has rational structures, attributed to the periods of these cusp forms.