I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar q)+...)$
${\it i.e.}$ only non-negative powers of $q$ and $\bar{q}$ arise, with possibly a logarithmic divergence (but no a pole divergence)
Does anyone know an example of such a function, or a theorem forbidding it?
Thank you very much!
Note added: Sorry for the non-standard notation. By $\bar{q}$ I mean $\bar{q}=\exp{(-2\pi I z^*)}$, the complex conjugated of $q$, and by real I mean invariant under complex conjugation. (they are non-holomorphic, the non-holomorphic Eisenstein series is an example of this, sorry for the language, it comes from Physics literature)