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)



Existence of real modular function with specific behavior as $q\to 0$
For any holomorphic cuspform $f$ vanishing to order exactly $1$ at $i\infty$, $F(x+iy)=\log|f(x+iy)|$ is of the form you want.


The extreme simple case of this construction is the (in)famous $f(z)=q\prod (1-q^n)^{24}$, the log of whose absolute value appears in the Laurent expansion of the Eisenstein series at $s=1$ (Kronecker's limit formula).


