On pseudo rational modular forms of weight 2 and level N So consider the $\mathbb{Q}$-vector space $V$ of functions which satisfy the following conditions
(1) $f:\mathbb{H}\rightarrow\mathbb{C}$ is holomorphic. Here $\mathbb{H}$ stands for the
upper half plane.
(2) $f(z+1)=f(z)$ 
(3) The Fourier series of $f$ at infinity has the form $\sum_{n\geq 1} a_nq^n$
where $q=e^{2\pi iz}$ where the $a_n$'s are rational numbers
(4) $\frac{1}{Nz^2}f(\frac{-1}{Nz})=\pm f(z)$ 
Examples of non-zero elements of $V$ are given for example by $\mathbb{Q}$-linear combinations of modular forms associated to rational elliptic curves of conductor $N$
that have the same sign in their functional equation.
Let us denote this sub vector space by $W$. Unless I made a mistake in my calculation, an example of an element of $V$ which is not in $W$ could be 

$$
\sum_{d|N} d a_d E_2(dz) 
$$

with $\sum d a_d=0$ and $a_d=a_{N/d}$. If $N$ is sufficiently composite then we may find such $a_d$'s. Here $E_2$ is the weight $2$ Eisenstein series suitably normalized.
Q: How big is $V$ and is it possible to describe it in some
interesting way? 
 A: To summarize some remarks in the comments. The function $f \cdot d \tau$ will be a differential on $Y_N:=\mathbf{H}/\Gamma$, where $\Gamma_N$ is the group generated by the two matrices
$$\left( \begin{matrix} 1 & 1 \\\ 0 & 1 \end{matrix} \right) \ , \ 
\left( \begin{matrix} 0 & -1 \\\ N & 0 \end{matrix} \right)$$
If $N > 4$, then $\Gamma_N$ will have infinite index in $\mathrm{SL}_2(\mathbf{Z})$. In particular, if $X_N$ is the complex curve obtained by filling in the puncture at $\infty$ (to account for the condition that the $q$-expansion has positive coefficients), then $X_N$ (for $N > 4$) will be an open curve, and so $H^1(X_N,\Omega^1)$ will be infinite dimensional (and not particularly interesting). It's a theorem of Hecke that $\Gamma_N$ is the normalizer of $\Gamma_0(N)$ for $N \le 4$. In this way, Hecke was able to show that an $L$-series $L(f,s) = \sum a_n n^{-s}$ which satisfied a functional equation of a certain kind (with "conductor" $N \le 4$) was exactly the Mellin transform of a modular form. Being modular is thus (for $N > 4$) a stronger condition than simply asking that the $L$-series satisfies the appropriate functional equation. Historically this was interesting, because one could conjecture that the $L$-series of elliptic curves satisfied a functional equation of a certain kind, which is (a priori) a weaker conjecture than asking that elliptic curves are modular. Weil, however, showed that if the $L$-series attached to an elliptic curve $E/\mathbf{Q}$ satisfied the right functional equation,  and the same was true of all the quadratic twists of $E$, then $E$ was actually modular (the so called converse theorem, which has been vastly generalized). 
