Suppose I am given a sequence $a_0, a_1,a_2 \cdots a_{k-1} \in \mathbb{Z}$. Does there exist a modular form $f$ of weight $k$ such that
$$ f(q) = \sum_{n=0}^{\infty}{a_n q^n} $$
is its Fourier series ?
Where $\Gamma = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : \; a,b,c,d \in \mathbb{Z}, \; ad-bc =1 \right\}$
My idea was to find a suitable basis of $M_k(\Gamma)$ but I didn't find any which would help here.
Would appreciate any hints (!) as to how to approach this problem.
Let $a_0,a_1,a_2,\dots a_{k-1}\in\mathbb Z$ be $k\geq 1$ integers. The modular forms on $\Gamma=SL(2,\mathbb Z)$ form a ring. For fixed $l\geq 4$ even, the dimension of the vector space $M_l(\Gamma)$ of modular forms of weight $l$ on $\Gamma$ has complex dimension $\lfloor\tfrac{l}{12}\rfloor$ if $l\equiv 2\mod 12$ and $\lfloor\tfrac{l}{12}\rfloor+1$ otherwise. Set now $l$ such that $\dim_{\mathbb C}M_l(\Gamma)\geq k$. As the ring of modular forms on $\Gamma$ is generated by the Eisenstein series $E_4$ and $E_6$, there is a polynomial $$P_l=\sum_{m,n\geq0: 4m+6n=l}E_4^mE_6^n$$ in $E_4$ and $E_6$ with weight $l$. By linear algebra, one can easily find the coefficients of $P_l$ such that $$P_l(\tau)=\sum_{n=0}^\infty a_nq^n,$$ with the first $k$ coefficients $a_n$ as given above.