TITLE: An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level
QUESTION [6 upvotes]: In Miyake's book on modular forms explicit formulas for the $q$-expansions of a basis of the space Eisenstein series of arbitrary level and weight were given. I guess similar formulas for a basis of the entire space cuspidal forms do not exist. Is there at least an explicit formula for the $q$-expansion of a non-zero cuspidal holomorphic modular form of level $\Gamma_0(p)$ and weight $2$ for arbitrarily large primes $p$?

REPLY [5 votes]: Borisov and Gunnells have defined and studied the so-called toric modular forms. The idea is to take products of Eisenstein series of lower weight. So in your case, (linear combinations of) products of two Eisenstein series of weight 1. More precisely, Borisov and Gunnells consider Eisenstein series $s_a$ for $a \in \mathbb{Z}/N\mathbb{Z}$, $a \neq 0$. They have weight 1 and level $\Gamma_1(N)$, and their $q$-expansions are completely explicit. They show that the pairwise products $s_a s_b \in M_2(\Gamma_1(N))$ behave very much like modular symbols. In particular, and modulo Eisenstein series of weight 2, they satisfy the nice 3-term Manin relations, one can compute the action of the Hecke operators, and so on.
You are interested with $\Gamma_0(p)$ so you would need to consider
\begin{equation*}
f_{a,b} = \sum_{k=1}^{p-1} s_{ka} s_{kb}
\end{equation*}
which belongs to $M_2(\Gamma_0(p))$. It may not be cuspidal, but since $\Gamma_0(p)$ has only two cusps, there is only one Eisenstein series and you can compute the Eisenstein component of $f_{a,b}$ just from its leading term. In this way you get explicit cusp forms of weight 2 for any $p$. Borisov and Gunnells have shown that the (cuspidal components of) $f_{a,b}$ span the space of cusp forms of analytic rank zero, i.e. the space generated by the newforms $f$ satisfying $L(f,1) \neq 0$.