An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level

Question

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 $p$?

Answer 1

You ask for the explicit Fourier expansion of a weight $2$ cusp form of level $p$ for $p$ arbitrarily large. This suggests you're OK with only certain primes $p$. If $p \equiv 11 \pmod{12}$ is prime, the function $$ f(z) = \eta^{2}(z) \eta^{2}(pz) = q^{\frac{p+1}{12}} \prod_{n=1}^{\infty} (1-q^{n})^{2} (1-q^{pn})^{2} $$ is a weight $2$ cusp form for $\Gamma_{0}(p)$. It's still not clear that this formula is explicit enough for all purposes. However, let me note that a weight $2$ modular form of level $p$ is uniquely determined by its first $\lfloor \frac{p+1}{6} \rfloor$ Fourier coefficients, and to know that, one only needs to know the expansion of $\prod_{n=1}^{\lfloor (p+1)/6 \rfloor} (1-q^{n})^{2}$, which doesn't depend on $p$ at least.

Answer 2

There are various ways of constructing explicit cuspforms, but unfortunately none of them really qualify as "explicit formulae". This is probably how it should be: cusp forms are deep objects and you shouldn't expect to get your hands on them for free, as it were.

• You can build cusp forms using Hecke characters of imaginary quadratic fields (CM-type modular forms). See Miyake section 4.8. These are perhaps the easiest examples of cusp forms, but unfortunately this method will never give you anything of level $\Gamma_0(p)$ -- you have to either allow non-trivial characters or non-squarefree levels.

• You can use quaternion algebras: there is a unique quaternion algebra over $\mathbb{Q}$ ramified at $\{p, \infty\}$, and you can write down power series in which the coefficient of $q^n$ is given by some sum over the elements of $D$ of norm $n$. These turn out to be modular forms of level $\Gamma_0(p)$ and certain linear combinations of them are cusp forms. Whether you count that as an "explicit formula" is very much a matter of taste! This approach goes back to Eichler; there is a nice survey article by Kimball Martin.

• You can use modular symbols. This is not so much an "explicit formula" as an algorithm for computing modular forms spaces; but it works very well in practice. The algorithm is described in William Stein's book, and is implemented in Sage and in Magma. So you can fire up one of these programs and ask for a basis of $S_2(\Gamma_0(593))$ and it will spit out something in a second or two.

Answer 3

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 $L(f,1) \neq 0$.