Example of holomorphic modular form of weight 2?

Question

I read that for holomorphic modular forms of weight 2, $f(q ) = \sum a_n q^n $ Hecke proved $|a_n| < Cn$.

Are there any holomorphic modular forms of weight 2? There certainly aren't any for the group $\Gamma = SL(2, \mathbb{Z})$. The Eisenstein series is quasimodular.

William Stein gives an elaborate computation for the congruence group $\Gamma_0(N)$.

Are there examples where all coefficients are explicit?

Answer 1

Here are two standard examples:

• As you note, the weight 2 Eisenstein series $E_2(z) = \frac{-1}{24} + \sum_{n \ge 1} \sigma(n) q^n$ is not a modular form (only a quasimodular form); but for any $N \ge 2$, the function $E_2(z) - N E_2(Nz)$ is a holomorphic modular form of weight 2 and level $\Gamma_0(N)$.

• The product $$ q \prod_{n \ge 1} (1 - q^n)^2(1 - q^{11n})^2 $$ is a holomorphic cuspidal modular form of weight 2 and level $\Gamma_0(11)$.