For $n \geq 3$, the series $$ G_n = \sum_{\omega \neq 0} \frac{1}{\omega^n} $$ is the Eisenstein series of order $n$. Let $g_2 = 60G_4$ and $g_3=140G_6$. Then, the number $\Delta = g_2^3 - 27g_3^2$ is called the discriminant.
For given periods $\omega_1, \omega_2$ with $\omega_2 / \omega_1 \notin \mathbb{R}$, we define $$ J(\omega_1, \omega_2) = \frac{g_2^3(\omega_1, \omega_2)}{\Delta(\omega_1, \omega_2)}, $$ which is called Klein's modular function. For any $\tau \in \mathcal{H}$, we know that $J(\tau)$ is analytic in $\mathcal{H}$. We also have the Fourier expansion $$ 12^3J(\tau) = e^{-2\pi i \tau} + 744 + \sum_{n=1}^{\infty} c(n)e^{2\pi i n \tau}, $$ where the $c(n)$ are integers.
In the literature, many studies have been done on the calculation of these coefficients $c(n)$.
My Question: Why is the calculation of these coefficients important? Is knowing these coefficients related to solving more concrete problems in different areas such as number theory?