Let the modular form of weight $2$ for the group $\Gamma_0(N)$ and for a certain $N$: \begin{equation} f(q)= q \prod_{n=1}^{\infty}(1-p^{3n})^2 \prod_{n=1}^{\infty}(1-q^{9n})^2=\sum_{n=1}^{\infty}t_nq^n, \:\: \text{with} \:\:t_1=1. \: \end{equation}
Let the elliptic curve $F_3: \: x^3+y^3+z^3=0$.
Then $\#F_3(\mathbb{F}_p)$ is the number of points on $F_3$ when reduced modulo a prime $p$. Studying $F_3(\mathbb{F}_p)$ for primes between $2$ and $100$, we see that $t_p=p+1-\#F_3(\mathbb{F}_p)$ is equal to 0.
Is it always true ? Do you know a simple way to demonstrate this ? I thank you in advance for any suggestion.