I'm reading 'Rational Points on Elliptic Curves' by Silverman and Tate, and the exercise 4.6 is about the following special case of Eichler-Shimura theorem. Let $C$ be the elliptic curve given by the equation $C: y^2 = x^3-4x^2+16$ and let $F(q)$ be the formal power series given by infinite product $F(q) = q \prod_{n=1}^\infty (1-q^n)^2(1-q^{11n})^2$. Let $N_n$ be the coefficient of $q^n$, so $F(q)=\sum_{n=1}^\infty N_n q^n$. The exercise is to prove that $M_p+N_p=p$, where $M_p$ is the number of points on $C$ over the field $\mathbb{F}_p$. So, what is the idea of proof? Because I have no ideas how $M_p$ and $N_p$ can be connected. Thanks in advance!

I'm reading ‘Rational Points on Elliptic Curves’ by Silverman and Tate, and the exercise 4.6 is about the following special case of the Eichler–Shimura theorem. Let $C$ be the elliptic curve given by the equation $C: y^2 = x^3-4x^2+16$ and let $F(q)$ be the formal power series given by the infinite product $F(q) = q \prod_{n=1}^\infty (1-q^n)^2(1-q^{11n})^2$. Let $N_n$ be the coefficient of $q^n$ in $F(q)$, so $F(q)=\sum_{n=1}^\infty N_n q^n$. The exercise is to prove that $M_p+N_p=p$, where $M_p$ is the number of points on $C$ over the field $\mathbb{F}_p$. So, what is the idea of proof? Because I have no ideas how $M_p$ and $N_p$ can be connected.