While reading the following theorem for Apostol's modular functions and dirichlet series in number theory, I have a question:
(Theorem 6.14, page 130)
Assume that k is even and $k\geq 4$. If the space $M_k$ contains a simultaneous eigenform $f$ with Fourier expansion $f(\tau)= \sum_{m=0}^{\infty} c(m) x^m$ ,where $x= e^{2πi\tau}$, then $c(1)\neq 0$.
Proof: The coefficient of $x$ in the Fourier expansion of $T_n(f)$ is $\gamma_n(1)= c(n)$. Since f is a simultaneous eigenform this coefficient is also equal to $\lambda(n) c(1)$, so $c(n) = \lambda(n) c(1)$ for all $ n\geq 1$.
I am not able to deduce how does f being a simultaneous eigenform implies that this coefficient will also be equal to $\lambda(n) c(1)$.
If f is an eigenform for every Hecke operator $T_n, n\geq 1$ then f is called a simultaneous eigenform.
If $ f\in M_k$ and has a Fourier expansion $f(\tau)= \sum_{m=0}^{\infty} c(m) x^m$ where $x= e^{2πi\tau} $, then $T_n f$ has the Fourier expansion $T_n(f)(\tau) = \sum_{m=0}^{\infty} \gamma_{n} (m) x^m $, where $\gamma_n(m) = \sum_{d |(m,n)} d^{k-1} c(\frac{m n}{d^2})$
Let $n\in \mathbb N$ be given. As $f$ is an eigenform for $T_n$, then $T_nf=\lambda (n)f$, and in particular the ceofficients of their Fourier series must be equal. In your notation, the Fourier series of the LHS is $$\sum _{m}\gamma _n(m)x^m$$ whilst the Fourier series of the RHS is $$\lambda (n)\sum _mc(m)x^m$$ so $\lambda (n)c(m)=\gamma _n(m)$ and in particular $\lambda (n)c(1)=\gamma _n(1)=c(n)$.