This question might seen as a duplicate of this, however my aim is to understand the theory which lies beneath the computations of Sage.
Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as
$$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ Dedekind eta-function is defined as $$\eta(z)=q^{1/24}\prod_{n\geq1}(1-q^n).$$
As it was beautifully explained by Noam Elkies, the ideal of cusp forms for $\Gamma_0(4)$ is principal and generated by
$$f(z)=\eta(2z)^{12}=q-12q^3+54q^5-...\,. \,\,\,(1)$$
My questuion is how to prove that the coefficients in the expansion $(1)$ are multiplicative, i.e. $c_nc_m=c_{nm}$ whenever $\text{g.c.d}(n,m)=1.$ As I know this should follows from the theory of Hecke operators of $\Gamma_0(4)$ and the fact that $f(z)$ is an eigenform.
Can anyone give some explanations of this?