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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?

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    $\begingroup$ It does follow from the usual argument with Hecke operators. Have you read one of these proofs? $\endgroup$ – davidlowryduda Jun 18 '15 at 21:54
  • $\begingroup$ But we need to consider Hecke operators exactly for $\Gamma_0(4)$ and prove that $f(z)$ is eigenform for all of them, right? $\endgroup$ – vitaliy Jun 18 '15 at 22:42
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    $\begingroup$ It is not very hard to prove that something is an eigenvector when the space it lives in is 1-dimensional. $\endgroup$ – David Loeffler Jun 19 '15 at 8:10

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