I am a beginner, in the theory of modular forms. I came across this identity:
$\dfrac{\eta(4\tau)^6\eta(8\tau)^4}{\eta(2\tau)^4}=\displaystyle\sum_{n\geq 1}\left(\sum_{d\mid n}\chi_{-4}(n/d)\;d^2\right)q^{2n}$. I have numerically checked the coefficients of both sides, and the first 100 coefficients agree.
By Newmann's criteria, I find that the left-hand side is a modular form in $M_3(8,\chi_{-4})$, and the right-hand side is the Eisenstein series $E_{3,\chi_{-4},1}(q^2)$ (also in $M_3(8,\chi_{-4})$) where $E_{k,\chi,\psi}(q)$ is the notation in Stein's book on modular forms.
The space $M_3(8,\chi_{-4})$ is 4 dimensional, and spanned only by the four Eisenstein series in this space. My question is about the behavior at the cusps of $\Gamma_0(8)$ which are $\{0,1/2,1/4,i\infty\}$ of both sides in the above identity. By using properties of the eta-function, it turns out that the eta-quotient above has finite values at the cusps $0$ and $1/2$, but vanishes at the other two cusps. However, the Eisenstein series in the right-hand side above vanishes at $0, 1/2$, and $i\infty$. I have checked this numerically, after I did some theoretical calculation as follows: the Eisenstein series in the right-hand side above is some constant times $G_3^{(2,0)}(\tau)$ where $G_k^{(a_1,a_2)}(\tau)$ is the notation for Koblitz Eisenstein series in his book (Chapter 3, Section 3, Page 131) which is an Eisenstein series of level $N$, and weight $k$ with $0\leq a_1, a_2<N$. Now, if we consider the matrix $\gamma=\begin{bmatrix}1&0\\2&1\end{bmatrix}$, we have $(G_3^{(2,0)}|_3\gamma)(\tau)=G_3^{(2,0)}(\tau)$. This implies that as $\tau\rightarrow i\infty$, $G_3^{(2,0)}(\tau)\rightarrow 0$ (since $a_1\neq 0$). This shows that $(G_3^{(2,0)}|_3\gamma)(i\infty)=0$, and hence the Eisenstein series in the identity above is zero at the cusp $1/2$.
Is the above identity true since the eta-quotient has a non-zero value at 1/2 and the Eisenstein series becomes zero at 1/2?
Any comments/suggestions is highly appreciated!
Thanks!