Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions 
$$
\theta\_s(z;\tau)\_k=\sum\_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, \hspace{20pt} s=0,...,k-1$$
form a basis of $H^0(E, L)$. I'm especially interested in the case $k=3$ when theta functions embed the elliptic curve as Hesse cubic in $\mathbb{P}^2$. 


I have the following basic question: are there formulas for derivatives of such theta functions at zero $\left.\frac{d\theta\_s(z,\tau)\_3}{dz}\right|\_{z=0}$?


The closest result I know is Jacobi formula for Jacobi theta functions
$$
\theta'\_{\frac{1}{2} \frac{1}{2}}=-\pi \theta\_{0 0} \theta \_{\frac{1}{2} 0} \theta \_{0 \frac{1}{2}},
$$
but (as far as I understand) it corresponds to the case $k=4$, so it does not answer my question.



Derivatives of theta functions at zero
An eta-product identity for $k=3$, $s=1$ similar to that given in my comment above may be found in ["Some eta-identities arising from theta series"](http://www.mscand.dk/article/viewFile/12299/10315) by Günter Köhler (Math. Scand. **66**, 1990, p. 146, identity (3)):
$$
\frac{\eta^2(\tau)\eta^2(4\tau)}{\eta(2\tau)}=\sum\_{n=1}^\infty\left(\frac n3\right)ne^\frac{2\pi in^2\tau}3,
$$
which is equivalent to
$$
\prod\_{n=1}^\infty\frac{(1-q^n)^2(1-q^{4n})^2}{1-q^{2n}}=\sum\_{n=1}^\infty\left(\frac n3\right)nq^{\frac{n^2-1}3}=1-2q+4q^5-5q^8+7q^{16}-8q^{21}+...
$$
The rhs is the same as $q^{-\frac13}\sum\_{n\in\mathbb Z}(3n+1)q^{\frac13{(3n+1)^2}}$, i. e., up to rescalings, $\frac\partial{\partial z}$ of your $\theta\_1(z;\tau)\_3$ at $z=0$.


I don't know whether such identities for other $k$ are systematically treated anywhere. Köhler also has a book ["Eta Products and Theta Series Identities"](http://www.springer.com/us/book/9783642161513) but I have never looked inside. From "About this book":



> 
> The author brings to the public the large number of identities that have been discovered over the past 20 years, the majority of which have not been published elsewhere.
> 
> 
> 


(Second part)
This might be a separate answer but I decided to just add it here. In "[Eta-quotients and theta functions](http://www.sciencedirect.com/science/article/pii/S0001870813001102)" by Robert J. Lemke Oliver ([free online version](http://web.stanford.edu/~rjlo/papers/06-EtaTheta.pdf) available), a classification of all those theta-functions of the form $\theta\_\psi(z)=\sum\_n\psi(n)n^\delta q^{n^2}$ which admit an eta-product decomposition is given. Here $\psi$ is a Dirichlet character, and $\delta=0$ or $1$ according to the parity of $\psi$. There turns out to be very few of this kind - just eight in the even case and five in the odd case. The case of your $k=3$, $s=1$ is the third identity of Theorem 1.1 (2) in that paper; I am not reproducing it here as it is essentially the same as the Köhler's case above. If one allows some twists of the characters, there are a little bit more such products.


