I was looking for a "general" modular transformation for the first kind of Jacobi theta function, $\theta_1(u,\tau)$. I do know how this theta function transforms under the two generators of modular transformations, $$T:\tau\to\tau+1\quad\mbox{and}\quad S:\tau\to-\frac{1}\tau,$$ so in order to find the general formula for $$\tau\to\frac{a\tau+b}{c\tau+d}\quad\mbox{where}~~a,b,c,d\in\mathbb Z~~\mbox{and}~~ad-bc=1,$$ I thought I need to figure out how the general modular transformation is written in terms of the above two generators. Using the Euclidean algorithm, I could find a general combination of $T$ and $S$ that gives $$\frac{a\tau+b}{c\tau+d}\to\frac{a'\tau}{c'\tau+d}$$ but that was the best I could do...
I would really appreciate if you help me to figure out the general modular transformation for $\theta_1(u,\tau)$! (I searched for the formula first but I couldn't find it anywhere.)
You can find the closed and quite explicit form for a general theta-transformation in https://doi.org/10.1017/S0308210512001023. I honestly don't know how the formula had been derived there. Seems there are also lot of new formulas there you may be interested in.