On the Relations among Characteristic Functions of Theta Functions

In this study, using the characteristic values $\begin{bmatrix} \varepsilon\\ \varepsilon' \end{bmatrix} = \begin{bmatrix} 1\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0 \end{bmatrix} \pmod 2$ a theorem on the $\frac{1}{2^r}$ coefficients of periods of first order theta function according to the $(1,τ)$ period pair (for $r \in N^+$) is established. The following equalities are also obtained.

1. $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{1}{2^r}-\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
2. $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{\pi i}{2^r}}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$