On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function 


$$\Theta\_{x\_{0}}(\tau, z) = \sum\_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, x\_{0})}$$


is a holomorphic Jacobi form of weight $N/2$ and index $Q(x\_{0})$ for some congruence subgroup of $SL\_{2}(\mathbb{Z})$. In this formula, $Q: \mathbb{Z}^{N} \to \mathbb{Z}$ is a positive-definite quadratic form, $B(x, x\_{0}) = \frac{1}{2}(Q(x + x\_{0}) - Q(x) - Q(x\_{0}))$ is the associated bilinear form, and $x\_{0}$ is some lattice vector. Finally, $q = e^{2 \pi i \tau}$ and $y = e^{2 \pi i z}$, and these are conventions I'd like to stay with for my purposes. 


I would like to apply this to the example of the four classical Jacobi theta functions, but I haven't found any conclusive references, and those references I have found, all seem to use different conventions. I believe the Jacobi theta functions are given for my definition above of $q$ and $y$ by


$$\vartheta\_{1}(\tau, z) = - \sum\_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}}(-y)^{n+\frac{1}{2}}$$


$$\vartheta\_{2}(\tau, z) = \sum\_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}} y^{n + \frac{1}{2}}$$


$$\vartheta\_{3}(\tau, z) = \sum\_{n \in \mathbb{Z}} q^{n^{2}/2} y^{n}$$


$$\vartheta\_{4}(\tau, z) = \sum\_{n \in \mathbb{Z}} q^{n^{2}/2} (-y)^{n}$$


My questions simply is, what is the weight and index of each of these theta functions, and with respect to what congruence subgroup? Is there a nice reference? 


I'm struggling to even reconcile these functions with the general theta series above from Eichler and Zagier. For example, $n^{2}/2$ is not an integer-valued quadratic form, and even if it were, $n$ has the wrong coefficient to be the corresponding bilinear form. This leads me to worry that I'm using the wrong conventions for $q$ and $y$. Moreover, don't the factors of $(-1)^{n+1/2}$ and $(-1)^{n}$ in $\vartheta\_{1}$ and $\vartheta\_{4}$ respectively, prevent us from putting these in the general form of $\Theta\_{x\_{0}}(\tau, z)$ above? 



Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions
To expand on the comments. All four of the functions in the question are somehow versions of the classical theta function \begin{align\*} \vartheta(\tau,z) &= \sum\_{n \in \mathbb{Z}} (-1)^n q^{\frac{1}{2}(n+1/2)^2} \zeta^{n + 1/2} \\ &= \zeta^{1/2} q^{1/8} \prod\_{n=1}^{\infty} (1 - q^n) (1 - q^n \zeta) (1 - q^{n-1} \zeta^{-1}), \end{align\*} where $q = e^{2\pi i \tau}$ and $\zeta = e^{2\pi i z}$. This is indeed a Jacobi form, of half-integer weight and half-integer index and it is not covered explicitly by Eichler-Zagier. To make sense of it you should interpret it as having a multiplier system under $SL\_2(\mathbb{Z})$ (or transforming under a character of the metaplectic group) as well as a character of the Heisenberg group.


If you let $\chi$ denote the character of Dedekind's eta function, so $$\eta((M,\sqrt{c \tau + d}) \cdot \tau) = \chi(M,\sqrt{c \tau + d}) \sqrt{c \tau + d} \eta(\tau)$$ then I believe the behavior of $\vartheta$ is $$\vartheta\Big( (M,\sqrt{c \tau + d}) \cdot \tau, \frac{z}{c \tau + d} \Big) = \chi(M,\sqrt{c \tau + d})^3 \sqrt{c \tau + d} e^{\pi i \frac{c z^2}{c \tau + d}} \vartheta(\tau,z)$$ and $$\vartheta(\tau,z+\lambda \tau + \mu) = (-1)^{\lambda +\mu} q^{-\lambda^2 / 2} \zeta^{-\lambda} \vartheta(\tau,z).$$


