Let $q$ be an odd prime, and $Q$ be the quadratic form on $\mathbb{R}^4$ defined as $$Q(x_0,x_1,x_2,x_3)=x_0^2+4q^2(x_1^2+x_2^2+x_3^2)$$ My aim is to prove that the theta function series $$\Theta_Q(z) = \sum \limits_{\vec{x} \in \mathbb{Z}^4} e^{2 \pi i Q(\vec{x}) z}$$ is a (weakly) modular form of weight $2$ and level $16q^2$.
Now suppose we want to show that a given function $f:\mathbb{H} \to \mathbb{C}$ is a (weakly) modular form of weight $k$ for the modular group $\Gamma$, we can exploit the fact that $\Gamma$ is generated by the functions $$z \mapsto z+1$$ $$z \mapsto -1/z$$ and so it is enough to show that $$f(z)=f(z+1)$$ $$f(-1/z)=z^kf(z)$$
But what do we do when the level is not $1$? Do we know of any generators for the congruence subgroup $\Gamma(16q^2)$?
It is easy to see, simply by definition, that $$\Theta_Q(z+1)=\Theta_Q(z)$$ I then tried using the Poisson summation formula on the function $$e^{- \pi \vec{v}.\vec{v}}$$ with the lattice $\Lambda \subset \mathbb{R}^4$ comprising vectors of the form $(v_0,v_1,v_2,v_3)^T$ where $v_0/\sqrt{2},v_1/2\sqrt{2}q,v_2/2\sqrt{2}q,v_3/2\sqrt{2}q \in \mathbb{Z}$ but that is not of much help since I am not even sure what to aim for. What I have is $$\Theta_{Q^*}\left( -\frac{1}{z} \right) = - 32q^3 z^2 \Theta_{Q}(z)$$ where $Q^*$ is a "dual" form defined as $$Q^{*}(x_0,x_1,x_2,x_3)=x_0^2+\frac{1}{4q^2}(x_1^2+x_2^2+x_3^2)$$
By the way, I am a computer science student trying to understand the Ramanujan graph construction by Lubotsky et al, so I am only now scratching the surface of modular forms. I tried looking up references, but they get in too deep and I lose direction.