The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$

In [Mazur-Swinnerton-Dyer][1] and [Zagier][2]'s construction, a theta function associated with a positive definite quadratic form is induced:

$$\theta(q)=\sum_{x\in\mathbb{Z}^4}q^{\frac{1}{2}x^{T}Ax}$$

where $$A=\left(\begin{matrix}2 & 0 & 1 & 1\\
0 & 4 & 1 &2\\
1 & 1 & 10 & 1\\
1 & 2 & 1 & 20
\end{matrix}\right)$$

$A$ is a positive definite matrix of determinant $37^2$, and we have $37A^{-1}=K^TAK$ where $K$ is an integral matrix of determinant $\pm 1$.

**Question:** Suppose $A$ is a positive definite matrix with integral entries. All diagonal entries are even numbers. The determinant of $A$ is a square number $N^2$. Is it true that for every $N>1$, there is $A$ that $NA^{-1}=K^TAK$, where $K$ is an integral matrix of determinant $\pm 1$?


  [1]: https://eudml.org/doc/142281
  [2]: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084592/fulltext.pdf