How to compute the power of an ideal of an order in the imaginary quadratic field? I want to compute the power of an ideal by hand and I am clueless on how to do so.
Let $\mathfrak{a}$ be an ideal of an order with discriminant $\Delta$ in the imaginary quadratic field such that $\mathfrak{a} = \left[a\mathbb{Z} + \frac{b+\sqrt{\Delta}}{2}\mathbb{Z}\right]$ and $\mathfrak{a} = (a,b)$ in standard representation.
Let $\mathfrak{b} = \left(\frac{p+1}{4},1\right) = \left[\frac{p+1}{4}\mathbb{Z} + \frac{1+\sqrt{\Delta}}{2}\mathbb{Z}\right]$. I want to calculate $\mathfrak{b}^p$. Here $p\equiv 3 \pmod 4$ and $\Delta = p^4\Delta_K$ where $\Delta_K = -p$. I do not have any work to show because I don't know how to compute this by hand.
On Sage, I have, $\mathfrak{b}^p = \left[\frac{p+1}{4}\mathbb{Z} + \frac{1+\sqrt{\Delta}}{2}\mathbb{Z}\right]^p = (p^2,-p)$. I need to show this on paper for a proof so I want to know how does one go about doing this by hand. 

Quadratic ideals can be represented as binary quadratic forms and computing the power of a form is easy with the use of an algorithm from Computational Number Theory by Henri Cohen.
