How to evaluate the quotient of Dedekind eta function in Pari/Gp

Question

This expression I found in some research paper, which connects quotient of Dedekind eta function and ray class field of conductor N, which in turn gives the value of j-invariants.

For $K=\mathbb Q(i)$ and $N=3$ (conductor), the discriminant of the order is -36. Now,$$j_{1,3}(\tau)=\frac{\eta(\tau)^{12}}{\eta(3\tau)^{12}}$$ where $\tau=i$. I evaluated this expression in Pari/Gp, and I got the answer in decimals which I found wrong, if some one knows how to evaluate the above quotient, their reply will be of great help

Answer 1

The Pari eta function has a flag that controls which of two variants is computed. To quote the Pari/GP help function:

eta(z,{flag=0}): if flag=0, returns prod(n=1,oo, 1-q^n), where q = exp(2 i Pi z) if z is a complex scalar (belonging to the upper half plane); q = z if z is a p-adic number or can be converted to a power series. If flag is non-zero, the function only applies to complex scalars and returns the true eta function, with the factor q^(1/24) included.

Here are the two different results for your argument:

eta(I)^12/eta(3*I)^12 = 0.9777785058350805514953449281 
eta(I,1)^12/eta(3*I,1)^12 = 523.5922308261581215514463074