Modular forms of level one half, and level of Dedekind eta-function

[MaximLeyenson]

The web page LMFDB : mf.half_integral_weight.dedekind_eta

currently states that

"The Dedekind eta function is a crucial example of a half-integral weight modular form, having weight $1/2$ and level 1".

It is very possible that I do not understand correctly the definition of modular forms of half-integer weight; I never studied them in detail. -- (My understanding is that it should have the same root of unity (of degree 4) in the functional equation as the theta-constant $\Theta(0,\tau)$, as defined by Neal Koblitz in his textbook; and the level $N$ should also be divisible by 4 for the transformation rule to be correctly defined.)

But, if I understand it correctly and my computation with the 24th root of unity $\epsilon(a,b,c,d)$ appearing in the functional equation of eta-function is correct, then the Dedekind eta-function $\eta(\tau)$ is of level $\Gamma(24)$, not of level one.

(If my computation is correct, $\epsilon$ restricts "in the correct way" to the group $\Gamma(24)$ ).

(Then, later, we can take $\eta(24 \tau)$ to change level to $\Gamma_0$ )

I would be thankful for your comments. -- Maxim Leyenson