I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as $$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(q) \frac{z^k}{k!}\right)}$$ and is the exponential for the sigma orientation of Tate K-theory over $\mathbb{Q}[[q]]$. Doesn't the sigma orientation correspond to the universal elliptic formal group law in the neighborhood of that cusp? And as such, shouldn't $\sigma_L^{-1}(z)(q)$ be a logarithm for the universal elliptic formal group law?

If so, its inverse (in $z$) should be some kind of weight-1 modular form in $q$, no? A logarithm for the universal formal group law has the same coefficients as an invariant differential fiberwise, so it should transform like a section of the pushforward of the relative differentials $\pi_* \Omega_{E/S}$, whose global sections down below are weight-1 modular forms.

However, the expression I wrote above doesn't really seem to transform in the correct way; I've tried computing the action of $\Gamma$ on $\sigma_L^{-1}$ with the usual substitution $\tau\mapsto \frac{a\tau + b}{c\tau + d}$ and $z\mapsto \frac{z}{c\tau + d}$, but all I end up getting is a mess. Is this coordinate $z$ not the usual coordinate coming from $(\tau, z)\in \mathbb{H} \times \mathbb{C}$?

Where am I going wrong?