Period lattice of cubic twists of CM modular forms

Question

Let $K=\mathbb{Q}(\sqrt{-3})$ be a CM field. Let $E_1:y^2=x^3+1/4$ and $E_p:y^2=x^3+p^2/4$ where $p\equiv 1\mod 3$ is a prime. Let $f_1$ and $f_p$ be the modular forms of $E_1$ and $E_p$. They are constructed from some Hecke characters related to cubic residue symboles of $K$. And $f_p$ is a cubic twist of $f_1$ over $K$. Computer computation shows that the period lattice of $f_1$ is $\sqrt[3]{p}$ multiple of the period lattice of $f_p$. How to prove this fact ? It is not hard to prove the same relation for the periods of $E_1$ and $E_p$. By Glenn Stevens' conjecture in the paper Stickelberger elements and modular parametrizations of elliptic curves, $E_1$ and $E_p$ is the optimal quotient of $\Gamma_1(N)$, and the Manin–Stevens constant is 1, so the result for $f_1$ and $f_p$ should also be right. But how to prove this? Inspired by Stevens' proof for the quadratic twist case in section 5 of his paper, we can reduce to a weaker result that the period lattice of $f_p$ is contained in $1/\sqrt[3]{p}$ times the period lattice of $f_1$. But still, does anyone have any idea to prove this?