# Twists of CM modular forms

Let $K$ be an imaginary quadratic field of class number 1 and $E$ an elliptic curve over $\mathbf{Q}$ with CM by $\mathcal{O}_K$. Let $\psi$ be the Groessencharacter of $K$ attached to $E$, and $$g_{\psi} = \sum_{\mathfrak{a}} \psi(\mathfrak{a}) q^{N(\mathfrak{a})}$$ the corresponding CM-type modular form, where the sum is over the integral ideals prime to the conductor $\mathfrak{f}$ of $\psi$. So the level of $g_{\psi}$ = the conductor of $E$ = $|\Delta_K| N(\mathfrak{f})$; let's call this integer $M$.

Up to some fudge-factor (the Manin constant), the $\Gamma_1(M)$-invariant differential $g_{\psi}(z) \mathrm{d}z$ on the upper half-plane is the pullback of the invariant differential $\omega_E$ on $E$ under a modular parametrization $\pi: X_1(M) \to E$.

Suppose I choose an auxilliary ideal $\mathfrak{b}$ of $K$ (coprime to everything in sight, i.e. to $\mathfrak{f}$ and the discriminant of $K / \mathbf{Q}$). Then I can consider, for each character $\eta$ of the ray class group $G_{\mathfrak{b}}$ of $K$ modulo $\mathfrak{b}$, the Groessencharacter $\psi\eta$ and hence the modular form $g_{\psi\eta}$. These all have level $M B$, where $B = N(\mathfrak{b})$.

Does there exist a morphism of algebraic varieties $\pi_\mathfrak{b}: X_1(M B) \to E$, defined over the ray class field $K(\mathfrak{b})$, such that the span of the conjugates of $\pi_\mathfrak{b}^*(\omega_E)$ under $\operatorname{Gal}(K(\mathfrak{b}) / K) \cong G_{\mathfrak{b}}$ is equal to the span of the forms $g_{\psi \eta}$ for all characters $\eta$ of $G_{\mathfrak{b}}$?

(By way of motivation: if one takes a general (not necessarily CM) modular form $g$ of level $N$ and a rational integer $m$, then the space of modular forms at level $m^2 N$ spanned by the twists of $g$ by Dirichlet characters mod $m$ is also the span of the Galois conjugates of the pullback of $g$ under a map $X_1(m^2 N) \to X_1(N)$ defined over $\mathbf{Q}(\mu_m)$. So I'm hoping for a sort of "CM version" of this.)