Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the algebraic part of the special $L$value of $E_f$ over $K$, a totally real quadratic field, in terms of the conjugacy classes of maximal orders in the definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$?. p.s: when $K$ is a quadratic imaginary field, there is such an expression.

Elaborating David H's hilariously laconic comment... :) Lacking any other answer: work of Shimura and Waldspurger from 20+ years ago may be the relevant stuff to see what is true. My notsowellinformed perception of the situation is that whatever one is likely to prove (or is known by this date, or at least notsolongago) goes by way of a theta correspondence. But/and a theta correspondence from an orthogonal group to $SL_2(\mathbb R)$ (or a twofold cover, if necessary) maps at archimedean places to holomorphic discrete series essentially only when the source repn of the orthogonal group is the trivial repn on a definite orthogonal group, such as $SO(2)$. In contrast, at archimedean places $SO(1,1)$ produces principal series, so such an orthogonal group's afms map to waveforms, not holomorphic things. That is, if things go by way of Weil (SegalShale) repns, then hitting holomorphic things is a significant constraint, whatever is going on at finite primes. 

