It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic. This process of normalization by complex periods is regularly attributed to Shimura, though I can't seem to find a concrete reference explaining this result.

More precisely, let $ \Gamma=\Gamma_0(N)$ and fix an eigenform $f\in S_k(\Gamma)$. The modular symbol $\xi_f\in \operatorname{Hom}_{\Gamma}(\operatorname{Div}^0(\mathbb{P}^1(\mathbb{Q})),V_{k-2}(\mathbb{C}))$, where $V_{k-2}(\mathbb{C})$ is the space of homogeneous polynomials with complex coefficients of degree $k-2$, attached to $f$ is defined by $$ \xi_f(\{r\}-\{s\})=2\pi i \int_s^r f(z)(zX+Y)^{k-2}dz. $$ One can expand this into a homogeneous polynomial $\sum_{j=0}^{k-2} c_jX^jY^{k-2-j}$ where $c_j=\binom{k-2}{j}2\pi i \int_s^rf(z)z^jdz$. The matrix $\begin{pmatrix} -1 &0\\ 0&1\end{pmatrix}$ normalizes $\Gamma$, so the modular symbols come equipped with an involution, and hence there is a unique eigenspace decomposition $\xi_f=\xi_f^++\xi_f^-$, with $\xi^\pm$ in the $\pm 1$-eigenspace.

The following theorem is stated in the literature (see, for example, [Greenberg-Stevens, 3.5.4], [Bertolini-Darmon,1.1], or [Pollack-Weston,page 7]).

**Theorem.** *There exists complex numbers $\Omega_f^\pm$ such that $\xi_f^\pm/\Omega_f^\pm$ takes values in $V_{k-2}(K_f)$, where $K_f$ is the number field generated by the Fourier coefficients of $f$.*

Greenberg-Stevens cite this 1977 paper of Shimura, Pollack-Weston cite Shimura's book on automorphic functions, and the Bertolini-Darmon does not give a reference. I could not find anything helpful in Shimura's automorphic function book, but I think theorem 1 from the 1977 paper is probably what we want. For simplicity, I state it below in the case where $f$ has rational coefficients.

**Theorem.** (Shimura, Theorem 1) *Fix a primitive Dirichlet character $\chi$. There exist complex numbers $u_f^\pm$ such that
$$
\frac{L(f_\chi,j)}{u_f^\epsilon\tau(\chi)(2\pi i)^j}\in K_fK_\chi
$$
where $0< j< k$, $\epsilon$ is the sign of $\chi(-1)(-1)^j$, $\tau(\chi)$ is the classical Gauss sum, and $L(f_\chi,s)=\sum\chi(n)a_nn^{-s}$ is the $L$-function of $f$ twisted by $\chi$.*

In fact, Shimura gives precise (though rather noncanonical) descriptions of these periods $u_f^\pm$: they are essentially the value of the $L$-function at $k-1$.

I would like to know how the first theorem stated above follows from this theorem 1 of Shimura.

It seems like a nontrivial exercise, or perhaps I am just having some trouble connecting the dots. I would also be content to see a reference which outlines a proof of the first theorem above.

My thoughts are roughly the following. With the notation as above, let $m$ be the conductor of $\chi$. I know that (see [Mazur-Tate-Teitelbaum, 8.6], for example) one has the following connection between coefficients of the modular symbols and special values of $L$-functions
$$
\frac{j!}{(-2\pi i)^{j+1}}\frac{m^{j+1}}{\tau(\bar \chi)}L(f_{\bar\chi},j+1) =\sum_{a\in (\mathbb{Z}/m\mathbb{Z})^\times}\chi(a)\int_{-a/m}^{i\infty}f(z)(mz+a)^j dz,
$$
for $0\leq j \leq k-2$. This tells us, for example, that certain *weighted sums* of the coefficients of $\xi_f(\{\infty\}-\{-a/m\})$ can be scaled to be algebraic. Even more, after writing down the symbols $\xi_f^\pm$, I can find periods $\Omega_f^\pm$ such that, roughly speaking, $$
\frac{1}{\Omega_f^\pm}\sum\chi(a)(\text{$j$th coefficient of $\xi_f^\pm(\{\infty\}-\{a/m\}$}) )
$$
is algebraic, but again, this only tells me that (a) certain weighted sums of the coefficients are algebraic, and (b) only gives information about the modular symbol evaluated at $\{\infty\}-\{a/m\}$, which as far as I can tell, is not the generality needed for the first theorem above.

(I posted this question on MSE a few days ago, but did not have much luck there. I hope re-posting it here is not too much of a faux pas.)