I'm following the notation in Chapter II of Cremona's book on modular symbol algorithms.
Let $f$ be a weight two cusp form for a congruence subgroup $G$. For any two points $\alpha, \beta \in \mathcal{H}$ in the extended upper half plane equivalent under the action of $G$, there is the notion of a modular symbol $[\alpha, \beta]$, given by projecting any smooth path from $\alpha$ to $\beta$ to the quotient $X_G = \mathcal{H} / G$, which determines a closed path and hence an integral homology class in $H_1(X_G, \mathbb{Z})$. Integrating the differential $2\pi if(z)\mathrm{d}z$ over such a modular symbol yields a period of $f$, denoted by $$\langle[\alpha, \beta], f\rangle := \int_\alpha^\beta 2\pi if(z)\mathrm{d}z.$$
Now let $G$ be the congruence subgroup $\Gamma_0(N)$. For each prime $p \nmid N$, the Hecke operator $T_p$ with eigenvalue $a_p$ acts on these modular symbols in such a way that integration with $2\pi if(z)\mathrm{d}z$ yields the following identity (2.8.9): $$(1 + p - a_p) \cdot \langle[0, \infty], f\rangle = \sum_{k = 1}^{p - 1} \langle[0, \tfrac{k}{p}], f\rangle.$$
I am interested in the setting where $f$ is the weight two cusp form associated to an optimal elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$. In this setting, the left hand side encodes arithmetic information of $E$: the integer $1 + p - a_p$ is just the number of points of $E$ over $\mathbb{F}_p$, while $\langle[0, \infty], f\rangle$ is the central $L$-value $L(E, 1)$ up to a sign. By dividing both sides by the real period $\Omega_E$, the factor $\langle[0, \infty], f\rangle / \Omega_E$ becomes rational, so the left hand side is also rational. In fact, the entire left hand side seemingly becomes integral, which I believe is a consequence of the optimality of $E$, possibly with further assumptions.
My question is: what is the arithmetic meaning of the individual terms $\langle[0, \tfrac{k}{p}], f\rangle / \Omega_E$ for each of $k = 1, \dots, p - 1$? The sum on the right hand side is somewhat symmetric about $k = \tfrac{p - 1}{2}$, in the sense that $\overline{\langle[0, \tfrac{k}{p}], f\rangle} = \langle[0, \tfrac{p - k}{p}], f\rangle$, so perhaps I should instead ask the question for $\Re(\langle[0, \tfrac{k}{p}], f\rangle) / \Omega_E$ for each of $k = 1, \dots, \tfrac{p - 1}{2}$. These also seem to be half-integers, which should also be a consequence of the optimality of $E$, again possibly with further assumptions. Is it possible to express these completely in terms of arithmetic invariants of $E$ without doing the standard algorithmic computation?
As an example, let $f$ be the weight two cusp form associated to the elliptic curve with Cremona label 11a1. Then $$(1 + p - a_p) \cdot \left(-\dfrac{L(E, 1)}{\Omega_E}\right) = (1 + 7 - (-2)) \cdot \left(-\dfrac{0.25384\dots}{1.2692\dots}\right) \approx -2,$$ while a period computation in SageMath shows $$\sum_{k = 1}^6 \dfrac{\langle[0, \tfrac{k}{7}], f\rangle}{\Omega_E} = 2\left(\sum_{k = 1}^3 \dfrac{\Re(\langle[0, \tfrac{k}{7}], f\rangle)}{\Omega_E}\right) = 2(\tfrac{1}{2} + \tfrac{1}{2} + (-2)) = -2,$$ as expected.
Context: weighting the periods in the right hand side appropriately gives the central $L$-value when twisted by a Dirichlet character of conductor $p$, which I am trying to understand in terms of the non-twisted central $L$-value and other arithmetic invariants. I am brand new to this area and I might be going down a bottomless rabbit hole, so any suggestions is appreciated.