Signed algebraic periods and a finite integral lattice

For every pair of natural numbers $N,k$ with $N>0$ and $k\ge 2$, every unital ring homomorphism $\iota:\overline{\mathbb Q}\to\mathbb C$ from the algebraic closure of $\mathbb Q$, and every collection consisting of a weight-$k$ cusp form $f$ on $\Gamma_1(N)$, a Dirichlet character $\varepsilon:\mathbb Z/N\mathbb Z\to\overline{\mathbb Q}$ (extended by zero on nonunits), and coefficients $a_n\in\overline{\mathbb Q}$ for all $n\in\mathbb N$, assume that the coefficient of degree $n$ of the width-one $q$-expansion of $f$ equals $\iota(a_n)$ for every $n$, that $a_1=1$, that for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and $z$ in the complex upper half-plane one has $f(\gamma z)=\iota(\varepsilon(d))(cz+d)^k f(z)$, and that for every prime natural number $\ell$ and every such $z$ one has $\ell^{-1}\sum_{b=0}^{\ell-1}f((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}f(\ell z)=\iota(a_\ell)f(z)$. Here $\Gamma_0(N)$ consists of determinant-one integer matrices with $c\equiv0\pmod N$, and $\Gamma_1(N)$ additionally has $a\equiv d\equiv1\pmod N$, acting on the upper half-plane by fractional linear transformations. Then there exist two nonzero complex numbers $\Omega_+,\Omega_-$ and a function $v:\{+,-\}\times\mathbb N\times\mathbb Q\to\overline{\mathbb Q}$ such that for both signs $s$, every integer $j$ with $0\le j\le k-2$, and every rational $r$, $\iota(v(s,j,r))=(I_j(r)+\sigma_s(-1)^jI_j(-r))/(2\Omega_s)$, where $\sigma_+=1$, $\sigma_-=-1$, and $I_j(r)=2\pi\int_{(0,\infty)}f(r+it)(r+it)^j\,dt$ is the complex Bochner integral with respect to real Lebesgue measure; moreover, the additive subgroup of $\overline{\mathbb Q}$ consisting of all finite integer linear combinations of the values $v(s,j,r)$ with $s\in\{+,-\}$, $0\le j\le k-2$, and $r\in\mathbb Q$ is finitely generated as a $\mathbb Z$-module. The function $v$ is defined also at every $j>k-2$, but those values satisfy no comparison or finite-generation requirement. The statement includes $N=1$, $k=2$ (when only $j=0$ is constrained), and $r=0$; at $r=0$ the signed numerator is zero whenever $\sigma_s(-1)^j=-1$. It excludes $N=0$ and $k<2$, imposes no uniqueness of the periods or values, and has no separate integrability hypothesis: the integrals are the total Bochner integrals, whose value is zero for a nonintegrable integrand.